2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 85.5%

Time: 23.8s

Alternatives: 19

Speedup: 1.8×

• could not determine a ground truth

  1. x = -4.285823992275737e-293
  2. n = 4.7716337508064637e+291

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% 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: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;n \leq -15000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 155:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= n -15000000000.0)
     t_0
     (if (<= n 155.0)
       (- (exp (/ x n)) (pow x (pow n -1.0)))
       (if (<= n 5.2e+97) (/ (/ (pow (pow x (/ -1.0 n)) -1.0) x) n) t_0)))))

C

double code(double x, double n) {
	double t_0 = log(((1.0 + x) / x)) / n;
	double tmp;
	if (n <= -15000000000.0) {
		tmp = t_0;
	} else if (n <= 155.0) {
		tmp = exp((x / n)) - pow(x, pow(n, -1.0));
	} else if (n <= 5.2e+97) {
		tmp = (pow(pow(x, (-1.0 / n)), -1.0) / x) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(((1.0d0 + x) / x)) / n
    if (n <= (-15000000000.0d0)) then
        tmp = t_0
    else if (n <= 155.0d0) then
        tmp = exp((x / n)) - (x ** (n ** (-1.0d0)))
    else if (n <= 5.2d+97) then
        tmp = (((x ** ((-1.0d0) / n)) ** (-1.0d0)) / x) / n
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if (n <= -15000000000.0) {
		tmp = t_0;
	} else if (n <= 155.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, Math.pow(n, -1.0));
	} else if (n <= 5.2e+97) {
		tmp = (Math.pow(Math.pow(x, (-1.0 / n)), -1.0) / x) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if n <= -15000000000.0:
		tmp = t_0
	elif n <= 155.0:
		tmp = math.exp((x / n)) - math.pow(x, math.pow(n, -1.0))
	elif n <= 5.2e+97:
		tmp = (math.pow(math.pow(x, (-1.0 / n)), -1.0) / x) / n
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (n <= -15000000000.0)
		tmp = t_0;
	elseif (n <= 155.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ (n ^ -1.0)));
	elseif (n <= 5.2e+97)
		tmp = Float64(Float64(((x ^ Float64(-1.0 / n)) ^ -1.0) / x) / n);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if (n <= -15000000000.0)
		tmp = t_0;
	elseif (n <= 155.0)
		tmp = exp((x / n)) - (x ^ (n ^ -1.0));
	elseif (n <= 5.2e+97)
		tmp = (((x ^ (-1.0 / n)) ^ -1.0) / x) / n;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -15000000000.0], t$95$0, If[LessEqual[n, 155.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.2e+97], N[(N[(N[Power[N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.5e10 or 5.2e97 < n

   1. Initial program 37.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.9

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

   5. Applied rewrites 78.9%

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

   7. Applied rewrites 78.9%

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

   if -1.5e10 < n < 155

   1. Initial program 82.1%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 155 < n < 5.2e97

   1. Initial program 11.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 63.1%

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

   9. Applied rewrites 63.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -15000000000:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 155:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{n \cdot n} \cdot 0.5, x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (pow (+ x 1.0) (pow n -1.0)) t_0)))
   (if (<= t_1 -2e-8)
     (- 1.0 t_0)
     (if (<= t_1 4e-16)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (fma (* (/ x (* n n)) 0.5) x 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = pow((x + 1.0), pow(n, -1.0)) - t_0;
	double tmp;
	if (t_1 <= -2e-8) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 4e-16) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(((x / (n * n)) * 0.5), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0)
	tmp = 0.0
	if (t_1 <= -2e-8)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 4e-16)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(fma(Float64(Float64(x / Float64(n * n)) * 0.5), x, 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-8], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 4e-16], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -2e-8

   1. Initial program 99.1%

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

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

   if -2e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 3.9999999999999999e-16

