2nthrt (problem 3.4.6)

Percentage Accurate: 52.6% → 87.7%

Time: 28.5s

Alternatives: 19

Speedup: 1.9×

• could not determine a ground truth

  1. x = -2.544299644442555e+220
  2. n = 1.734273582304435e+164

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: 52.6% 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: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 19000000:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n} - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 19000000.0)
   (/
    (-
     (/
      (+
       (* 0.5 (* (log (* x (+ x 1.0))) (log (/ (+ x 1.0) x))))
       (* (- (pow (log1p x) 3.0) (pow (log x) 3.0)) (/ 0.16666666666666666 n)))
      n)
     (log (/ x (+ x 1.0))))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 19000000.0) {
		tmp = ((((0.5 * (log((x * (x + 1.0))) * log(((x + 1.0) / x)))) + ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) * (0.16666666666666666 / n))) / n) - log((x / (x + 1.0)))) / n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 19000000.0) {
		tmp = ((((0.5 * (Math.log((x * (x + 1.0))) * Math.log(((x + 1.0) / x)))) + ((Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0)) * (0.16666666666666666 / n))) / n) - Math.log((x / (x + 1.0)))) / n;
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 19000000.0:
		tmp = ((((0.5 * (math.log((x * (x + 1.0))) * math.log(((x + 1.0) / x)))) + ((math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0)) * (0.16666666666666666 / n))) / n) - math.log((x / (x + 1.0)))) / n
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 19000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * Float64(log(Float64(x * Float64(x + 1.0))) * log(Float64(Float64(x + 1.0) / x)))) + Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) * Float64(0.16666666666666666 / n))) / n) - log(Float64(x / Float64(x + 1.0)))) / n);
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 19000000.0], N[(N[(N[(N[(N[(0.5 * N[(N[Log[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9e7

   1. Initial program 47.8%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{-\frac{\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}}{n}} \]

   if 1.9e7 < x

   1. Initial program 72.3%

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

   3. Taylor expanded in x around inf

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), n\right), x\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right), n\right), x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right), n\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right), n\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right), n\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{1 \cdot \log x}{n}}\right), n\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\log x \cdot 1}{n}}\right), n\right), x\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), n\right), x\right) \]

      11. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      13. /-lowering-/.f64 99.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   5. Simplified 99.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 19000000:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n} - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;\frac{\frac{\frac{{\log x}^{3} \cdot -0.16666666666666666}{n} + -0.5 \cdot {\log x}^{2}}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.48)
   (/
    (-
     (/
      (+
       (/ (* (pow (log x) 3.0) -0.16666666666666666) n)
       (* -0.5 (pow (log x) 2.0)))
      n)
     (log x))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.48) {
		tmp = (((((pow(log(x), 3.0) * -0.16666666666666666) / n) + (-0.5 * pow(log(x), 2.0))) / n) - log(x)) / n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.48d0) then
        tmp = ((((((log(x) ** 3.0d0) * (-0.16666666666666666d0)) / n) + ((-0.5d0) * (log(x) ** 2.0d0))) / n) - log(x)) / n
    else
        tmp = ((x ** (1.0d0 / n)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.48) {
		tmp = (((((Math.pow(Math.log(x), 3.0) * -0.16666666666666666) / n) + (-0.5 * Math.pow(Math.log(x), 2.0))) / n) - Math.log(x)) / n;
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.48:
		tmp = (((((math.pow(math.log(x), 3.0) * -0.16666666666666666) / n) + (-0.5 * math.pow(math.log(x), 2.0))) / n) - math.log(x)) / n
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.48)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64((log(x) ^ 3.0) * -0.16666666666666666) / n) + Float64(-0.5 * (log(x) ^ 2.0))) / n) - log(x)) / n);
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.48)
		tmp = ((((((log(x) ^ 3.0) * -0.16666666666666666) / n) + (-0.5 * (log(x) ^ 2.0))) / n) - log(x)) / n;
	else
		tmp = ((x ^ (1.0 / n)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.48], N[(N[(N[(N[(N[(N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] / n), $MachinePrecision] + N[(-0.5 * N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.47999999999999998

   1. Initial program 48.5%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 48.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 48.2%

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

   6. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\log x}^{3}}{n} - \frac{1}{2} \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n}} \]

   8. Simplified 75.9%

      \[\leadsto \color{blue}{0 - \frac{\log x - \frac{\frac{-0.16666666666666666 \cdot {\log x}^{3}}{n} + -0.5 \cdot {\log x}^{2}}{n}}{n}} \]

   if 0.47999999999999998 < x

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), n\right), x\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right), n\right), x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right), n\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right), n\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right), n\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{1 \cdot \log x}{n}}\right), n\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\log x \cdot 1}{n}}\right), n\right), x\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), n\right), x\right) \]

