2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 86.2%

Time: 43.4s

Alternatives: 19

Speedup: 1.9×

• could not determine a ground truth

  1. x = -2
  2. n = 4.94066e-324

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -210000:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{\frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) + -0.16666666666666666 \cdot \left({\log x}^{3} - {\left(\mathsf{log1p}\left(x\right)\right)}^{3}\right)}{n}}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{elif}\;n \leq 25:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -210000.0)
   (/
    (+
     (/
      (+
       (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
       (/
        (+
         (*
          (/ 0.041666666666666664 n)
          (- (pow (log1p x) 4.0) (pow (log x) 4.0)))
         (* -0.16666666666666666 (- (pow (log x) 3.0) (pow (log1p x) 3.0))))
        n))
      n)
     (- (log1p x) (log x)))
    n)
   (if (<= n 25.0)
     (- (exp (/ x n)) (pow x (/ 1.0 n)))
     (* (/ -1.0 n) (log (/ x (+ x 1.0)))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -210000.0) {
		tmp = ((((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) + ((((0.041666666666666664 / n) * (pow(log1p(x), 4.0) - pow(log(x), 4.0))) + (-0.16666666666666666 * (pow(log(x), 3.0) - pow(log1p(x), 3.0)))) / n)) / n) + (log1p(x) - log(x))) / n;
	} else if (n <= 25.0) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = (-1.0 / n) * log((x / (x + 1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -210000.0) {
		tmp = ((((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) + ((((0.041666666666666664 / n) * (Math.pow(Math.log1p(x), 4.0) - Math.pow(Math.log(x), 4.0))) + (-0.16666666666666666 * (Math.pow(Math.log(x), 3.0) - Math.pow(Math.log1p(x), 3.0)))) / n)) / n) + (Math.log1p(x) - Math.log(x))) / n;
	} else if (n <= 25.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (-1.0 / n) * Math.log((x / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -210000.0:
		tmp = ((((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) + ((((0.041666666666666664 / n) * (math.pow(math.log1p(x), 4.0) - math.pow(math.log(x), 4.0))) + (-0.16666666666666666 * (math.pow(math.log(x), 3.0) - math.pow(math.log1p(x), 3.0)))) / n)) / n) + (math.log1p(x) - math.log(x))) / n
	elif n <= 25.0:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = (-1.0 / n) * math.log((x / (x + 1.0)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -210000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) + Float64(Float64(Float64(Float64(0.041666666666666664 / n) * Float64((log1p(x) ^ 4.0) - (log(x) ^ 4.0))) + Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) - (log1p(x) ^ 3.0)))) / n)) / n) + Float64(log1p(x) - log(x))) / n);
	elseif (n <= 25.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-1.0 / n) * log(Float64(x / Float64(x + 1.0))));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -210000.0], N[(N[(N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(0.041666666666666664 / n), $MachinePrecision] * N[(N[Power[N[Log[1 + x], $MachinePrecision], 4.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 25.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / n), $MachinePrecision] * N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.1e5

   1. Initial program 35.3%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 73.4%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   if -2.1e5 < n < 25