   1. Initial program 42.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 76.4%

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

   if 3.9999999999999999e-16 < (-.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 48.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 85.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -2 \cdot 10^{-8}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{n \cdot n} \cdot 0.5, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\frac{n}{x}, 0.3333333333333333, -0.5 \cdot n\right)}{x} + n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (pow (+ x 1.0) (pow n -1.0)) t_0)))
   (if (<= t_1 -2e-8)
     (- 1.0 t_0)
     (if (<= t_1 4e-16)
       (/ (log (/ (+ 1.0 x) x)) n)
       (/
        (/ (+ (/ (fma (/ n x) 0.3333333333333333 (* -0.5 n)) x) n) x)
        (* n n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = pow((x + 1.0), pow(n, -1.0)) - t_0;
	double tmp;
	if (t_1 <= -2e-8) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 4e-16) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = (((fma((n / x), 0.3333333333333333, (-0.5 * n)) / x) + n) / x) / (n * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0)
	tmp = 0.0
	if (t_1 <= -2e-8)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 4e-16)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(n / x), 0.3333333333333333, Float64(-0.5 * n)) / x) + n) / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-8], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 4e-16], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(n / x), $MachinePrecision] * 0.3333333333333333 + N[(-0.5 * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + n), $MachinePrecision] / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -2e-8

   1. Initial program 99.1%

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

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

   if -2e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 3.9999999999999999e-16

   1. Initial program 42.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 76.4%

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

   if 3.9999999999999999e-16 < (-.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 48.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 9.3

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

   5. Applied rewrites 9.3%

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

   7. Applied rewrites 52.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 53.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -2 \cdot 10^{-8}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\frac{n}{x}, 0.3333333333333333, -0.5 \cdot n\right)}{x} + n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+160}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -2e+14)
   (/ (/ 0.3333333333333333 (pow x 3.0)) n)
   (if (<= (pow n -1.0) -5e-45)
     (/ (- (log x)) n)
     (if (<= (pow n -1.0) 0.002)
       (/ (pow x -1.0) n)
       (if (<= (pow n -1.0) 2e+160)
         (- 1.0 (pow x (pow n -1.0)))
         (/ (/ n x) (* n n)))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -2e+14) {
		tmp = (0.3333333333333333 / pow(x, 3.0)) / n;
	} else if (pow(n, -1.0) <= -5e-45) {
		tmp = -log(x) / n;
	} else if (pow(n, -1.0) <= 0.002) {
		tmp = pow(x, -1.0) / n;
	} else if (pow(n, -1.0) <= 2e+160) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n ** (-1.0d0)) <= (-2d+14)) then
        tmp = (0.3333333333333333d0 / (x ** 3.0d0)) / n
    else if ((n ** (-1.0d0)) <= (-5d-45)) then
        tmp = -log(x) / n
    else if ((n ** (-1.0d0)) <= 0.002d0) then
        tmp = (x ** (-1.0d0)) / n
    else if ((n ** (-1.0d0)) <= 2d+160) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -2e+14) {
		tmp = (0.3333333333333333 / Math.pow(x, 3.0)) / n;
	} else if (Math.pow(n, -1.0) <= -5e-45) {
		tmp = -Math.log(x) / n;
	} else if (Math.pow(n, -1.0) <= 0.002) {
		tmp = Math.pow(x, -1.0) / n;
	} else if (Math.pow(n, -1.0) <= 2e+160) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -2e+14:
		tmp = (0.3333333333333333 / math.pow(x, 3.0)) / n
	elif math.pow(n, -1.0) <= -5e-45:
		tmp = -math.log(x) / n
	elif math.pow(n, -1.0) <= 0.002:
		tmp = math.pow(x, -1.0) / n
	elif math.pow(n, -1.0) <= 2e+160:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -2e+14)
		tmp = Float64(Float64(0.3333333333333333 / (x ^ 3.0)) / n);
	elseif ((n ^ -1.0) <= -5e-45)
		tmp = Float64(Float64(-log(x)) / n);
	elseif ((n ^ -1.0) <= 0.002)
		tmp = Float64((x ^ -1.0) / n);
	elseif ((n ^ -1.0) <= 2e+160)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -2e+14)
		tmp = (0.3333333333333333 / (x ^ 3.0)) / n;
	elseif ((n ^ -1.0) <= -5e-45)
		tmp = -log(x) / n;
	elseif ((n ^ -1.0) <= 0.002)
		tmp = (x ^ -1.0) / n;
	elseif ((n ^ -1.0) <= 2e+160)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e+14], N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-45], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 0.002], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e+160], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e14