      11. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      13. /-lowering-/.f64 98.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   5. Simplified 98.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;\frac{\frac{\frac{{\log x}^{3} \cdot -0.16666666666666666}{n} + -0.5 \cdot {\log x}^{2}}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{if}\;n \leq -30500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 28500000000:\\ \;\;\;\;e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ x (+ x 1.0))) (- 0.0 n))))
   (if (<= n -30500000000.0)
     t_0
     (if (<= n 28500000000.0)
       (- (exp (* (log1p x) (/ 1.0 n))) (pow x (/ 1.0 n)))
       t_0))))

C

double code(double x, double n) {
	double t_0 = log((x / (x + 1.0))) / (0.0 - n);
	double tmp;
	if (n <= -30500000000.0) {
		tmp = t_0;
	} else if (n <= 28500000000.0) {
		tmp = exp((log1p(x) * (1.0 / n))) - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log((x / (x + 1.0))) / (0.0 - n);
	double tmp;
	if (n <= -30500000000.0) {
		tmp = t_0;
	} else if (n <= 28500000000.0) {
		tmp = Math.exp((Math.log1p(x) * (1.0 / n))) - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((x / (x + 1.0))) / (0.0 - n)
	tmp = 0
	if n <= -30500000000.0:
		tmp = t_0
	elif n <= 28500000000.0:
		tmp = math.exp((math.log1p(x) * (1.0 / n))) - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n))
	tmp = 0.0
	if (n <= -30500000000.0)
		tmp = t_0;
	elseif (n <= 28500000000.0)
		tmp = Float64(exp(Float64(log1p(x) * Float64(1.0 / n))) - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -30500000000.0], t$95$0, If[LessEqual[n, 28500000000.0], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.05e10 or 2.85e10 < n

   1. Initial program 34.8%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 78.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 78.4%

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

   if -3.05e10 < n < 2.85e10

   1. Initial program 92.8%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left(\frac{x \cdot x - 1 \cdot 1}{x - 1}\right)}^{\left(\frac{1}{n}\right)}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left(\frac{1}{\frac{x - 1}{x \cdot x - 1 \cdot 1}}\right)}^{\left(\frac{1}{n}\right)}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left({\left(\frac{x - 1}{x \cdot x - 1 \cdot 1}\right)}^{-1}\right)}^{\left(\frac{1}{n}\right)}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. pow-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left(\frac{x - 1}{x \cdot x - 1 \cdot 1}\right)}^{\left(-1 \cdot \frac{1}{n}\right)}\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({\left(\frac{x - 1}{x \cdot x - 1 \cdot 1}\right)}^{\left(\frac{-1}{n}\right)}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. pow-to-exp N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(e^{\log \left(\frac{x - 1}{x \cdot x - 1 \cdot 1}\right) \cdot \frac{-1}{n}}\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{x - 1}{x \cdot x - 1 \cdot 1}\right) \cdot \frac{-1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{x - 1}{x \cdot x - 1 \cdot 1}\right), \left(\frac{-1}{n}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{1}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}\right), \left(\frac{-1}{n}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      10. flip-+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{1}{x + 1}\right), \left(\frac{-1}{n}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      11. log-div N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(\log 1 - \log \left(x + 1\right)\right), \left(\frac{-1}{n}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(0 - \log \left(x + 1\right)\right), \left(\frac{-1}{n}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \log \left(x + 1\right)\right), \left(\frac{-1}{n}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \log \left(1 + x\right)\right), \left(\frac{-1}{n}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      15. accelerator-lowering-log1p.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log1p.f64}\left(x\right)\right), \left(\frac{-1}{n}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      16. /-lowering-/.f64 98.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log1p.f64}\left(x\right)\right), \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{e^{\left(0 - \mathsf{log1p}\left(x\right)\right) \cdot \frac{-1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -30500000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;n \leq 28500000000:\\ \;\;\;\;e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{if}\;n \leq -34500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 52000000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ x (+ x 1.0))) (- 0.0 n))))
   (if (<= n -34500000000.0)
     t_0
     (if (<= n 52000000000.0)
       (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))
       t_0))))

C

double code(double x, double n) {
	double t_0 = log((x / (x + 1.0))) / (0.0 - n);
	double tmp;
	if (n <= -34500000000.0) {
		tmp = t_0;
	} else if (n <= 52000000000.0) {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log((x / (x + 1.0))) / (0.0 - n);
	double tmp;
	if (n <= -34500000000.0) {
		tmp = t_0;
	} else if (n <= 52000000000.0) {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((x / (x + 1.0))) / (0.0 - n)
	tmp = 0
	if n <= -34500000000.0:
		tmp = t_0
	elif n <= 52000000000.0:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n))
	tmp = 0.0
	if (n <= -34500000000.0)
		tmp = t_0;
	elseif (n <= 52000000000.0)
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -34500000000.0], t$95$0, If[LessEqual[n, 52000000000.0], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.45e10 or 5.2e10 < n