   1. Initial program 81.4%

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

      1. 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. log1p-define N/A

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

      7. log1p-lowering-log1p.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.9%

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

   7. Simplified 99.9%

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

   if 25 < n

   1. Initial program 25.0%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 83.4%

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

   10. Simplified 83.4%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -210000:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{\frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) + -0.16666666666666666 \cdot \left({\log x}^{3} - {\left(\mathsf{log1p}\left(x\right)\right)}^{3}\right)}{n}}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{elif}\;n \leq 25:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -62000000:\\ \;\;\;\;\frac{1}{t\_0 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \frac{2 \cdot \left(\mathsf{log1p}\left(x\right) + \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n}\right) - 2 \cdot \left(\log x + \frac{{\log x}^{2}}{n}\right)}{n}\\ \mathbf{elif}\;n \leq 25:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= n -62000000.0)
     (*
      (/ 1.0 (+ t_0 (pow (+ x 1.0) (/ 1.0 n))))
      (/
       (-
        (* 2.0 (+ (log1p x) (/ (pow (log1p x) 2.0) n)))
        (* 2.0 (+ (log x) (/ (pow (log x) 2.0) n))))
       n))
     (if (<= n 25.0)
       (- (exp (/ x n)) t_0)
       (* (/ -1.0 n) (log (/ x (+ x 1.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (n <= -62000000.0) {
		tmp = (1.0 / (t_0 + pow((x + 1.0), (1.0 / n)))) * (((2.0 * (log1p(x) + (pow(log1p(x), 2.0) / n))) - (2.0 * (log(x) + (pow(log(x), 2.0) / n)))) / n);
	} else if (n <= 25.0) {
		tmp = exp((x / n)) - t_0;
	} else {
		tmp = (-1.0 / n) * log((x / (x + 1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (n <= -62000000.0) {
		tmp = (1.0 / (t_0 + Math.pow((x + 1.0), (1.0 / n)))) * (((2.0 * (Math.log1p(x) + (Math.pow(Math.log1p(x), 2.0) / n))) - (2.0 * (Math.log(x) + (Math.pow(Math.log(x), 2.0) / n)))) / n);
	} else if (n <= 25.0) {
		tmp = Math.exp((x / n)) - t_0;
	} else {
		tmp = (-1.0 / n) * Math.log((x / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if n <= -62000000.0:
		tmp = (1.0 / (t_0 + math.pow((x + 1.0), (1.0 / n)))) * (((2.0 * (math.log1p(x) + (math.pow(math.log1p(x), 2.0) / n))) - (2.0 * (math.log(x) + (math.pow(math.log(x), 2.0) / n)))) / n)
	elif n <= 25.0:
		tmp = math.exp((x / n)) - t_0
	else:
		tmp = (-1.0 / n) * math.log((x / (x + 1.0)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (n <= -62000000.0)
		tmp = Float64(Float64(1.0 / Float64(t_0 + (Float64(x + 1.0) ^ Float64(1.0 / n)))) * Float64(Float64(Float64(2.0 * Float64(log1p(x) + Float64((log1p(x) ^ 2.0) / n))) - Float64(2.0 * Float64(log(x) + Float64((log(x) ^ 2.0) / n)))) / n));
	elseif (n <= 25.0)
		tmp = Float64(exp(Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(-1.0 / n) * log(Float64(x / Float64(x + 1.0))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -62000000.0], N[(N[(1.0 / N[(t$95$0 + N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[(N[Log[1 + x], $MachinePrecision] + N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[Log[x], $MachinePrecision] + N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 25.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 / n), $MachinePrecision] * N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.2e7

   1. Initial program 34.5%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 34.5%

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

   5. Taylor expanded in n around inf

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

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

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

   7. Simplified 73.0%

      \[\leadsto \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}} \cdot \color{blue}{\frac{2 \cdot \left(\mathsf{log1p}\left(x\right) + \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n}\right) - 2 \cdot \left(\log x + \frac{{\log x}^{2}}{n}\right)}{n}} \]

   if -6.2e7 < n < 25

   1. Initial program 81.4%

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

      1. 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. log1p-define N/A

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

      7. log1p-lowering-log1p.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.8%

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

   7. Simplified 99.8%

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

   if 25 < n

   1. Initial program 25.0%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 83.4%

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

   10. Simplified 83.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -62000000:\\ \;\;\;\;\frac{1}{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \frac{2 \cdot \left(\mathsf{log1p}\left(x\right) + \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n}\right) - 2 \cdot \left(\log x + \frac{{\log x}^{2}}{n}\right)}{n}\\ \mathbf{elif}\;n \leq 25:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -45000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} - \log x\right)}{n}\\ \mathbf{elif}\;n \leq 25:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -45000000.0)
   (/
    (+
     (log1p x)
     (- (/ (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))) n) (log x)))
    n)
   (if (<= n 25.0)
     (- (exp (/ x n)) (pow x (/ 1.0 n)))
     (* (/ -1.0 n) (log (/ x (+ x 1.0)))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -45000000.0) {
		tmp = (log1p(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) / n) - log(x))) / n;
	} else if (n <= 25.0) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = (-1.0 / n) * log((x / (x + 1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -45000000.0) {
		tmp = (Math.log1p(x) + (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) / n) - Math.log(x))) / n;
	} else if (n <= 25.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (-1.0 / n) * Math.log((x / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -45000000.0:
		tmp = (math.log1p(x) + (((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) / n) - math.log(x))) / n
	elif n <= 25.0:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = (-1.0 / n) * math.log((x / (x + 1.0)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -45000000.0)
		tmp = Float64(Float64(log1p(x) + Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) / n) - log(x))) / n);
	elseif (n <= 25.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-1.0 / n) * log(Float64(x / Float64(x + 1.0))));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -45000000.0], N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 25.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / n), $MachinePrecision] * N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.5e7