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 81.0%

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

   if -2e14 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999976e-45

   1. Initial program 18.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.7%

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

   if -4.99999999999999976e-45 < (/.f64 #s(literal 1 binary64) n) < 2e-3

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 60.7%

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

   if 2e-3 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000001e160

   1. Initial program 75.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.6%

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

   if 2.00000000000000001e160 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 23.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 11.7

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

   5. Applied rewrites 11.7%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 94.9%

       \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+160}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+160}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -2e+14)
   (/ (/ (/ -0.3333333333333333 (* x x)) n) (- x))
   (if (<= (pow n -1.0) -5e-45)
     (/ (- (log x)) n)
     (if (<= (pow n -1.0) 0.002)
       (/ (pow x -1.0) n)
       (if (<= (pow n -1.0) 2e+160)
         (- 1.0 (pow x (pow n -1.0)))
         (/ (/ n x) (* n n)))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -2e+14) {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	} else if (pow(n, -1.0) <= -5e-45) {
		tmp = -log(x) / n;
	} else if (pow(n, -1.0) <= 0.002) {
		tmp = pow(x, -1.0) / n;
	} else if (pow(n, -1.0) <= 2e+160) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n ** (-1.0d0)) <= (-2d+14)) then
        tmp = (((-0.3333333333333333d0) / (x * x)) / n) / -x
    else if ((n ** (-1.0d0)) <= (-5d-45)) then
        tmp = -log(x) / n
    else if ((n ** (-1.0d0)) <= 0.002d0) then
        tmp = (x ** (-1.0d0)) / n
    else if ((n ** (-1.0d0)) <= 2d+160) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -2e+14) {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	} else if (Math.pow(n, -1.0) <= -5e-45) {
		tmp = -Math.log(x) / n;
	} else if (Math.pow(n, -1.0) <= 0.002) {
		tmp = Math.pow(x, -1.0) / n;
	} else if (Math.pow(n, -1.0) <= 2e+160) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -2e+14:
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x
	elif math.pow(n, -1.0) <= -5e-45:
		tmp = -math.log(x) / n
	elif math.pow(n, -1.0) <= 0.002:
		tmp = math.pow(x, -1.0) / n
	elif math.pow(n, -1.0) <= 2e+160:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -2e+14)
		tmp = Float64(Float64(Float64(-0.3333333333333333 / Float64(x * x)) / n) / Float64(-x));
	elseif ((n ^ -1.0) <= -5e-45)
		tmp = Float64(Float64(-log(x)) / n);
	elseif ((n ^ -1.0) <= 0.002)
		tmp = Float64((x ^ -1.0) / n);
	elseif ((n ^ -1.0) <= 2e+160)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -2e+14)
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	elseif ((n ^ -1.0) <= -5e-45)
		tmp = -log(x) / n;
	elseif ((n ^ -1.0) <= 0.002)
		tmp = (x ^ -1.0) / n;
	elseif ((n ^ -1.0) <= 2e+160)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e+14], N[(N[(N[(-0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-45], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 0.002], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e+160], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e14

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.0%

       \[\leadsto -\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{x} \]

   if -2e14 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999976e-45

   1. Initial program 18.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.7%

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

   if -4.99999999999999976e-45 < (/.f64 #s(literal 1 binary64) n) < 2e-3

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 60.7%

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

   if 2e-3 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000001e160

   1. Initial program 75.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.6%

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

   if 2.00000000000000001e160 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 23.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 11.7

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

   5. Applied rewrites 11.7%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 94.9%

       \[\leadsto \frac{\frac{n}{x}}{\color{blue}{n} \cdot n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{+160}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{n \cdot n} \cdot 0.5, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ (pow (pow x (/ -1.0 n)) -1.0) x) n)))
   (if (<= (pow n -1.0) -2e+14)
     t_0
     (if (<= (pow n -1.0) 5e-114)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (pow n -1.0) 0.002)
         t_0
         (- (fma (* (/ x (* n n)) 0.5) x 1.0) (pow x (pow n -1.0))))))))