   1. Initial program 34.8%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 78.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 78.4%

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

   if -3.45e10 < n < 5.2e10

   1. Initial program 92.8%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(x + 1\right) \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\frac{\log \left(x + 1\right)}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log \left(x + 1\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log \left(1 + x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. accelerator-lowering-log1p.f64 98.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log1p.f64}\left(x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -34500000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;n \leq 52000000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-10}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + \frac{x \cdot \left(-0.5 + x \cdot 0.3333333333333333\right) + \frac{x \cdot \left(0.5 + x \cdot -0.5\right)}{n}}{n}\right) + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-10)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
     (if (<= (/ 1.0 n) 2000000000000.0)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (-
        (+
         (*
          x
          (+
           (/ 1.0 n)
           (/
            (+
             (* x (+ -0.5 (* x 0.3333333333333333)))
             (/ (* x (+ 0.5 (* x -0.5))) n))
            n)))
         1.0)
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-10) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 2000000000000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((x * ((1.0 / n) + (((x * (-0.5 + (x * 0.3333333333333333))) + ((x * (0.5 + (x * -0.5))) / n)) / n))) + 1.0) - 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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-10)) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    else if ((1.0d0 / n) <= 2000000000000.0d0) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else
        tmp = ((x * ((1.0d0 / n) + (((x * ((-0.5d0) + (x * 0.3333333333333333d0))) + ((x * (0.5d0 + (x * (-0.5d0)))) / n)) / n))) + 1.0d0) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-10) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 2000000000000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((x * ((1.0 / n) + (((x * (-0.5 + (x * 0.3333333333333333))) + ((x * (0.5 + (x * -0.5))) / n)) / n))) + 1.0) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-10:
		tmp = math.pow((x + 1.0), (1.0 / n)) - t_0
	elif (1.0 / n) <= 2000000000000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	else:
		tmp = ((x * ((1.0 / n) + (((x * (-0.5 + (x * 0.3333333333333333))) + ((x * (0.5 + (x * -0.5))) / n)) / n))) + 1.0) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-10)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	elseif (Float64(1.0 / n) <= 2000000000000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * Float64(-0.5 + Float64(x * 0.3333333333333333))) + Float64(Float64(x * Float64(0.5 + Float64(x * -0.5))) / n)) / n))) + 1.0) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-10)
		tmp = ((x + 1.0) ^ (1.0 / n)) - t_0;
	elseif ((1.0 / n) <= 2000000000000.0)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	else
		tmp = ((x * ((1.0 / n) + (((x * (-0.5 + (x * 0.3333333333333333))) + ((x * (0.5 + (x * -0.5))) / n)) / n))) + 1.0) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-10], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000000000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * N[(-0.5 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000007e-10

   1. Initial program 99.5%

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

   if -2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n) < 2e12

   1. Initial program 34.7%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 77.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 77.5%

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

   if 2e12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 60.4%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \color{blue}{\left(\frac{x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}}{n}\right)}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x + \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot x\right), \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{3}\right), \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{2} \cdot x\right)\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{-1}{2}\right)\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      13. *-lowering-*.f64 96.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   8. Simplified 96.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-10}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + \frac{x \cdot \left(-0.5 + x \cdot 0.3333333333333333\right) + \frac{x \cdot \left(0.5 + x \cdot -0.5\right)}{n}}{n}\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + \frac{x \cdot \left(-0.5 + x \cdot 0.3333333333333333\right) + \frac{x \cdot \left(0.5 + x \cdot -0.5\right)}{n}}{n}\right) + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-83)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2000000000000.0)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (-
        (+
         (*
          x
          (+
           (/ 1.0 n)
           (/
            (+
             (* x (+ -0.5 (* x 0.3333333333333333)))
             (/ (* x (+ 0.5 (* x -0.5))) n))
            n)))
         1.0)
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-83) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2000000000000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((x * ((1.0 / n) + (((x * (-0.5 + (x * 0.3333333333333333))) + ((x * (0.5 + (x * -0.5))) / n)) / n))) + 1.0) - 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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-83)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2000000000000.0d0) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else
        tmp = ((x * ((1.0d0 / n) + (((x * ((-0.5d0) + (x * 0.3333333333333333d0))) + ((x * (0.5d0 + (x * (-0.5d0)))) / n)) / n))) + 1.0d0) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-83) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2000000000000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((x * ((1.0 / n) + (((x * (-0.5 + (x * 0.3333333333333333))) + ((x * (0.5 + (x * -0.5))) / n)) / n))) + 1.0) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-83:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2000000000000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	else:
		tmp = ((x * ((1.0 / n) + (((x * (-0.5 + (x * 0.3333333333333333))) + ((x * (0.5 + (x * -0.5))) / n)) / n))) + 1.0) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-83)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2000000000000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * Float64(-0.5 + Float64(x * 0.3333333333333333))) + Float64(Float64(x * Float64(0.5 + Float64(x * -0.5))) / n)) / n))) + 1.0) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-83)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2000000000000.0)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	else
		tmp = ((x * ((1.0 / n) + (((x * (-0.5 + (x * 0.3333333333333333))) + ((x * (0.5 + (x * -0.5))) / n)) / n))) + 1.0) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-83], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000000000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * N[(-0.5 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e-83