   1. Initial program 34.5%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 73.0%

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

   if -4.5e7 < n < 25

   1. Initial program 81.4%

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

      1. 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. log1p-define N/A

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

      7. log1p-lowering-log1p.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.8%

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

   7. Simplified 99.8%

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

   if 25 < n

   1. Initial program 25.0%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 83.4%

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

   10. Simplified 83.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -45000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} - \log x\right)}{n}\\ \mathbf{elif}\;n \leq 25:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -22000000000:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 25:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -22000000000.0)
   (/ (log (/ (+ x 1.0) x)) n)
   (if (<= n 25.0)
     (- (exp (/ x n)) (pow x (/ 1.0 n)))
     (* (/ -1.0 n) (log (/ x (+ x 1.0)))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -22000000000.0) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if (n <= 25.0) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = (-1.0 / n) * log((x / (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-22000000000.0d0)) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if (n <= 25.0d0) then
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    else
        tmp = ((-1.0d0) / n) * log((x / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -22000000000.0) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if (n <= 25.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (-1.0 / n) * Math.log((x / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -22000000000.0:
		tmp = math.log(((x + 1.0) / x)) / n
	elif n <= 25.0:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = (-1.0 / n) * math.log((x / (x + 1.0)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -22000000000.0)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (n <= 25.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-1.0 / n) * log(Float64(x / Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -22000000000.0)
		tmp = log(((x + 1.0) / x)) / n;
	elseif (n <= 25.0)
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	else
		tmp = (-1.0 / n) * log((x / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -22000000000.0], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 25.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / n), $MachinePrecision] * N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.2e10

   1. Initial program 34.5%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 72.2%

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

   10. Simplified 72.2%

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

       1. associate-*l/ N/A

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

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

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

       3. mul-1-neg N/A

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

       4. neg-log N/A

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

       5. clear-num N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. +-lowering-+.f64 72.2%

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

   12. Applied egg-rr 72.2%

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

   if -2.2e10 < n < 25

   1. Initial program 81.4%

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

      1. 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. log1p-define N/A

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

      7. log1p-lowering-log1p.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.8%

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

   7. Simplified 99.8%

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

   if 25 < n

   1. Initial program 25.0%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 83.4%

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

   10. Simplified 83.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -22000000000:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 25:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot \log \left(\frac{x}{x + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right) + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 0.05)
       (/ (log (/ (+ x 1.0) x)) n)
       (-
        (+ (* x (+ (/ 1.0 n) (* x (+ (/ 0.5 (* n n)) (/ -0.5 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-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.05) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / 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-11)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 0.05d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = ((x * ((1.0d0 / n) + (x * ((0.5d0 / (n * n)) + ((-0.5d0) / 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-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.05) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / 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-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 0.05:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / 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-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.05)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / 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-11)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 0.05)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / 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-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.05], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $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) < -1.99999999999999988e-11

   1. Initial program 98.4%

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

      1. 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. log1p-define N/A

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

      7. log1p-lowering-log1p.f64 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 96.0%

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

   7. Simplified 96.0%

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

      1. div-inv N/A

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

      2. exp-to-pow N/A

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

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

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

      4. /-lowering-/.f64 96.0%

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

   9. Applied egg-rr 96.0%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 0.050000000000000003

   1. Initial program 29.4%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 78.8%

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

   10. Simplified 78.8%

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

       1. associate-*l/ N/A

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

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

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

       3. mul-1-neg N/A

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

       4. neg-log N/A

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

       5. clear-num N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. +-lowering-+.f64 78.8%

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

   12. Applied egg-rr 78.8%

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

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 48.6%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 81.0%