C

double code(double x, double n) {
	double t_0 = (pow(pow(x, (-1.0 / n)), -1.0) / x) / n;
	double tmp;
	if (pow(n, -1.0) <= -2e+14) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 5e-114) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (pow(n, -1.0) <= 0.002) {
		tmp = t_0;
	} else {
		tmp = fma(((x / (n * n)) * 0.5), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(((x ^ Float64(-1.0 / n)) ^ -1.0) / x) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -2e+14)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 5e-114)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif ((n ^ -1.0) <= 0.002)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(Float64(x / Float64(n * n)) * 0.5), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e+14], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-114], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 0.002], t$95$0, N[(N[(N[(N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e14 or 4.99999999999999989e-114 < (/.f64 #s(literal 1 binary64) n) < 2e-3

   1. Initial program 72.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   7. Applied rewrites 88.7%

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

   9. Applied rewrites 88.7%

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

   if -2e14 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999989e-114

   1. Initial program 37.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 78.7%

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

   if 2e-3 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 48.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 85.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 0.002:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{n \cdot n} \cdot 0.5, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-172}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-135}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\frac{n}{x}, 0.3333333333333333, -0.5 \cdot n\right)}{x} + n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= (pow n -1.0) -2e+14)
     (/ (/ (/ -0.3333333333333333 (* x x)) n) (- x))
     (if (<= (pow n -1.0) -5e-45)
       t_0
       (if (<= (pow n -1.0) 1e-172)
         (/ (pow n -1.0) x)
         (if (<= (pow n -1.0) 1e-135)
           t_0
           (/
            (/ (+ (/ (fma (/ n x) 0.3333333333333333 (* -0.5 n)) x) n) x)
            (* n n))))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (pow(n, -1.0) <= -2e+14) {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	} else if (pow(n, -1.0) <= -5e-45) {
		tmp = t_0;
	} else if (pow(n, -1.0) <= 1e-172) {
		tmp = pow(n, -1.0) / x;
	} else if (pow(n, -1.0) <= 1e-135) {
		tmp = t_0;
	} else {
		tmp = (((fma((n / x), 0.3333333333333333, (-0.5 * n)) / x) + n) / x) / (n * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if ((n ^ -1.0) <= -2e+14)
		tmp = Float64(Float64(Float64(-0.3333333333333333 / Float64(x * x)) / n) / Float64(-x));
	elseif ((n ^ -1.0) <= -5e-45)
		tmp = t_0;
	elseif ((n ^ -1.0) <= 1e-172)
		tmp = Float64((n ^ -1.0) / x);
	elseif ((n ^ -1.0) <= 1e-135)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(n / x), 0.3333333333333333, Float64(-0.5 * n)) / x) + n) / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e+14], N[(N[(N[(-0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-45], t$95$0, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-172], N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-135], t$95$0, N[(N[(N[(N[(N[(N[(n / x), $MachinePrecision] * 0.3333333333333333 + N[(-0.5 * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + n), $MachinePrecision] / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e14

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.0%

       \[\leadsto -\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{x} \]

   if -2e14 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999976e-45 or 1e-172 < (/.f64 #s(literal 1 binary64) n) < 1e-135

