2nthrt (problem 3.4.6)

Percentage Accurate: 54.0% → 92.3%

Time: 23.3s

Alternatives: 15

Speedup: 1.9×

• could not determine a ground truth

  1. x = 2.4328761757586566e+113
  2. n = 1.2721866338700554e-220

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: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 48.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}} \]
   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 90.2%

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

   if 1 < x

   1. Initial program 72.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 99.0%

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

Alternative 2: 78.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ 1.0 x) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 -5000.0)
     t_2
     (if (<= t_1 0.0) (/ (log (/ (+ 1.0 x) x)) n) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((1.0 + x), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((1.0d0 + x) ** (1.0d0 / n)) - t_0
    t_2 = 1.0d0 - t_0
    if (t_1 <= (-5000.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((1.0 + x), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -5000.0:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((1.0 + x) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -5000.0)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -5e3 or 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 82.6%

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

   if -5e3 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 44.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 83.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -5000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-38)
   (/ (/ (/ 1.0 (pow x (/ -1.0 n))) x) n)
   (if (<= (/ 1.0 n) 5e-10)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 2e+251)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (/ 1.0 (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-38) {
		tmp = ((1.0 / pow(x, (-1.0 / n))) / x) / n;
	} else if ((1.0 / n) <= 5e-10) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+251) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-38)) then
        tmp = ((1.0d0 / (x ** ((-1.0d0) / n))) / x) / n
    else if ((1.0d0 / n) <= 5d-10) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+251) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-38) {
		tmp = ((1.0 / Math.pow(x, (-1.0 / n))) / x) / n;
	} else if ((1.0 / n) <= 5e-10) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+251) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-38:
		tmp = ((1.0 / math.pow(x, (-1.0 / n))) / x) / n
	elif (1.0 / n) <= 5e-10:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+251:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-38)
		tmp = Float64(Float64(Float64(1.0 / (x ^ Float64(-1.0 / n))) / x) / n);
	elseif (Float64(1.0 / n) <= 5e-10)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+251)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-38)
		tmp = ((1.0 / (x ^ (-1.0 / n))) / x) / n;
	elseif ((1.0 / n) <= 5e-10)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+251)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-38], N[(N[(N[(1.0 / N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-10], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+251], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-38

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

   if -1.9999999999999999e-38 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000031e-10

   1. Initial program 34.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 83.4%

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

   if 5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e251

   1. Initial program 76.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if 2.0000000000000001e251 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{x \cdot \color{blue}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{\left({x}^{\left(\frac{-1}{n}\right)} \cdot x\right) \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-38)
   (/ 1.0 (* (* (pow x (/ -1.0 n)) x) n))
   (if (<= (/ 1.0 n) 5e-10)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 2e+251)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (/ 1.0 (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-38) {
		tmp = 1.0 / ((pow(x, (-1.0 / n)) * x) * n);
	} else if ((1.0 / n) <= 5e-10) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+251) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-38)) then
        tmp = 1.0d0 / (((x ** ((-1.0d0) / n)) * x) * n)
    else if ((1.0d0 / n) <= 5d-10) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+251) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-38) {
		tmp = 1.0 / ((Math.pow(x, (-1.0 / n)) * x) * n);
	} else if ((1.0 / n) <= 5e-10) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+251) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-38:
		tmp = 1.0 / ((math.pow(x, (-1.0 / n)) * x) * n)
	elif (1.0 / n) <= 5e-10:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+251:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-38)
		tmp = Float64(1.0 / Float64(Float64((x ^ Float64(-1.0 / n)) * x) * n));
	elseif (Float64(1.0 / n) <= 5e-10)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+251)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-38)
		tmp = 1.0 / (((x ^ (-1.0 / n)) * x) * n);
	elseif ((1.0 / n) <= 5e-10)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+251)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-38], N[(1.0 / N[(N[(N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-10], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+251], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-38

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 98.7%

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

   if -1.9999999999999999e-38 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000031e-10

   1. Initial program 34.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 83.4%

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

   if 5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e251

   1. Initial program 76.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if 2.0000000000000001e251 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{x \cdot \color{blue}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{\left({x}^{\left(\frac{-1}{n}\right)} \cdot x\right) \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-38)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)
   (if (<= (/ 1.0 n) 5e-10)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 2e+251)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (/ 1.0 (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-38) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-10) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+251) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-38)) then
        tmp = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    else if ((1.0d0 / n) <= 5d-10) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+251) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-38) {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-10) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+251) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-38:
		tmp = math.pow(x, (-1.0 - (-1.0 / n))) / n
	elif (1.0 / n) <= 5e-10:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+251:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-38)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 5e-10)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+251)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-38)
		tmp = (x ^ (-1.0 - (-1.0 / n))) / n;
	elseif ((1.0 / n) <= 5e-10)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+251)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-38], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-10], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+251], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-38

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 98.7%

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

   if -1.9999999999999999e-38 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000031e-10