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

   5. Simplified 81.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + 0.5 \cdot \frac{x}{n \cdot n}\right) + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 0.05)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (+ (* x (+ (/ 1.0 n) (* 0.5 (/ x (* 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-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.05) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = ((x * ((1.0 / n) + (0.5 * (x / (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-11)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 0.05d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = ((x * ((1.0d0 / n) + (0.5d0 * (x / (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-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 0.05) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = ((x * ((1.0 / n) + (0.5 * (x / (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-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 0.05:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = ((x * ((1.0 / n) + (0.5 * (x / (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-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.05)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(1.0 / n) + Float64(0.5 * Float64(x / Float64(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-11)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 0.05)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = ((x * ((1.0 / n) + (0.5 * (x / (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-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.05], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(0.5 * N[(x / N[(n * n), $MachinePrecision]), $MachinePrecision]), $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) < -1.99999999999999988e-11

   1. Initial program 98.4%

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

      1. 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. log1p-define N/A

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

      7. log1p-lowering-log1p.f64 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 96.0%

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

   7. Simplified 96.0%

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

      1. div-inv N/A

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

      2. exp-to-pow N/A

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

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

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

      4. /-lowering-/.f64 96.0%

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

   9. Applied egg-rr 96.0%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 0.050000000000000003

   1. Initial program 29.4%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 78.8%

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

   10. Simplified 78.8%

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

       1. associate-*l/ N/A

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

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

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

       3. mul-1-neg N/A

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

       4. neg-log N/A

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

       5. clear-num N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. +-lowering-+.f64 78.8%

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

   12. Applied egg-rr 78.8%

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

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 48.6%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 81.0%

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

   5. Simplified 81.0%

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

   6. Taylor expanded in n around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 81.0%

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

   8. Simplified 81.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.05:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + 0.5 \cdot \frac{x}{n \cdot n}\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e+18)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e+178)
         (- (+ (/ x n) 1.0) t_0)
         (*
          (/ 1.0 n)
          (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e+18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	}
	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-11)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 1d+18) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+178) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = (1.0d0 / n) * ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x)
    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-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e+18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e+18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+178:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e+18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+178)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(1.0 / n) * Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x));
	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-11)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 1e+18)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+178)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	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-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+178], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999988e-11

   1. Initial program 98.4%

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

      1. 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. log1p-define N/A

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

      7. log1p-lowering-log1p.f64 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 96.0%

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

   7. Simplified 96.0%

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

      1. div-inv N/A

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

      2. exp-to-pow N/A

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

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

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

      4. /-lowering-/.f64 96.0%

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

   9. Applied egg-rr 96.0%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 1e18

   1. Initial program 29.9%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 78.3%

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

   10. Simplified 78.3%

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

       1. associate-*l/ N/A

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

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

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

       3. mul-1-neg N/A

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

       4. neg-log N/A

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

       5. clear-num N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. +-lowering-+.f64 78.3%

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

   12. Applied egg-rr 78.3%

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

   if 1e18 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e178

   1. Initial program 77.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   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 79.5%

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

   5. Simplified 79.5%

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

   if 2.0000000000000001e178 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 7.2%

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

   10. Simplified 7.2%

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

   11. Taylor expanded in x around inf

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

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

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

   13. Simplified 100.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e+18)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+163)
         (- 1.0 t_0)
         (*
          (/ 1.0 n)
          (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e+18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	}
	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-11)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 1d+18) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+163) then
        tmp = 1.0d0 - t_0
    else
        tmp = (1.0d0 / n) * ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x)
    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-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e+18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e+18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+163:
		tmp = 1.0 - t_0
	else:
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e+18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+163)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(1.0 / n) * Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x));
	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-11)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 1e+18)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+163)
		tmp = 1.0 - t_0;
	else
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	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-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+163], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999988e-11

   1. Initial program 98.4%

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

      1. 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. log1p-define N/A

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

      7. log1p-lowering-log1p.f64 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 96.0%

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

   7. Simplified 96.0%

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

      1. div-inv N/A

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

      2. exp-to-pow N/A

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

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

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

      4. /-lowering-/.f64 96.0%

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

   9. Applied egg-rr 96.0%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 1e18

   1. Initial program 29.9%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 78.3%

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

   10. Simplified 78.3%

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

       1. associate-*l/ N/A

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

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

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

       3. mul-1-neg N/A

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

       4. neg-log N/A

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

       5. clear-num N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. +-lowering-+.f64 78.3%

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

   12. Applied egg-rr 78.3%

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

   if 1e18 < (/.f64 #s(literal 1 binary64) n) < 5e163

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 80.4%

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

   if 5e163 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 8.5%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 6.9%