   1. Initial program 89.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), n\right), x\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right), n\right), x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right), n\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right), n\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right), n\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{1 \cdot \log x}{n}}\right), n\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\log x \cdot 1}{n}}\right), n\right), x\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), n\right), x\right) \]

      11. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      13. /-lowering-/.f64 93.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   5. Simplified 93.0%

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

   if -5e-83 < (/.f64 #s(literal 1 binary64) n) < 2e12

   1. Initial program 35.5%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 79.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 79.6%

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

   if 2e12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 60.4%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \color{blue}{\left(\frac{x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}}{n}\right)}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x + \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot x\right), \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{3}\right), \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{2} \cdot x\right)\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{-1}{2}\right)\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      13. *-lowering-*.f64 96.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   8. Simplified 96.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + \frac{x \cdot \left(-0.5 + x \cdot 0.3333333333333333\right) + \frac{x \cdot \left(0.5 + x \cdot -0.5\right)}{n}}{n}\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + \frac{x \cdot \left(-0.5 + \frac{0.5}{n}\right)}{n}\right) + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-83)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2000000000000.0)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (- (+ (* x (+ (/ 1.0 n) (/ (* x (+ -0.5 (/ 0.5 n))) n))) 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-83) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2000000000000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((x * ((1.0 / n) + ((x * (-0.5 + (0.5 / n))) / n))) + 1.0) - 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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-83)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2000000000000.0d0) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else
        tmp = ((x * ((1.0d0 / n) + ((x * ((-0.5d0) + (0.5d0 / n))) / n))) + 1.0d0) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-83) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2000000000000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((x * ((1.0 / n) + ((x * (-0.5 + (0.5 / n))) / n))) + 1.0) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-83:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2000000000000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	else:
		tmp = ((x * ((1.0 / n) + ((x * (-0.5 + (0.5 / n))) / n))) + 1.0) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-83)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2000000000000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(x * Float64(-0.5 + Float64(0.5 / n))) / n))) + 1.0) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-83)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2000000000000.0)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	else
		tmp = ((x * ((1.0 / n) + ((x * (-0.5 + (0.5 / n))) / n))) + 1.0) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-83], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000000000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(x * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e-83

   1. Initial program 89.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), n\right), x\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right), n\right), x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right), n\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right), n\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right), n\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{1 \cdot \log x}{n}}\right), n\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\log x \cdot 1}{n}}\right), n\right), x\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), n\right), x\right) \]

      11. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      13. /-lowering-/.f64 93.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   5. Simplified 93.0%

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

   if -5e-83 < (/.f64 #s(literal 1 binary64) n) < 2e12

   1. Initial program 35.5%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 79.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 79.6%

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

   if 2e12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 60.4%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \color{blue}{\left(\frac{x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}}{n}\right)}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x + \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot x\right), \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{3}\right), \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{2} \cdot x\right)\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{-1}{2}\right)\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      13. *-lowering-*.f64 96.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \frac{-1}{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right), n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   8. Simplified 96.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{n} + x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{\frac{1}{2} \cdot 1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{\frac{1}{2}}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{\frac{1}{2}}{n \cdot n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{\frac{\frac{1}{2}}{n}}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{\frac{\frac{1}{2} \cdot 1}{n}}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{\frac{1}{2} \cdot \frac{1}{n}}{n} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       12. associate-*r/ N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{\frac{1}{2} \cdot \frac{1}{n}}{n} - \frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{\frac{1}{2} \cdot \frac{1}{n}}{n} - \frac{\frac{1}{2}}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       14. div-sub N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \frac{\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}}{n}\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{x \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}\right)}{n}\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   11. Simplified 96.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + \frac{x \cdot \left(-0.5 + \frac{0.5}{n}\right)}{n}\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-83)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2000000000000.0)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (- (+ (/ x n) 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-83) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2000000000000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((x / n) + 1.0) - 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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-83)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2000000000000.0d0) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else
        tmp = ((x / n) + 1.0d0) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-83) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2000000000000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((x / n) + 1.0) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-83:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2000000000000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	else:
		tmp = ((x / n) + 1.0) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-83)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2000000000000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	else
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-83)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2000000000000.0)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	else
		tmp = ((x / n) + 1.0) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-83], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000000000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e-83