   1. Initial program 34.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 83.4%

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

   if 5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e251

   1. Initial program 76.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if 2.0000000000000001e251 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{x \cdot \color{blue}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-38)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)
   (if (<= (/ 1.0 n) 5e-10)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 2e+251) (- 1.0 (pow x (/ 1.0 n))) (/ 1.0 (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-38) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-10) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+251) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-38)) then
        tmp = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    else if ((1.0d0 / n) <= 5d-10) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+251) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-38) {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-10) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+251) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-38:
		tmp = math.pow(x, (-1.0 - (-1.0 / n))) / n
	elif (1.0 / n) <= 5e-10:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+251:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-38)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 5e-10)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+251)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-38)
		tmp = (x ^ (-1.0 - (-1.0 / n))) / n;
	elseif ((1.0 / n) <= 5e-10)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+251)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-38], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-10], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+251], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-38

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 98.7%

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

   if -1.9999999999999999e-38 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000031e-10

   1. Initial program 34.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 83.4%

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

   if 5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e251

   1. Initial program 76.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.2%

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

   if 2.0000000000000001e251 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{x \cdot \color{blue}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\left(\frac{\mathsf{fma}\left(0.5, n, \frac{\mathsf{fma}\left(-0.08333333333333333, \frac{n}{x} - n, \left(\frac{n}{x} \cdot -0.08333333333333333\right) \cdot -0.5\right)}{-x}\right)}{x} + n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{{x}^{4} \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.5e-176)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.5)
     (/ (- x (log x)) n)
     (if (<= x 1.05e+99)
       (/
        1.0
        (*
         (+
          (/
           (fma
            0.5
            n
            (/
             (fma
              -0.08333333333333333
              (- (/ n x) n)
              (* (* (/ n x) -0.08333333333333333) -0.5))
             (- x)))
           x)
          n)
         x))
       (/ -0.25 (* (pow x 4.0) n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.5e-176) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.05e+99) {
		tmp = 1.0 / (((fma(0.5, n, (fma(-0.08333333333333333, ((n / x) - n), (((n / x) * -0.08333333333333333) * -0.5)) / -x)) / x) + n) * x);
	} else {
		tmp = -0.25 / (pow(x, 4.0) * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.5e-176)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.05e+99)
		tmp = Float64(1.0 / Float64(Float64(Float64(fma(0.5, n, Float64(fma(-0.08333333333333333, Float64(Float64(n / x) - n), Float64(Float64(Float64(n / x) * -0.08333333333333333) * -0.5)) / Float64(-x))) / x) + n) * x));
	else
		tmp = Float64(-0.25 / Float64((x ^ 4.0) * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.5e-176], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.05e+99], N[(1.0 / N[(N[(N[(N[(0.5 * n + N[(N[(-0.08333333333333333 * N[(N[(n / x), $MachinePrecision] - n), $MachinePrecision] + N[(N[(N[(n / x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + n), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(-0.25 / N[(N[Power[x, 4.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.5e-176

   1. Initial program 61.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 61.1%

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

   if 6.5e-176 < x < 0.5

   1. Initial program 35.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.5%

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

   if 0.5 < x < 1.05000000000000005e99

   1. Initial program 48.1%

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

   3. Taylor expanded in 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 48.7

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

   5. Applied rewrites 48.7%

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

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 64.4%

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

   11. Taylor expanded in x around -inf

       \[\leadsto \frac{1}{-1 \cdot \color{blue}{\left(x \cdot \left(-1 \cdot n + -1 \cdot \frac{-1 \cdot \frac{\left(\frac{-1}{2} \cdot \frac{\frac{-1}{3} \cdot n + \frac{1}{4} \cdot n}{x} + \left(\frac{-1}{4} \cdot \frac{n}{x} + \frac{1}{6} \cdot \frac{n}{x}\right)\right) - \left(\frac{-1}{3} \cdot n + \frac{1}{4} \cdot n\right)}{x} - \frac{-1}{2} \cdot n}{x}\right)\right)}} \]

   13. Applied rewrites 65.9%

       \[\leadsto \frac{1}{\left(\frac{\mathsf{fma}\left(0.5, n, \frac{\mathsf{fma}\left(-0.08333333333333333, \frac{n}{x} - n, \left(-0.08333333333333333 \cdot \frac{n}{x}\right) \cdot -0.5\right)}{-x}\right)}{x} + n\right) \cdot \color{blue}{x}} \]

   if 1.05000000000000005e99 < x

   1. Initial program 82.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.5

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 71.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n} + \frac{\frac{1}{4}}{n \cdot {x}^{2}}\right)}{x} + \frac{1}{n}}{x} \]

   11. Applied rewrites 71.4%

       \[\leadsto \frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{n}}{x} + \frac{1}{n}}{x} \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 82.5%

       \[\leadsto \frac{-0.25}{{x}^{4} \cdot \color{blue}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\left(\frac{\mathsf{fma}\left(0.5, n, \frac{\mathsf{fma}\left(-0.08333333333333333, \frac{n}{x} - n, \left(\frac{n}{x} \cdot -0.08333333333333333\right) \cdot -0.5\right)}{-x}\right)}{x} + n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{{x}^{4} \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x \cdot x\right)}^{-0.5}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.5e-176)
   (- 1.0 (pow x (/ 1.0 n)))
   (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 <= 6.5e-176) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else 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 <= 6.5d-176) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else 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 <= 6.5e-176) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else 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 <= 6.5e-176:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif 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 <= 6.5e-176)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (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 <= 6.5e-176)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (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, 6.5e-176], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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 3 regimes