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

   10. Simplified 6.9%

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

   11. Taylor expanded in x around inf

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

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

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

   13. Simplified 94.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-11)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 1e+18)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+163)
         (- 1.0 t_0)
         (*
          (/ 1.0 n)
          (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-11) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e+18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	}
	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-11)) then
        tmp = (t_0 / x) / n
    else if ((1.0d0 / n) <= 1d+18) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+163) then
        tmp = 1.0d0 - t_0
    else
        tmp = (1.0d0 / n) * ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x)
    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-11) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e+18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-11:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 1e+18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+163:
		tmp = 1.0 - t_0
	else:
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-11)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e+18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+163)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(1.0 / n) * Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x));
	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-11)
		tmp = (t_0 / x) / n;
	elseif ((1.0 / n) <= 1e+18)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+163)
		tmp = 1.0 - t_0;
	else
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	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-11], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+163], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999988e-11

   1. Initial program 98.4%

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

      1. 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. log1p-define N/A

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

      7. log1p-lowering-log1p.f64 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 96.0%

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

   7. Simplified 96.0%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. div-inv N/A

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

      5. exp-to-pow N/A

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

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

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

      7. /-lowering-/.f64 96.0%

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

   9. Applied egg-rr 96.0%

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

   if -1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n) < 1e18

   1. Initial program 29.9%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 78.3%

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

   10. Simplified 78.3%

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

       1. associate-*l/ N/A

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

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

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

       3. mul-1-neg N/A

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

       4. neg-log N/A

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

       5. clear-num N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. +-lowering-+.f64 78.3%

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

   12. Applied egg-rr 78.3%

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

   if 1e18 < (/.f64 #s(literal 1 binary64) n) < 5e163

   1. Initial program 80.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 80.4%

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

   if 5e163 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 8.5%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 6.9%

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

   10. Simplified 6.9%

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

   11. Taylor expanded in x around inf

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

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

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

   13. Simplified 94.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e-7)
     t_0
     (if (<= (/ 1.0 n) 1e+18)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+163)
         t_0
         (*
          (/ 1.0 n)
          (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-7) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = t_0;
	} else {
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	}
	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 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-2d-7)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d+18) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+163) then
        tmp = t_0
    else
        tmp = (1.0d0 / n) * ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-7) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = t_0;
	} else {
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-7:
		tmp = t_0
	elif (1.0 / n) <= 1e+18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+163:
		tmp = t_0
	else:
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-7)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e+18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+163)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / n) * Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -2e-7)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e+18)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+163)
		tmp = t_0;
	else
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-7], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+163], t$95$0, N[(N[(1.0 / n), $MachinePrecision] * N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-7 or 1e18 < (/.f64 #s(literal 1 binary64) n) < 5e163

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 64.1%

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

   if -1.9999999999999999e-7 < (/.f64 #s(literal 1 binary64) n) < 1e18

   1. Initial program 29.8%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 58.7%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 77.8%

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

   10. Simplified 77.8%

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

       1. associate-*l/ N/A

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

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

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

       3. mul-1-neg N/A

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

       4. neg-log N/A

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

       5. clear-num N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. +-lowering-+.f64 77.8%

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

   12. Applied egg-rr 77.8%

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

   if 5e163 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 8.5%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 6.9%

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

   10. Simplified 6.9%

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

   11. Taylor expanded in x around inf

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

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

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

   13. Simplified 94.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-207}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4e-207)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.0038)
     (/ (- x (log x)) n)
     (if (<= x 4.8e+231)
       (/ (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* n x))) x)) x)
       0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4e-207) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.0038) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.8e+231) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / 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 <= 4d-207) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.0038d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4.8d+231) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (n * x))) / 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 <= 4e-207) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.0038) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.8e+231) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4e-207:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.0038:
		tmp = (x - math.log(x)) / n
	elif x <= 4.8e+231:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4e-207)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.0038)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.8e+231)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(n * x))) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4e-207)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.0038)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.8e+231)
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4e-207], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0038], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.8e+231], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.9999999999999997e-207

   1. Initial program 57.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 57.3%

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

   if 3.9999999999999997e-207 < x < 0.00379999999999999999

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 35.6%

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

   5. Simplified 35.6%

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

   6. Taylor expanded in n around inf

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. log-lowering-log.f64 55.7%

