2nthrt (problem 3.4.6)

Percentage Accurate: 54.2% → 86.2%

Time: 23.2s

Alternatives: 17

Speedup: 1.9×

• could not determine a ground truth

  1. x = -9.318999802818611e-126
  2. n = -1.469369025808478e-30

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: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-14)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-24)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-14) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-24) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-14) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-24) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-14:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 2e-24:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-14)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-24)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-14], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-24], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000002e-14

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

      1. 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 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 96.0%

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

   9. Applied rewrites 96.0%

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

   if -5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999985e-24

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 87.6

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

   5. Applied rewrites 87.6%

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

   if 1.99999999999999985e-24 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.4%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-log1p.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -0.001)
     (- 1.0 t_0)
     (if (<= t_1 0.005)
       (/ (log (/ (+ x 1.0) x)) n)
       (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -0.001) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.005) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    if (t_1 <= (-0.001d0)) then
        tmp = 1.0d0 - t_0
    else if (t_1 <= 0.005d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -0.001) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.005) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -0.001:
		tmp = 1.0 - t_0
	elif t_1 <= 0.005:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.005)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -0.001)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 0.005)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.001], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.005], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $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))) < -1e-3

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.3%

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

   if -1e-3 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0050000000000000001

   1. Initial program 47.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 86.3

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

   5. Applied rewrites 86.3%

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

   7. Applied rewrites 86.3%

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

   if 0.0050000000000000001 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 43.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 9.5

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

   5. Applied rewrites 9.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -0.001:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0.005:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5 - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{x}{n}, \mathsf{fma}\left(-0.5, x, 0.5\right)\right)}{n}\right)}{-n}, x, \frac{1}{n}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-14)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-24)
       (/ (- (log1p x) (log x)) n)
       (-
        (fma
         (fma
          (/
           (fma
            -0.3333333333333333
            x
            (- 0.5 (/ (fma 0.16666666666666666 (/ x n) (fma -0.5 x 0.5)) n)))
           (- 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-14) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-24) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma(fma((fma(-0.3333333333333333, x, (0.5 - (fma(0.16666666666666666, (x / n), fma(-0.5, x, 0.5)) / n))) / -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-14)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-24)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(fma(Float64(fma(-0.3333333333333333, x, Float64(0.5 - Float64(fma(0.16666666666666666, Float64(x / n), fma(-0.5, x, 0.5)) / n))) / Float64(-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-14], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-24], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(-0.3333333333333333 * x + N[(0.5 - N[(N[(0.16666666666666666 * N[(x / n), $MachinePrecision] + N[(-0.5 * x + 0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-n)), $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-14

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

      1. 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 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 96.0%

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

   9. Applied rewrites 96.0%

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

   if -5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999985e-24

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 87.6

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

   5. Applied rewrites 87.6%

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

   if 1.99999999999999985e-24 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 89.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5 - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{x}{n}, \mathsf{fma}\left(-0.5, x, 0.5\right)\right)}{n}\right)}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5 - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{x}{n}, \mathsf{fma}\left(-0.5, x, 0.5\right)\right)}{n}\right)}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% 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 36.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}} \]
   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 83.1%

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

   if 1 < x

   1. Initial program 79.4%

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

   3. Taylor expanded in x around inf

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

      1. 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 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 5: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5 - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{x}{n}, \mathsf{fma}\left(-0.5, x, 0.5\right)\right)}{n}\right)}{-n}, x, \frac{1}{n}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-14)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-24)
       (/ (log (/ (+ x 1.0) x)) n)
       (-
        (fma
         (fma
          (/
           (fma
            -0.3333333333333333
            x
            (- 0.5 (/ (fma 0.16666666666666666 (/ x n) (fma -0.5 x 0.5)) n)))
           (- 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-14) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-24) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(fma((fma(-0.3333333333333333, x, (0.5 - (fma(0.16666666666666666, (x / n), fma(-0.5, x, 0.5)) / n))) / -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-14)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-24)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(fma(-0.3333333333333333, x, Float64(0.5 - Float64(fma(0.16666666666666666, Float64(x / n), fma(-0.5, x, 0.5)) / n))) / Float64(-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-14], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-24], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(-0.3333333333333333 * x + N[(0.5 - N[(N[(0.16666666666666666 * N[(x / n), $MachinePrecision] + N[(-0.5 * x + 0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-n)), $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-14

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

      1. 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 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 96.0%