   1. Initial program 89.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), n\right), x\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right), n\right), x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right), n\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right), n\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right), n\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{1 \cdot \log x}{n}}\right), n\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\log x \cdot 1}{n}}\right), n\right), x\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), n\right), x\right) \]

      11. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      13. /-lowering-/.f64 93.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   5. Simplified 93.0%

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

   if -5e-83 < (/.f64 #s(literal 1 binary64) n) < 2e12

   1. Initial program 35.5%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 79.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 79.6%

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

   if 2e12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{x \cdot 1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + x \cdot \frac{1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x \cdot 1}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. /-lowering-/.f64 80.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 80.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-83)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2000000000000.0)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (- 1.0 t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-83) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2000000000000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = 1.0 - 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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-83)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2000000000000.0d0) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else
        tmp = 1.0d0 - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-83) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2000000000000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-83:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2000000000000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-83)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2000000000000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-83)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2000000000000.0)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	else
		tmp = 1.0 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-83], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000000000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], N[(1.0 - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e-83

   1. Initial program 89.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), n\right), x\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right), n\right), x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right), n\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right), n\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right), n\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{1 \cdot \log x}{n}}\right), n\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\log x \cdot 1}{n}}\right), n\right), x\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), n\right), x\right) \]

      11. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      13. /-lowering-/.f64 93.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   5. Simplified 93.0%

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

   if -5e-83 < (/.f64 #s(literal 1 binary64) n) < 2e12

   1. Initial program 35.5%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 79.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 79.6%

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

   if 2e12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 76.4%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 76.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 2000000000000.0)
   (/ (log (/ x (+ x 1.0))) (- 0.0 n))
   (- 1.0 (pow x (/ 1.0 n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2000000000000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = 1.0 - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 2000000000000.0d0) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2000000000000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 2000000000000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	else:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 2000000000000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	else
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 2000000000000.0)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	else
		tmp = 1.0 - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000000000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < 2e12

   1. Initial program 57.7%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 68.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 68.7%

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

   if 2e12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 76.4%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 76.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2000000000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-247}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.06 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.65e-247)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.6)
     (- 0.0 (/ (log x) n))
     (if (<= x 2.06e+167)
       (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)
       0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.65e-247) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.6) {
		tmp = 0.0 - (log(x) / n);
	} else if (x <= 2.06e+167) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.65d-247) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.6d0) then
        tmp = 0.0d0 - (log(x) / n)
    else if (x <= 2.06d+167) then
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.65e-247) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.6) {
		tmp = 0.0 - (Math.log(x) / n);
	} else if (x <= 2.06e+167) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.65e-247:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.6:
		tmp = 0.0 - (math.log(x) / n)
	elif x <= 2.06e+167:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.65e-247)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.6)
		tmp = Float64(0.0 - Float64(log(x) / n));
	elseif (x <= 2.06e+167)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.65e-247)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.6)
		tmp = 0.0 - (log(x) / n);
	elseif (x <= 2.06e+167)
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.65e-247], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.6], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.06e+167], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.64999999999999986e-247

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 65.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 65.9%

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

   if 1.64999999999999986e-247 < x < 0.599999999999999978

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 44.1%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 44.1%

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

   6. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log x}{n}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{\log x}{n}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\log x}{n}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\log x, \color{blue}{n}\right)\right) \]

      5. log-lowering-log.f64 54.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right) \]

   8. Simplified 54.7%

      \[\leadsto \color{blue}{0 - \frac{\log x}{n}} \]

   if 0.599999999999999978 < x < 2.06e167

   1. Initial program 49.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 77.2%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}\right)}, x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), n\right), x\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), n\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), n\right), x\right) \]

      11. /-lowering-/.f64 71.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), n\right), x\right) \]

   8. Simplified 71.1%

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

   if 2.06e167 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 57.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 57.2%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 93.4%

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

      1. metadata-eval 93.4%

         \[\leadsto 0 \]