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

   8. Simplified 55.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x - \log x\right)}, n\right) \]
   10. Step-by-step derivation

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

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

       2. log-lowering-log.f64 55.3%

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

   11. Simplified 55.3%

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

   if 0.00379999999999999999 < x < 4.80000000000000013e231

   1. Initial program 58.1%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 58.1%

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

   10. Simplified 58.1%

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

   11. Taylor expanded in x around -inf

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}\right)\right)}{x} \]

       3. sub-neg N/A

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

       4. mul-1-neg N/A

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

       5. distribute-neg-out N/A

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

       6. remove-double-neg N/A

          \[\leadsto \frac{\frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} + \frac{1}{n}}{x} \]

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

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

   13. Simplified 71.4%

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

   if 4.80000000000000013e231 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.1%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 95.2%

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

      1. metadata-eval 95.2%

         \[\leadsto 0 \]

   10. Applied egg-rr 95.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-207}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0038)
   (/ (- x (log x)) n)
   (if (<= x 4.4e+231)
     (/ (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* n x))) x)) x)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0038) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.4e+231) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / 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 <= 0.0038d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4.4d+231) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (n * x))) / 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 <= 0.0038) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.4e+231) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0038:
		tmp = (x - math.log(x)) / n
	elif x <= 4.4e+231:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0038)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.4e+231)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(n * x))) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0038)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.4e+231)
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0038], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.4e+231], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.00379999999999999999

   1. Initial program 41.8%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 38.0%

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

   5. Simplified 38.0%

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

   6. Taylor expanded in n around inf

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. log-lowering-log.f64 50.7%

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

   8. Simplified 50.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x - \log x\right)}, n\right) \]
   10. Step-by-step derivation

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

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

       2. log-lowering-log.f64 50.4%

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

   11. Simplified 50.4%

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

   if 0.00379999999999999999 < x < 4.39999999999999983e231

   1. Initial program 58.1%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 58.1%

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

   10. Simplified 58.1%

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

   11. Taylor expanded in x around -inf

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}\right)\right)}{x} \]

       3. sub-neg N/A

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

       4. mul-1-neg N/A

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

       5. distribute-neg-out N/A

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

       6. remove-double-neg N/A

          \[\leadsto \frac{\frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} + \frac{1}{n}}{x} \]

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

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

   13. Simplified 71.4%

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

   if 4.39999999999999983e231 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.1%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 95.2%

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

      1. metadata-eval 95.2%

         \[\leadsto 0 \]

   10. Applied egg-rr 95.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0038)
   (- 0.0 (/ (log x) n))
   (if (<= x 4.4e+231)
     (/ (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* n x))) x)) x)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0038) {
		tmp = 0.0 - (log(x) / n);
	} else if (x <= 4.4e+231) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / 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 <= 0.0038d0) then
        tmp = 0.0d0 - (log(x) / n)
    else if (x <= 4.4d+231) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (n * x))) / 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 <= 0.0038) {
		tmp = 0.0 - (Math.log(x) / n);
	} else if (x <= 4.4e+231) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0038:
		tmp = 0.0 - (math.log(x) / n)
	elif x <= 4.4e+231:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0038)
		tmp = Float64(0.0 - Float64(log(x) / n));
	elseif (x <= 4.4e+231)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(n * x))) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0038)
		tmp = 0.0 - (log(x) / n);
	elseif (x <= 4.4e+231)
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0038], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.4e+231], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.00379999999999999999

   1. Initial program 41.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 41.2%

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

   6. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

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

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. log-lowering-log.f64 49.6%

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

   8. Simplified 49.6%

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

   if 0.00379999999999999999 < x < 4.39999999999999983e231

   1. Initial program 58.1%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 58.1%

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

   10. Simplified 58.1%

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

   11. Taylor expanded in x around -inf

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}\right)\right)}{x} \]

       3. sub-neg N/A

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

       4. mul-1-neg N/A

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

       5. distribute-neg-out N/A

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

       6. remove-double-neg N/A

          \[\leadsto \frac{\frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} + \frac{1}{n}}{x} \]

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

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

   13. Simplified 71.4%

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

   if 4.39999999999999983e231 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.1%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 95.2%

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

      1. metadata-eval 95.2%

         \[\leadsto 0 \]