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

   9. Applied rewrites 96.0%

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

   if -5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999985e-24

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.6%

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

   if 1.99999999999999985e-24 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 89.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5 - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{x}{n}, \mathsf{fma}\left(-0.5, x, 0.5\right)\right)}{n}\right)}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5 - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{x}{n}, \mathsf{fma}\left(-0.5, x, 0.5\right)\right)}{n}\right)}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5 - \frac{0.16666666666666666 \cdot \frac{x}{n}}{n}\right)}{-n}, x, \frac{1}{n}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-14)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-24)
       (/ (log (/ (+ x 1.0) x)) n)
       (-
        (fma
         (fma
          (/
           (fma
            -0.3333333333333333
            x
            (- 0.5 (/ (* 0.16666666666666666 (/ x n)) n)))
           (- 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-14) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-24) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(fma((fma(-0.3333333333333333, x, (0.5 - ((0.16666666666666666 * (x / n)) / n))) / -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-14)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-24)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(fma(-0.3333333333333333, x, Float64(0.5 - Float64(Float64(0.16666666666666666 * Float64(x / n)) / n))) / Float64(-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-14], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-24], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(-0.3333333333333333 * x + N[(0.5 - N[(N[(0.16666666666666666 * N[(x / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-n)), $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-14

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

      1. 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 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 96.0%

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

   9. Applied rewrites 96.0%

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

   if -5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999985e-24

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.6%

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

   if 1.99999999999999985e-24 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 89.9%

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

   9. Taylor expanded in n around 0

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

   11. Applied rewrites 89.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5 - \frac{\frac{x}{n} \cdot 0.16666666666666666}{n}\right)}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x, 0.5 - \frac{0.16666666666666666 \cdot \frac{x}{n}}{n}\right)}{-n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.5}{n} + -0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-14)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-24)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (fma (fma (/ (+ (/ 0.5 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-14) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-24) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(fma((((0.5 / 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-14)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-24)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(Float64(0.5 / n) + -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-14], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-24], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision] / n), $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-14

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

      1. 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 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 96.0%

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

   9. Applied rewrites 96.0%

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

   if -5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999985e-24

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.6%

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

   if 1.99999999999999985e-24 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in n around -inf

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

   8. Applied rewrites 89.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   10. 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

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

   11. Applied rewrites 87.4%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.5}{n} + -0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-14)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-24)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e+123)
         (- (+ (/ x n) 1.0) t_0)
         (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-14) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-24) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-14)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 2d-24) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+123) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-14) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-24) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-14:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 2e-24:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+123:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-14)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-24)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+123)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-14)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 2e-24)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+123)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-14], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-24], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+123], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000002e-14

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

      1. 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 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 96.0%

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

   9. Applied rewrites 96.0%

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

   if -5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999985e-24

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.6%

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

   if 1.99999999999999985e-24 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e123

   1. Initial program 85.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

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

   1. Initial program 21.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 10.7

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

   5. Applied rewrites 10.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.3%

      \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-14)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-24)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e+123)
         (- 1.0 t_0)
         (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-14) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-24) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-14)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 2d-24) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+123) then
        tmp = 1.0d0 - t_0
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-14) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-24) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-14:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 2e-24:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+123:
		tmp = 1.0 - t_0
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-14)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-24)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+123)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-14)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 2e-24)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+123)
		tmp = 1.0 - t_0;
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-14], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-24], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+123], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000002e-14

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

      1. 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 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 96.0%

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

   9. Applied rewrites 96.0%

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

   if -5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999985e-24

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.6%

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

   if 1.99999999999999985e-24 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999996e123

   1. Initial program 85.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 84.4%

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

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

   1. Initial program 21.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 10.7

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

   5. Applied rewrites 10.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.3%

      \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 36.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.2%

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

   if 1 < x

   1. Initial program 79.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.7%

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

   10. Applied rewrites 79.3%

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

Alternative 11: 60.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.95)
   (/ (- x (log x)) n)
   (if (<= x 1.05e+162) (/ (/ (- 1.0 (/ 0.5 x)) x) n) (/ (/ -0.5 (* x x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.05e+162) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.05d+162) then
        tmp = ((1.0d0 - (0.5d0 / x)) / x) / n
    else
        tmp = ((-0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.05e+162) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.95:
		tmp = (x - math.log(x)) / n
	elif x <= 1.05e+162:
		tmp = ((1.0 - (0.5 / x)) / x) / n
	else:
		tmp = (-0.5 / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.05e+162)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.95)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.05e+162)
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	else
		tmp = (-0.5 / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.05e+162], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.94999999999999996