   10. Applied egg-rr 93.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-247}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.06 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.06 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.6)
   (- 0.0 (/ (log x) n))
   (if (<= x 2.06e+167)
     (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.6) {
		tmp = 0.0 - (log(x) / n);
	} else if (x <= 2.06e+167) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = 0.0d0 - (log(x) / n)
    else if (x <= 2.06d+167) then
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.6) {
		tmp = 0.0 - (Math.log(x) / n);
	} else if (x <= 2.06e+167) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.6:
		tmp = 0.0 - (math.log(x) / n)
	elif x <= 2.06e+167:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(0.0 - Float64(log(x) / n));
	elseif (x <= 2.06e+167)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.6)
		tmp = 0.0 - (log(x) / n);
	elseif (x <= 2.06e+167)
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.6], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.06e+167], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.599999999999999978

   1. Initial program 48.5%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 48.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 48.2%

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

   6. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log x}{n}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{\log x}{n}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\log x}{n}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\log x, \color{blue}{n}\right)\right) \]

      5. log-lowering-log.f64 52.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right) \]

   8. Simplified 52.1%

      \[\leadsto \color{blue}{0 - \frac{\log x}{n}} \]

   if 0.599999999999999978 < x < 2.06e167

   1. Initial program 49.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 77.2%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}\right)}, x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), n\right), x\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), n\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), n\right), x\right) \]

      11. /-lowering-/.f64 71.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), n\right), x\right) \]

   8. Simplified 71.1%

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

   if 2.06e167 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 57.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 57.2%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 93.4%

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

      1. metadata-eval 93.4%

         \[\leadsto 0 \]

   10. Applied egg-rr 93.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.06 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.05e+167)
   (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.05e+167) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.05d+167) then
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.05e+167) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.05e+167:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.05e+167)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.05e+167)
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.05e+167], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.05e167

   1. Initial program 48.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 34.8%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}\right)}, x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), n\right), x\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), n\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), n\right), x\right) \]

      11. /-lowering-/.f64 44.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), n\right), x\right) \]

   8. Simplified 44.6%

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

   if 2.05e167 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 57.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 57.2%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 93.4%

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

      1. metadata-eval 93.4%

         \[\leadsto 0 \]

   10. Applied egg-rr 93.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.7% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.06 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\frac{x + \left(\frac{0.3333333333333333}{x} - 0.5\right)}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.06e+167)
   (/ (/ (/ (+ x (- (/ 0.3333333333333333 x) 0.5)) n) x) x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.06e+167) {
		tmp = (((x + ((0.3333333333333333 / x) - 0.5)) / n) / x) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.06d+167) then
        tmp = (((x + ((0.3333333333333333d0 / x) - 0.5d0)) / n) / x) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.06e+167) {
		tmp = (((x + ((0.3333333333333333 / x) - 0.5)) / n) / x) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.06e+167:
		tmp = (((x + ((0.3333333333333333 / x) - 0.5)) / n) / x) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.06e+167)
		tmp = Float64(Float64(Float64(Float64(x + Float64(Float64(0.3333333333333333 / x) - 0.5)) / n) / x) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.06e+167)
		tmp = (((x + ((0.3333333333333333 / x) - 0.5)) / n) / x) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.06e+167], N[(N[(N[(N[(x + N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.06e167

   1. Initial program 48.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 34.8%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + \frac{{x}^{\left(\frac{1}{n}\right)}}{x} \cdot \left(\left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}\right)}{x}} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + \frac{{x}^{\left(\frac{1}{n}\right)} \cdot \left(\left(\frac{\frac{1}{2}}{n \cdot n} + \frac{\frac{-1}{2}}{n}\right) + \frac{\frac{\frac{1}{6}}{n \cdot \left(n \cdot n\right)} + \left(\frac{\frac{1}{3}}{n} + \frac{\frac{-1}{2}}{n \cdot n}\right)}{x}\right)}{x}\right), x\right) \]

      2. frac-add N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)} \cdot x + n \cdot \left({x}^{\left(\frac{1}{n}\right)} \cdot \left(\left(\frac{\frac{1}{2}}{n \cdot n} + \frac{\frac{-1}{2}}{n}\right) + \frac{\frac{\frac{1}{6}}{n \cdot \left(n \cdot n\right)} + \left(\frac{\frac{1}{3}}{n} + \frac{\frac{-1}{2}}{n \cdot n}\right)}{x}\right)\right)}{n \cdot x}\right), x\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)} \cdot x + n \cdot \left({x}^{\left(\frac{1}{n}\right)} \cdot \left(\left(\frac{\frac{1}{2}}{n \cdot n} + \frac{\frac{-1}{2}}{n}\right) + \frac{\frac{\frac{1}{6}}{n \cdot \left(n \cdot n\right)} + \left(\frac{\frac{1}{3}}{n} + \frac{\frac{-1}{2}}{n \cdot n}\right)}{x}\right)\right)\right), \left(n \cdot x\right)\right), x\right) \]