   10. Applied egg-rr 95.2%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0038)
   (* (log x) (/ -1.0 n))
   (if (<= x 4.4e+231)
     (/ (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* n x))) x)) x)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0038) {
		tmp = log(x) * (-1.0 / n);
	} else if (x <= 4.4e+231) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / 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 <= 0.0038d0) then
        tmp = log(x) * ((-1.0d0) / n)
    else if (x <= 4.4d+231) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (n * x))) / 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 <= 0.0038) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if (x <= 4.4e+231) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0038:
		tmp = math.log(x) * (-1.0 / n)
	elif x <= 4.4e+231:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0038)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (x <= 4.4e+231)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(n * x))) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0038)
		tmp = log(x) * (-1.0 / n);
	elseif (x <= 4.4e+231)
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0038], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.4e+231], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.00379999999999999999

   1. Initial program 41.8%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 50.8%

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

   10. Simplified 50.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, n\right), \color{blue}{\log x}\right) \]
   12. Step-by-step derivation

       1. log-lowering-log.f64 49.5%

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

   13. Simplified 49.5%

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

   if 0.00379999999999999999 < x < 4.39999999999999983e231

   1. Initial program 58.1%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 58.1%

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

   10. Simplified 58.1%

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

   11. Taylor expanded in x around -inf

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}\right)\right)}{x} \]

       3. sub-neg N/A

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

       4. mul-1-neg N/A

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

       5. distribute-neg-out N/A

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

       6. remove-double-neg N/A

          \[\leadsto \frac{\frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} + \frac{1}{n}}{x} \]

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

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

   13. Simplified 71.4%

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

   if 4.39999999999999983e231 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.1%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 95.2%

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

      1. metadata-eval 95.2%

         \[\leadsto 0 \]

   10. Applied egg-rr 95.2%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+234}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.1e+234)
   (/ (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* n x))) x)) x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.1e+234) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / 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 <= 2.1d+234) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (n * x))) / 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 <= 2.1e+234) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.1e+234:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.1e+234)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(n * x))) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.1e+234)
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (n * x))) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.1e+234], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.1e234

   1. Initial program 47.0%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 55.5%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 53.1%

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

   10. Simplified 53.1%

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

   11. Taylor expanded in x around -inf

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}\right)\right)}{x} \]

       3. sub-neg N/A

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

       4. mul-1-neg N/A

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

       5. distribute-neg-out N/A

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

       6. remove-double-neg N/A

          \[\leadsto \frac{\frac{\frac{1}{3} \cdot \frac{1}{n \cdot x} - \frac{1}{2} \cdot \frac{1}{n}}{x} + \frac{1}{n}}{x} \]

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

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

   13. Simplified 47.2%

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

   if 2.1e234 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.1%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 95.2%

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

      1. metadata-eval 95.2%

         \[\leadsto 0 \]

   10. Applied egg-rr 95.2%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+234}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{n \cdot x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 9.5e+231)
   (* (/ 1.0 n) (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x))
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 9.5e+231) {
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / 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 <= 9.5d+231) then
        tmp = (1.0d0 / n) * ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / 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 <= 9.5e+231) {
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 9.5e+231:
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 9.5e+231)
		tmp = Float64(Float64(1.0 / n) * Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 9.5e+231)
		tmp = (1.0 / n) * ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 9.5e+231], N[(N[(1.0 / n), $MachinePrecision] * N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000002e231

   1. Initial program 47.0%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 55.5%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 53.1%

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

   10. Simplified 53.1%

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

   11. Taylor expanded in x around inf

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

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

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

   13. Simplified 47.1%

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

   if 9.5000000000000002e231 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.1%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 95.2%

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

      1. metadata-eval 95.2%

         \[\leadsto 0 \]

   10. Applied egg-rr 95.2%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.0% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -9.5 \cdot 10^{-234}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -3.4e-6) (/ (/ 1.0 n) x) (if (<= n -9.5e-234) 0.0 (/ 1.0 (* n x)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -3.4e-6) {
		tmp = (1.0 / n) / x;
	} else if (n <= -9.5e-234) {
		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 (n <= (-3.4d-6)) then
        tmp = (1.0d0 / n) / x
    else if (n <= (-9.5d-234)) 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 (n <= -3.4e-6) {
		tmp = (1.0 / n) / x;
	} else if (n <= -9.5e-234) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -3.4e-6:
		tmp = (1.0 / n) / x
	elif n <= -9.5e-234:
		tmp = 0.0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -3.4e-6)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (n <= -9.5e-234)
		tmp = 0.0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -3.4e-6)
		tmp = (1.0 / n) / x;
	elseif (n <= -9.5e-234)
		tmp = 0.0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -3.4e-6], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -9.5e-234], 0.0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.40000000000000006e-6