   1. Initial program 36.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.2%

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

   if 0.94999999999999996 < x < 1.05e162

   1. Initial program 66.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.0%

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

   if 1.05e162 < x

   1. Initial program 94.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 94.3%

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

Alternative 12: 60.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.5e-7)
   (/ (- (log x)) n)
   (if (<= x 1.05e+162)
     (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)
     (/ (/ -0.5 (* x x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.5e-7) {
		tmp = -log(x) / n;
	} else if (x <= 1.05e+162) {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.5d-7) then
        tmp = -log(x) / n
    else if (x <= 1.05d+162) then
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    else
        tmp = ((-0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.5e-7) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.05e+162) {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.5e-7:
		tmp = -math.log(x) / n
	elif x <= 1.05e+162:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	else:
		tmp = (-0.5 / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.5e-7)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.05e+162)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.5e-7)
		tmp = -log(x) / n;
	elseif (x <= 1.05e+162)
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	else
		tmp = (-0.5 / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.5e-7], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.05e+162], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.4999999999999999e-7

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.2%

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

   if 1.4999999999999999e-7 < x < 1.05e162

   1. Initial program 64.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.9%

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

   if 1.05e162 < x

   1. Initial program 94.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 94.3%

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

Alternative 13: 50.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} + \frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.05e+162)
   (/ (+ (/ (- (/ (/ 0.3333333333333333 n) x) (/ 0.5 n)) x) (/ 1.0 n)) x)
   (/ (/ -0.5 (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.05e+162) {
		tmp = (((((0.3333333333333333 / n) / x) - (0.5 / n)) / x) + (1.0 / n)) / x;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.05d+162) then
        tmp = (((((0.3333333333333333d0 / n) / x) - (0.5d0 / n)) / x) + (1.0d0 / n)) / x
    else
        tmp = ((-0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.05e+162) {
		tmp = (((((0.3333333333333333 / n) / x) - (0.5 / n)) / x) + (1.0 / n)) / x;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.05e+162:
		tmp = (((((0.3333333333333333 / n) / x) - (0.5 / n)) / x) + (1.0 / n)) / x
	else:
		tmp = (-0.5 / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.05e+162)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / n) / x) - Float64(0.5 / n)) / x) + Float64(1.0 / n)) / x);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.05e+162)
		tmp = (((((0.3333333333333333 / n) / x) - (0.5 / n)) / x) + (1.0 / n)) / x;
	else
		tmp = (-0.5 / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.05e+162], N[(N[(N[(N[(N[(N[(0.3333333333333333 / n), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05e162

   1. Initial program 44.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 39.5%

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

   if 1.05e162 < x

   1. Initial program 94.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 94.3%

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

Alternative 14: 50.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.05e+162)
   (/ (/ (+ (/ (- (/ 0.3333333333333333 x) 0.5) x) 1.0) x) n)
   (/ (/ -0.5 (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.05e+162) {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.05d+162) then
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) + 1.0d0) / x) / n
    else
        tmp = ((-0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.05e+162) {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.05e+162:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n
	else:
		tmp = (-0.5 / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.05e+162)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.05e+162)
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) + 1.0) / x) / n;
	else
		tmp = (-0.5 / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.05e+162], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05e162

   1. Initial program 44.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.5%

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

   if 1.05e162 < x

   1. Initial program 94.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 94.3%

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

Alternative 15: 44.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.05e+162) (/ (/ 1.0 x) n) (/ (/ -0.5 (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.05e+162) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.05d+162) then
        tmp = (1.0d0 / x) / n
    else
        tmp = ((-0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.05e+162) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (-0.5 / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.05e+162:
		tmp = (1.0 / x) / n
	else:
		tmp = (-0.5 / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.05e+162)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.05e+162)
		tmp = (1.0 / x) / n;
	else
		tmp = (-0.5 / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.05e+162], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05e162

   1. Initial program 44.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 34.3%

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

   if 1.05e162 < x

   1. Initial program 94.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 94.3%

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

Alternative 16: 40.4% 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 53.8%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-log.f64 64.5

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

5. Applied rewrites 64.5%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 40.1%

   \[\leadsto \frac{\frac{1}{x}}{n} \]
9. Add Preprocessing

Alternative 17: 39.8% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
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 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 54.4%

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

8. Taylor expanded in n around inf

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

10. Applied rewrites 39.8%

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

11. Final simplification 39.8%

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

Reproduce

herbie shell --seed 2024308 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))