2nthrt (problem 3.4.6)

Percentage Accurate: 53.9% → 86.1%

Time: 42.5s

Alternatives: 20

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: 53.9% 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: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 13500:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 13500.0)
   (/
    (+
     (/
      (+
       (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
       (/ (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0))) n))
      n)
     (- (log1p x) (log x)))
    n)
   (/ (exp (/ (log x) n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 13500.0) {
		tmp = ((((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) + ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n)) / n) + (log1p(x) - log(x))) / n;
	} else {
		tmp = exp((log(x) / n)) / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 13500.0) {
		tmp = ((((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) + ((0.16666666666666666 * (Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0))) / n)) / n) + (Math.log1p(x) - Math.log(x))) / n;
	} else {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 13500.0:
		tmp = ((((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) + ((0.16666666666666666 * (math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0))) / n)) / n) + (math.log1p(x) - math.log(x))) / n
	else:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 13500.0)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) + Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n)) / n) + Float64(log1p(x) - log(x))) / n);
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 13500.0], 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[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 13500

   1. Initial program 33.3%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 83.5%

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

   if 13500 < x

   1. Initial program 61.1%

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

      1. 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 61.1%

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

   4. Applied egg-rr 61.1%

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

   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. /-lowering-/.f64 N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 13500:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 700000000:\\ \;\;\;\;\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 700000000.0)
   (*
    (/ -1.0 n)
    (-
     (log (/ x (+ x 1.0)))
     (/
      (+
       (* 0.5 (* (log (* x (+ x 1.0))) (log (/ (+ x 1.0) x))))
       (* (- (pow (log1p x) 3.0) (pow (log x) 3.0)) (/ 0.16666666666666666 n)))
      n)))
   (/ (exp (/ (log x) n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 700000000.0) {
		tmp = (-1.0 / n) * (log((x / (x + 1.0))) - (((0.5 * (log((x * (x + 1.0))) * log(((x + 1.0) / x)))) + ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) * (0.16666666666666666 / n))) / n));
	} else {
		tmp = exp((log(x) / n)) / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 700000000.0) {
		tmp = (-1.0 / n) * (Math.log((x / (x + 1.0))) - (((0.5 * (Math.log((x * (x + 1.0))) * Math.log(((x + 1.0) / x)))) + ((Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0)) * (0.16666666666666666 / n))) / n));
	} else {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 700000000.0:
		tmp = (-1.0 / n) * (math.log((x / (x + 1.0))) - (((0.5 * (math.log((x * (x + 1.0))) * math.log(((x + 1.0) / x)))) + ((math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0)) * (0.16666666666666666 / n))) / n))
	else:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 700000000.0)
		tmp = Float64(Float64(-1.0 / n) * Float64(log(Float64(x / Float64(x + 1.0))) - Float64(Float64(Float64(0.5 * Float64(log(Float64(x * Float64(x + 1.0))) * log(Float64(Float64(x + 1.0) / x)))) + Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) * Float64(0.16666666666666666 / n))) / n)));
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 700000000.0], N[(N[(-1.0 / n), $MachinePrecision] * N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(0.5 * N[(N[Log[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7e8

   1. Initial program 33.1%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 83.3%

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

   7. Applied egg-rr 83.3%

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

   if 7e8 < x

   1. Initial program 61.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 61.7%

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

   4. Applied egg-rr 61.7%

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

   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. /-lowering-/.f64 N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 700000000:\\ \;\;\;\;\frac{-1}{n} \cdot \left(\log \left(\frac{x}{x + 1}\right) - \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;\frac{\frac{{\log x}^{2} \cdot -0.5 + \frac{{\log x}^{3} \cdot -0.16666666666666666}{n}}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.36)
   (/
    (-
     (/
      (+
       (* (pow (log x) 2.0) -0.5)
       (/ (* (pow (log x) 3.0) -0.16666666666666666) n))
      n)
     (log x))
    n)
   (/ (exp (/ (log x) n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.36) {
		tmp = ((((pow(log(x), 2.0) * -0.5) + ((pow(log(x), 3.0) * -0.16666666666666666) / n)) / n) - log(x)) / n;
	} else {
		tmp = exp((log(x) / n)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.36d0) then
        tmp = (((((log(x) ** 2.0d0) * (-0.5d0)) + (((log(x) ** 3.0d0) * (-0.16666666666666666d0)) / n)) / n) - log(x)) / n
    else
        tmp = exp((log(x) / n)) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.36) {
		tmp = ((((Math.pow(Math.log(x), 2.0) * -0.5) + ((Math.pow(Math.log(x), 3.0) * -0.16666666666666666) / n)) / n) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.36:
		tmp = ((((math.pow(math.log(x), 2.0) * -0.5) + ((math.pow(math.log(x), 3.0) * -0.16666666666666666) / n)) / n) - math.log(x)) / n
	else:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.36)
		tmp = Float64(Float64(Float64(Float64(Float64((log(x) ^ 2.0) * -0.5) + Float64(Float64((log(x) ^ 3.0) * -0.16666666666666666) / n)) / n) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.36)
		tmp = (((((log(x) ^ 2.0) * -0.5) + (((log(x) ^ 3.0) * -0.16666666666666666) / n)) / n) - log(x)) / n;
	else
		tmp = exp((log(x) / n)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.36], N[(N[(N[(N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.35999999999999999