2. if x < 6.5e-176

   1. Initial program 61.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 61.1%

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

   if 6.5e-176 < x < 1

   1. Initial program 35.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.5%

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

   if 1 < x

   1. Initial program 72.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 69.3%

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

   10. Applied rewrites 74.3%

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

Alternative 9: 56.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x} + 1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.5e-176)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.9)
     (/ (- x (log x)) n)
     (/
      (/ (+ (/ (- -0.5 (/ (- (/ 0.25 x) 0.3333333333333333) x)) x) 1.0) n)
      x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.5e-176) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x) + 1.0) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 6.5d-176) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (((((-0.5d0) - (((0.25d0 / x) - 0.3333333333333333d0) / x)) / x) + 1.0d0) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6.5e-176) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x) + 1.0) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6.5e-176:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.9:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x) + 1.0) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.5e-176)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / x) - 0.3333333333333333) / x)) / x) + 1.0) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6.5e-176)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.9)
		tmp = (x - log(x)) / n;
	else
		tmp = ((((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x) + 1.0) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.5e-176], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 - N[(N[(N[(0.25 / x), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.5e-176

   1. Initial program 61.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 61.1%

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

   if 6.5e-176 < x < 0.900000000000000022

   1. Initial program 35.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.5%

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

   if 0.900000000000000022 < x

   1. Initial program 72.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 69.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n} + \frac{\frac{1}{4}}{n \cdot {x}^{2}}\right)}{x} + \frac{1}{n}}{x} \]

   11. Applied rewrites 69.7%

       \[\leadsto \frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{n}}{x} + \frac{1}{n}}{x} \]

   12. Taylor expanded in n around 0

       \[\leadsto \frac{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{n}}{x} \]

   14. Applied rewrites 69.7%

       \[\leadsto \frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x} + 1}{n}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 48.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.5%

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

   if 0.900000000000000022 < x

   1. Initial program 72.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 69.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n} + \frac{\frac{1}{4}}{n \cdot {x}^{2}}\right)}{x} + \frac{1}{n}}{x} \]

   11. Applied rewrites 69.7%

       \[\leadsto \frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{n}}{x} + \frac{1}{n}}{x} \]

   12. Taylor expanded in n around 0

       \[\leadsto \frac{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{n}}{x} \]

   14. Applied rewrites 69.7%

       \[\leadsto \frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x} + 1}{n}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x} + 1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.72)
   (/ (- (log x)) n)
   (/ (/ (+ (/ (- -0.5 (/ (- (/ 0.25 x) 0.3333333333333333) x)) x) 1.0) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = -log(x) / n;
	} else {
		tmp = ((((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x) + 1.0) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = -log(x) / n
    else
        tmp = (((((-0.5d0) - (((0.25d0 / x) - 0.3333333333333333d0) / x)) / x) + 1.0d0) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = ((((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x) + 1.0) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.72:
		tmp = -math.log(x) / n
	else:
		tmp = ((((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x) + 1.0) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.72)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / x) - 0.3333333333333333) / x)) / x) + 1.0) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = -log(x) / n;
	else
		tmp = ((((-0.5 - (((0.25 / x) - 0.3333333333333333) / x)) / x) + 1.0) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.72], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 - N[(N[(N[(0.25 / x), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 48.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.9%

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

   if 0.71999999999999997 < x

   1. Initial program 72.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 69.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n} + \frac{\frac{1}{4}}{n \cdot {x}^{2}}\right)}{x} + \frac{1}{n}}{x} \]

   11. Applied rewrites 69.7%

       \[\leadsto \frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{n}}{x} + \frac{1}{n}}{x} \]

   12. Taylor expanded in n around 0

       \[\leadsto \frac{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{n}}{x} \]

   14. Applied rewrites 69.7%

       \[\leadsto \frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{x} + 1}{n}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 46.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 59.1

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

5. Applied rewrites 59.1%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 45.5%

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

Alternative 13: 46.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 59.1

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

5. Applied rewrites 59.1%

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

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 45.4%

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

Alternative 14: 40.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 59.1

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

5. Applied rewrites 59.1%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 28.3%

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

9. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n} + \frac{\frac{1}{4}}{n \cdot {x}^{2}}\right)}{x} + \frac{1}{n}}{x} \]

11. Applied rewrites 28.3%

    \[\leadsto \frac{\frac{\frac{-0.5 - \frac{\frac{0.25}{x} - 0.3333333333333333}{x}}{n}}{x} + \frac{1}{n}}{x} \]

12. Taylor expanded in x around inf

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

14. Applied rewrites 41.6%

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

Alternative 15: 40.1% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 59.1

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

5. Applied rewrites 59.1%

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

7. Applied rewrites 59.1%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 41.3%

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

11. Final simplification 41.3%

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

Reproduce

herbie shell --seed 2024325 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))