   1. Initial program 14.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 80.4

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.9%

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

   if -4.99999999999999976e-45 < (/.f64 #s(literal 1 binary64) n) < 1e-172

   1. Initial program 44.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 65.2%

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

   if 1e-135 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 30.1

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

   5. Applied rewrites 30.1%

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

   7. Applied rewrites 51.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 55.2%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\frac{n}{x}, 0.3333333333333333, -0.5 \cdot n\right)}{x} + n}{x}}{\color{blue}{n} \cdot n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \mathbf{elif}\;{n}^{-1} \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-172}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-135}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\frac{n}{x}, 0.3333333333333333, -0.5 \cdot n\right)}{x} + n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + \frac{\mathsf{fma}\left(\frac{{\log x}^{2}}{n}, -0.5, -\log x\right)}{x}}{n} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.5e-5)
   (* (/ (+ 1.0 (/ (fma (/ (pow (log x) 2.0) n) -0.5 (- (log x))) x)) n) x)
   (/ (/ (pow (pow x (/ -1.0 n)) -1.0) x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.5e-5) {
		tmp = ((1.0 + (fma((pow(log(x), 2.0) / n), -0.5, -log(x)) / x)) / n) * x;
	} else {
		tmp = (pow(pow(x, (-1.0 / n)), -1.0) / x) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.5e-5)
		tmp = Float64(Float64(Float64(1.0 + Float64(fma(Float64((log(x) ^ 2.0) / n), -0.5, Float64(-log(x))) / x)) / n) * x);
	else
		tmp = Float64(Float64(((x ^ Float64(-1.0 / n)) ^ -1.0) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.5e-5], N[(N[(N[(1.0 + N[(N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision] * -0.5 + (-N[Log[x], $MachinePrecision])), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[Power[N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.49999999999999943e-5

   1. Initial program 43.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 5.4%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 70.9%

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

   if 6.49999999999999943e-5 < x

   1. Initial program 64.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 98.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + \frac{\mathsf{fma}\left(\frac{{\log x}^{2}}{n}, -0.5, -\log x\right)}{x}}{n} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left(\frac{-1}{n}\right)}\right)}^{-1}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -2e+14)
   (/ (/ (/ -0.3333333333333333 (* x x)) n) (- x))
   (if (<= (pow n -1.0) 1e+14)
     (/ (pow n -1.0) x)
     (/
      (/ (+ (/ (fma (/ n x) 0.3333333333333333 (* -0.5 n)) x) n) x)
      (* n n)))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -2e+14) {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	} else if (pow(n, -1.0) <= 1e+14) {
		tmp = pow(n, -1.0) / x;
	} else {
		tmp = (((fma((n / x), 0.3333333333333333, (-0.5 * n)) / x) + n) / x) / (n * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -2e+14)
		tmp = Float64(Float64(Float64(-0.3333333333333333 / Float64(x * x)) / n) / Float64(-x));
	elseif ((n ^ -1.0) <= 1e+14)
		tmp = Float64((n ^ -1.0) / x);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(n / x), 0.3333333333333333, Float64(-0.5 * n)) / x) + n) / x) / Float64(n * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e+14], N[(N[(N[(-0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e+14], N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(N[(n / x), $MachinePrecision] * 0.3333333333333333 + N[(-0.5 * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + n), $MachinePrecision] / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e14

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.0%

       \[\leadsto -\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{x} \]

   if -2e14 < (/.f64 #s(literal 1 binary64) n) < 1e14

   1. Initial program 34.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 55.2%

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

   if 1e14 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 40.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 8.9

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

   5. Applied rewrites 8.9%

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 60.9%

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

4. Final simplification 61.9%

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

Alternative 10: 53.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -2e+14)
   (/ (/ (/ -0.3333333333333333 (* x x)) n) (- x))
   (if (<= (pow n -1.0) 5e-48) (/ (pow n -1.0) x) (/ (/ n x) (* n n)))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -2e+14) {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	} else if (pow(n, -1.0) <= 5e-48) {
		tmp = pow(n, -1.0) / x;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n ** (-1.0d0)) <= (-2d+14)) then
        tmp = (((-0.3333333333333333d0) / (x * x)) / n) / -x
    else if ((n ** (-1.0d0)) <= 5d-48) then
        tmp = (n ** (-1.0d0)) / x
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -2e+14) {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	} else if (Math.pow(n, -1.0) <= 5e-48) {
		tmp = Math.pow(n, -1.0) / x;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -2e+14:
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x
	elif math.pow(n, -1.0) <= 5e-48:
		tmp = math.pow(n, -1.0) / x
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -2e+14)
		tmp = Float64(Float64(Float64(-0.3333333333333333 / Float64(x * x)) / n) / Float64(-x));
	elseif ((n ^ -1.0) <= 5e-48)
		tmp = Float64((n ^ -1.0) / x);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -2e+14)
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	elseif ((n ^ -1.0) <= 5e-48)
		tmp = (n ^ -1.0) / x;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e+14], N[(N[(N[(-0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-48], N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e14