   7. Applied egg-rr 34.3%

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

   8. Taylor expanded in n around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x + -1 \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)}{n \cdot x}\right)}, x\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x + -1 \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)}{n}}{x}\right), x\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x + -1 \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)}{n}\right), x\right), x\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x + -1 \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right), n\right), x\right), x\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right)\right)\right), n\right), x\right), x\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right), n\right), x\right), x\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right), n\right), x\right), x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{3} \cdot \frac{1}{x}\right)\right)\right), n\right), x\right), x\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{3} \cdot 1}{x}\right)\right)\right), n\right), x\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{3}}{x}\right)\right)\right), n\right), x\right), x\right) \]

      10. /-lowering-/.f64 44.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{3}, x\right)\right)\right), n\right), x\right), x\right) \]

   10. Simplified 44.6%

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

   if 2.06e167 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 57.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 57.2%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 93.4%

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

      1. metadata-eval 93.4%

         \[\leadsto 0 \]

   10. Applied egg-rr 93.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.06 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\frac{x + \left(\frac{0.3333333333333333}{x} - 0.5\right)}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.4% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{x + \left(\frac{0.3333333333333333}{x} - 0.5\right)}{x \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.95e+167)
   (/ (/ (+ x (- (/ 0.3333333333333333 x) 0.5)) (* x n)) x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.95e+167) {
		tmp = ((x + ((0.3333333333333333 / x) - 0.5)) / (x * n)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.95d+167) then
        tmp = ((x + ((0.3333333333333333d0 / x) - 0.5d0)) / (x * n)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.95e+167) {
		tmp = ((x + ((0.3333333333333333 / x) - 0.5)) / (x * n)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.95e+167:
		tmp = ((x + ((0.3333333333333333 / x) - 0.5)) / (x * n)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.95e+167)
		tmp = Float64(Float64(Float64(x + Float64(Float64(0.3333333333333333 / x) - 0.5)) / Float64(x * n)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.95e+167)
		tmp = ((x + ((0.3333333333333333 / x) - 0.5)) / (x * n)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.95e+167], N[(N[(N[(x + N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.9499999999999999e167

   1. Initial program 48.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 34.8%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + \frac{{x}^{\left(\frac{1}{n}\right)}}{x} \cdot \left(\left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right) + \frac{\frac{0.16666666666666666}{n \cdot \left(n \cdot n\right)} + \left(\frac{0.3333333333333333}{n} + \frac{-0.5}{n \cdot n}\right)}{x}\right)}{x}} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + \frac{{x}^{\left(\frac{1}{n}\right)} \cdot \left(\left(\frac{\frac{1}{2}}{n \cdot n} + \frac{\frac{-1}{2}}{n}\right) + \frac{\frac{\frac{1}{6}}{n \cdot \left(n \cdot n\right)} + \left(\frac{\frac{1}{3}}{n} + \frac{\frac{-1}{2}}{n \cdot n}\right)}{x}\right)}{x}\right), x\right) \]

      2. frac-add N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)} \cdot x + n \cdot \left({x}^{\left(\frac{1}{n}\right)} \cdot \left(\left(\frac{\frac{1}{2}}{n \cdot n} + \frac{\frac{-1}{2}}{n}\right) + \frac{\frac{\frac{1}{6}}{n \cdot \left(n \cdot n\right)} + \left(\frac{\frac{1}{3}}{n} + \frac{\frac{-1}{2}}{n \cdot n}\right)}{x}\right)\right)}{n \cdot x}\right), x\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)} \cdot x + n \cdot \left({x}^{\left(\frac{1}{n}\right)} \cdot \left(\left(\frac{\frac{1}{2}}{n \cdot n} + \frac{\frac{-1}{2}}{n}\right) + \frac{\frac{\frac{1}{6}}{n \cdot \left(n \cdot n\right)} + \left(\frac{\frac{1}{3}}{n} + \frac{\frac{-1}{2}}{n \cdot n}\right)}{x}\right)\right)\right), \left(n \cdot x\right)\right), x\right) \]

   7. Applied egg-rr 34.3%

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

   8. Taylor expanded in n around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(x + -1 \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right)}, \mathsf{*.f64}\left(n, x\right)\right), x\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right)\right)\right), \mathsf{*.f64}\left(n, x\right)\right), x\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x - \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right), \mathsf{*.f64}\left(n, x\right)\right), x\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right), \mathsf{*.f64}\left(n, x\right)\right), x\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{3} \cdot \frac{1}{x}\right)\right)\right), \mathsf{*.f64}\left(n, x\right)\right), x\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{3} \cdot 1}{x}\right)\right)\right), \mathsf{*.f64}\left(n, x\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{3}}{x}\right)\right)\right), \mathsf{*.f64}\left(n, x\right)\right), x\right) \]