   1. Initial program 41.4%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 68.0%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - \frac{0.041666666666666664}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right)}{n}}{n}}{0 - n}} \]

   7. Applied egg-rr 46.5%

      \[\leadsto \color{blue}{\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) - \frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}\right) \cdot \frac{-0.041666666666666664}{n}}{n}}{n}\right)} \]

   8. Taylor expanded in n around inf

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

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

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

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

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

      3. +-lowering-+.f64 65.7%

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

   10. Simplified 65.7%

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

   11. Taylor expanded in x around inf

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

       1. associate-/r* N/A

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

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

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

       3. /-lowering-/.f64 53.3%

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

   13. Simplified 53.3%

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

   if -3.40000000000000006e-6 < n < -9.4999999999999999e-234

   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

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

   5. Simplified 49.4%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 53.0%

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

      1. metadata-eval 53.0%

         \[\leadsto 0 \]

   10. Applied egg-rr 53.0%

       \[\leadsto \color{blue}{0} \]

   if -9.4999999999999999e-234 < n

   1. Initial program 42.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. log1p-define N/A

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

      7. log1p-lowering-log1p.f64 57.3%

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

   4. Applied egg-rr 57.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\log x}{n}\right)\right)\right)}}{n \cdot x} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{n \cdot x} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\frac{\log x}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\frac{\log x}{n}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log x, n\right)\right), \left(n \cdot x\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right), \left(n \cdot x\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      12. *-lowering-*.f64 38.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   7. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, n\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 41.3%

       \[\leadsto \frac{\color{blue}{1}}{x \cdot n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -9.5 \cdot 10^{-234}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.7% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;n \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4.8 \cdot 10^{-234}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))))
   (if (<= n -3.3e-6) t_0 (if (<= n -4.8e-234) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -3.3e-6) {
		tmp = t_0;
	} else if (n <= -4.8e-234) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    if (n <= (-3.3d-6)) then
        tmp = t_0
    else if (n <= (-4.8d-234)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -3.3e-6) {
		tmp = t_0;
	} else if (n <= -4.8e-234) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	tmp = 0
	if n <= -3.3e-6:
		tmp = t_0
	elif n <= -4.8e-234:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (n <= -3.3e-6)
		tmp = t_0;
	elseif (n <= -4.8e-234)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	tmp = 0.0;
	if (n <= -3.3e-6)
		tmp = t_0;
	elseif (n <= -4.8e-234)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.3e-6], t$95$0, If[LessEqual[n, -4.8e-234], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.30000000000000017e-6 or -4.7999999999999998e-234 < n

   1. Initial program 42.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 51.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log1p.f64}\left(x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   4. Applied egg-rr 51.8%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}{n \cdot x} \]

      2. log-rec N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}}{n \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}}{n \cdot x} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(-1 \cdot \frac{\log x}{n}\right)}}{n \cdot x} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\log x}{n}\right)\right)\right)}}{n \cdot x} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{n \cdot x} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\frac{\log x}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\frac{\log x}{n}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log x, n\right)\right), \left(n \cdot x\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right), \left(n \cdot x\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      12. *-lowering-*.f64 45.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   7. Simplified 45.4%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, n\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 44.9%

       \[\leadsto \frac{\color{blue}{1}}{x \cdot n} \]

   if -3.30000000000000017e-6 < n < -4.7999999999999998e-234

   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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 49.4%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 53.0%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 53.0%

         \[\leadsto 0 \]

   10. Applied egg-rr 53.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -4.8 \cdot 10^{-234}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.7% 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 52.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
4. Step-by-step derivation

5. Simplified 40.3%

   \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

6. Taylor expanded in n around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
7. Step-by-step derivation

8. Simplified 28.3%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 28.3%

      \[\leadsto 0 \]

10. Applied egg-rr 28.3%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))