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.0%

       \[\leadsto -\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{x} \]

   if -2e14 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-48

   1. Initial program 35.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-*.f64 56.7

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 58.2%

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

   if 4.9999999999999999e-48 < (/.f64 #s(literal 1 binary64) n)

   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 21.0

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

   5. Applied rewrites 21.0%

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

   7. Applied rewrites 50.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 49.4%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} - -1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -5e+21)
   (/ (/ (/ -0.3333333333333333 (* x x)) n) (- x))
   (/ (/ (- (/ (- (/ 0.3333333333333333 x) 0.5) x) -1.0) x) n)))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -5e+21) {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n ** (-1.0d0)) <= (-5d+21)) then
        tmp = (((-0.3333333333333333d0) / (x * x)) / n) / -x
    else
        tmp = (((((0.3333333333333333d0 / x) - 0.5d0) / x) - (-1.0d0)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -5e+21) {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -5e+21:
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x
	else:
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -5e+21)
		tmp = Float64(Float64(Float64(-0.3333333333333333 / Float64(x * x)) / n) / Float64(-x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -5e+21)
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	else
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e+21], N[(N[(N[(-0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e21

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 45.7

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

   5. Applied rewrites 45.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 51.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.4%

       \[\leadsto -\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{x} \]

   if -5e21 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 59.0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 54.2%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} - -1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} - \frac{-1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -5e+21)
   (/ (/ (/ -0.3333333333333333 (* x x)) n) (- x))
   (/ (- (/ 0.3333333333333333 (* (* x x) n)) (/ -1.0 n)) x)))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -5e+21) {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	} else {
		tmp = ((0.3333333333333333 / ((x * x) * n)) - (-1.0 / n)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n ** (-1.0d0)) <= (-5d+21)) then
        tmp = (((-0.3333333333333333d0) / (x * x)) / n) / -x
    else
        tmp = ((0.3333333333333333d0 / ((x * x) * n)) - ((-1.0d0) / n)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -5e+21) {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	} else {
		tmp = ((0.3333333333333333 / ((x * x) * n)) - (-1.0 / n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -5e+21:
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x
	else:
		tmp = ((0.3333333333333333 / ((x * x) * n)) - (-1.0 / n)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -5e+21)
		tmp = Float64(Float64(Float64(-0.3333333333333333 / Float64(x * x)) / n) / Float64(-x));
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) - Float64(-1.0 / n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -5e+21)
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	else
		tmp = ((0.3333333333333333 / ((x * x) * n)) - (-1.0 / n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e+21], N[(N[(N[(-0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e21

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 45.7

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

   5. Applied rewrites 45.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 51.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.4%

       \[\leadsto -\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{x} \]

   if -5e21 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 59.0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 54.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 54.1%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} - \frac{-1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\left(x - \log x\right) \cdot n}{n \cdot n}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (* (- x (log x)) n) (* n n))
   (if (<= x 1.2e+189)
     (/ (/ (- 1.0 (/ 0.5 x)) x) n)
     (/ (/ (/ -0.3333333333333333 (* x x)) n) (- x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = ((x - log(x)) * n) / (n * n);
	} else if (x <= 1.2e+189) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} else {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = ((x - log(x)) * n) / (n * n)
    else if (x <= 1.2d+189) then
        tmp = ((1.0d0 - (0.5d0 / x)) / x) / n
    else
        tmp = (((-0.3333333333333333d0) / (x * x)) / n) / -x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = ((x - Math.log(x)) * n) / (n * n);
	} else if (x <= 1.2e+189) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} else {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = ((x - math.log(x)) * n) / (n * n)
	elif x <= 1.2e+189:
		tmp = ((1.0 - (0.5 / x)) / x) / n
	else:
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(Float64(x - log(x)) * n) / Float64(n * n));
	elseif (x <= 1.2e+189)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / x) / n);
	else
		tmp = Float64(Float64(Float64(-0.3333333333333333 / Float64(x * x)) / n) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = ((x - log(x)) * n) / (n * n);
	elseif (x <= 1.2e+189)
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	else
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e+189], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(-0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 44.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 49.6