      7. /-lowering-/.f64 44.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{3}, x\right)\right)\right), \mathsf{*.f64}\left(n, x\right)\right), x\right) \]

   10. Simplified 44.2%

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

   if 1.9499999999999999e167 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 57.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 57.2%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 93.4%

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

      1. metadata-eval 93.4%

         \[\leadsto 0 \]

   10. Applied egg-rr 93.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{x + \left(\frac{0.3333333333333333}{x} - 0.5\right)}{x \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.8% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+32}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e+32) 0.0 (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+32) {
		tmp = 0.0;
	} else {
		tmp = (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 ((1.0d0 / n) <= (-2d+32)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+32) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e+32:
		tmp = 0.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+32)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e+32)
		tmp = 0.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+32], 0.0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000011e32

   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 x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 47.7%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 47.7%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 54.7%

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

      1. metadata-eval 54.7%

         \[\leadsto 0 \]

   10. Applied egg-rr 54.7%

       \[\leadsto \color{blue}{0} \]

   if -2.00000000000000011e32 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 42.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), n\right), x\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right), n\right), x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right), n\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right), n\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right), n\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{1 \cdot \log x}{n}}\right), n\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\log x \cdot 1}{n}}\right), n\right), x\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), n\right), x\right) \]

      11. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      13. /-lowering-/.f64 49.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   5. Simplified 49.2%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}\right), \color{blue}{n}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), x\right), n\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), x\right), n\right) \]

      6. /-lowering-/.f64 49.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), x\right), n\right) \]

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, n\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 49.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), n\right) \]

   10. Simplified 49.4%

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

Alternative 17: 45.8% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+32}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e+32) 0.0 (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+32) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d+32)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+32) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e+32:
		tmp = 0.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+32)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e+32)
		tmp = 0.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+32], 0.0, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000011e32

   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 x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 47.7%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 47.7%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 54.7%

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

      1. metadata-eval 54.7%

         \[\leadsto 0 \]

   10. Applied egg-rr 54.7%

       \[\leadsto \color{blue}{0} \]

   if -2.00000000000000011e32 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 42.4%

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

   3. Taylor expanded in x around inf

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), n\right), x\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right), n\right), x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right), n\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right), n\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right), n\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{1 \cdot \log x}{n}}\right), n\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\log x \cdot 1}{n}}\right), n\right), x\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), n\right), x\right) \]

      11. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      13. /-lowering-/.f64 49.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   5. Simplified 49.2%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{n}\right)}, x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 49.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), x\right) \]

   8. Simplified 49.4%

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

Alternative 18: 43.8% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{+167}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 2.3e+167) (/ 1.0 (* x n)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.3e+167) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.3d+167) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.3e+167) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.3e+167:
		tmp = 1.0 / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.3e+167)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.3e+167)
		tmp = 1.0 / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.3e+167], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.29999999999999988e167

   1. Initial program 48.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}\right), \color{blue}{x}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), n\right), x\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right), n\right), x\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right), n\right), x\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right), n\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right), n\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{1 \cdot \log x}{n}}\right), n\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\log x \cdot 1}{n}}\right), n\right), x\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), n\right), x\right) \]

      11. exp-to-pow N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      13. /-lowering-/.f64 52.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   5. Simplified 52.9%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(n \cdot x\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{n}\right)\right) \]

      3. *-lowering-*.f64 36.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 36.8%

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

   if 2.29999999999999988e167 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

      15. exp-to-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      17. /-lowering-/.f64 57.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   5. Simplified 57.2%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 93.4%

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

      1. metadata-eval 93.4%

         \[\leadsto 0 \]

   10. Applied egg-rr 93.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 30.8% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in x around 0

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

   1. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-frac N/A

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

   4. mul-1-neg N/A

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

   5. log-rec N/A

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

   6. mul-1-neg N/A

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

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right)}\right) \]

   8. log-rec N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{\mathsf{neg}\left(\log x\right)}{n}}\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{-1 \cdot \frac{-1 \cdot \log x}{n}}\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\left(-1 \cdot -1\right) \cdot \log x}{n}}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{1 \cdot \log x}{n}}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\frac{\log x \cdot 1}{n}}\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \left(e^{\log x \cdot \frac{1}{n}}\right)\right) \]

   15. exp-to-pow N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{1}{n}\right)}}\right)\right) \]

   16. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

   17. /-lowering-/.f64 43.8%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

5. Simplified 43.8%

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

6. Taylor expanded in n around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
7. Step-by-step derivation

8. Simplified 36.5%

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

   1. metadata-eval 36.5%

      \[\leadsto 0 \]

10. Applied egg-rr 36.5%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))