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

   5. Applied rewrites 49.6%

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

   7. Applied rewrites 58.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 58.8%

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

   if 1 < x < 1.2e189

   1. Initial program 46.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.0%

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

   if 1.2e189 < x

   1. Initial program 86.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\left(x - \log x\right) \cdot n}{n \cdot n}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\left(-n\right) \cdot \log x}{n \cdot n}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (* (- n) (log x)) (* n n))
   (if (<= x 1.2e+189)
     (/ (/ (- 1.0 (/ 0.5 x)) x) n)
     (/ (/ (/ -0.3333333333333333 (* x x)) n) (- x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (-n * log(x)) / (n * n);
	} else if (x <= 1.2e+189) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} else {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-n * log(x)) / (n * n)
    else if (x <= 1.2d+189) then
        tmp = ((1.0d0 - (0.5d0 / x)) / x) / n
    else
        tmp = (((-0.3333333333333333d0) / (x * x)) / n) / -x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (-n * Math.log(x)) / (n * n);
	} else if (x <= 1.2e+189) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} else {
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (-n * math.log(x)) / (n * n)
	elif x <= 1.2e+189:
		tmp = ((1.0 - (0.5 / x)) / x) / n
	else:
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(Float64(-n) * log(x)) / Float64(n * n));
	elseif (x <= 1.2e+189)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / x) / n);
	else
		tmp = Float64(Float64(Float64(-0.3333333333333333 / Float64(x * x)) / n) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (-n * log(x)) / (n * n);
	elseif (x <= 1.2e+189)
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	else
		tmp = ((-0.3333333333333333 / (x * x)) / n) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[((-n) * N[Log[x], $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e+189], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(-0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 44.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 49.6

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

   5. Applied rewrites 49.6%

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

   7. Applied rewrites 58.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 58.6%

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

   if 1 < x < 1.2e189

   1. Initial program 46.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.0%

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

   if 1.2e189 < x

   1. Initial program 86.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 86.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\left(-n\right) \cdot \log x}{n \cdot n}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.3333333333333333}{x \cdot x}}{n}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 57000000000000:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 57000000000000.0) (/ (/ n x) (* n n)) (/ (pow n -1.0) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 57000000000000.0) {
		tmp = (n / x) / (n * n);
	} else {
		tmp = pow(n, -1.0) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 57000000000000.0d0) then
        tmp = (n / x) / (n * n)
    else
        tmp = (n ** (-1.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 57000000000000.0) {
		tmp = (n / x) / (n * n);
	} else {
		tmp = Math.pow(n, -1.0) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 57000000000000.0:
		tmp = (n / x) / (n * n)
	else:
		tmp = math.pow(n, -1.0) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 57000000000000.0)
		tmp = Float64(Float64(n / x) / Float64(n * n));
	else
		tmp = Float64((n ^ -1.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 57000000000000.0)
		tmp = (n / x) / (n * n);
	else
		tmp = (n ^ -1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 57000000000000.0], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.7e13

   1. Initial program 44.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 49.2

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

   5. Applied rewrites 49.2%

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

   7. Applied rewrites 58.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 36.3%

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

   if 5.7e13 < x

   1. Initial program 64.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 74.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 57000000000000:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. mul-1-neg N/A

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

   8. distribute-lft-neg-in N/A

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

   9. metadata-eval N/A

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

   10. *-lft-identity N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-*.f64 60.6

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

5. Applied rewrites 60.6%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 48.2%

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

9. Final simplification 48.2%

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

Alternative 17: 40.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. mul-1-neg N/A

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

   8. distribute-lft-neg-in N/A

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

   9. metadata-eval N/A

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

   10. *-lft-identity N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-*.f64 60.6

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

5. Applied rewrites 60.6%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 48.2%

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

10. Applied rewrites 47.4%

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

11. Final simplification 47.4%

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

Alternative 18: 43.5% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 44.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 49.6

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

   5. Applied rewrites 49.6%

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

   7. Applied rewrites 58.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 36.2%

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

   if 1 < x

   1. Initial program 63.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.7%

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

Alternative 19: 4.5% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 63.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 37.2%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 4.5%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))