   1. Initial program 33.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 31.9%

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

   6. Taylor expanded in n around inf

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

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

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

   8. Simplified 82.2%

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

   9. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x + -1 \cdot \frac{\frac{-1}{2} \cdot {\log x}^{2} + \frac{-1}{6} \cdot \frac{{\log x}^{3}}{n}}{n}}{n}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

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

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

   11. Simplified 82.9%

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

   if 0.35999999999999999 < x

   1. Initial program 61.1%

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

      1. 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 61.1%

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

   4. Applied egg-rr 61.1%

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

   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. /-lowering-/.f64 N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;\frac{\frac{{\log x}^{2} \cdot -0.5 + \frac{{\log x}^{3} \cdot -0.16666666666666666}{n}}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-71}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* x n))))
   (if (<= (/ 1.0 n) -4e-103)
     t_0
     (if (<= (/ 1.0 n) 1e-71)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (if (<= (/ 1.0 n) 2e-6)
         t_0
         (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (x * n);
	double tmp;
	if ((1.0 / n) <= -4e-103) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-71) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_0;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (x * n);
	double tmp;
	if ((1.0 / n) <= -4e-103) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-71) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_0;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (x * n)
	tmp = 0
	if (1.0 / n) <= -4e-103:
		tmp = t_0
	elif (1.0 / n) <= 1e-71:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 2e-6:
		tmp = t_0
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(x * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-103)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-71)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = t_0;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-103], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-71], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], t$95$0, N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999983e-103 or 9.9999999999999992e-72 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 65.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 65.2%

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

   4. Applied egg-rr 65.2%

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

   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. /-lowering-/.f64 N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 84.4%

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

   7. Simplified 84.4%

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

   if -3.99999999999999983e-103 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999992e-72

   1. Initial program 27.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 84.9%

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

   7. Applied egg-rr 69.3%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 84.9%

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

   10. Simplified 84.9%

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

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. 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 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-103}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-71}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-71}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* x n))))
   (if (<= (/ 1.0 n) -4e-103)
     t_0
     (if (<= (/ 1.0 n) 1e-71)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (if (<= (/ 1.0 n) 2e-6)
         t_0
         (-
          (+ (* x (+ (/ 1.0 n) (* x (+ (/ 0.5 (* n n)) (/ -0.5 n))))) 1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (x * n);
	double tmp;
	if ((1.0 / n) <= -4e-103) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-71) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_0;
	} else {
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((log(x) / n)) / (x * n)
    if ((1.0d0 / n) <= (-4d-103)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d-71) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 2d-6) then
        tmp = t_0
    else
        tmp = ((x * ((1.0d0 / n) + (x * ((0.5d0 / (n * n)) + ((-0.5d0) / n))))) + 1.0d0) - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (x * n);
	double tmp;
	if ((1.0 / n) <= -4e-103) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-71) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_0;
	} else {
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (x * n)
	tmp = 0
	if (1.0 / n) <= -4e-103:
		tmp = t_0
	elif (1.0 / n) <= 1e-71:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 2e-6:
		tmp = t_0
	else:
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(x * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-103)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-71)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = t_0;
	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) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = exp((log(x) / n)) / (x * n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-103)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e-71)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 2e-6)
		tmp = t_0;
	else
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-103], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-71], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], t$95$0, 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] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999983e-103 or 9.9999999999999992e-72 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 65.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 65.2%

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

   4. Applied egg-rr 65.2%

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

   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. /-lowering-/.f64 N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 84.4%

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

   7. Simplified 84.4%

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

   if -3.99999999999999983e-103 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999992e-72

   1. Initial program 27.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 84.9%

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

   7. Applied egg-rr 69.3%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 84.9%

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

   10. Simplified 84.9%

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

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.4%

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

   3. Taylor expanded in 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 79.0%

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

   5. Simplified 79.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-103}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-71}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-71}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 (pow x (/ -1.0 n))) (* x n))))
   (if (<= (/ 1.0 n) -4e-103)
     t_0
     (if (<= (/ 1.0 n) 1e-71)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (if (<= (/ 1.0 n) 2e-6)
         t_0
         (-
          (+ (* x (+ (/ 1.0 n) (* x (+ (/ 0.5 (* n n)) (/ -0.5 n))))) 1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / pow(x, (-1.0 / n))) / (x * n);
	double tmp;
	if ((1.0 / n) <= -4e-103) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-71) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_0;
	} else {
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (x ** ((-1.0d0) / n))) / (x * n)
    if ((1.0d0 / n) <= (-4d-103)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d-71) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 2d-6) then
        tmp = t_0
    else
        tmp = ((x * ((1.0d0 / n) + (x * ((0.5d0 / (n * n)) + ((-0.5d0) / n))))) + 1.0d0) - (x ** (1.0d0 / n))
    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))) / (x * n);
	double tmp;
	if ((1.0 / n) <= -4e-103) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-71) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_0;
	} else {
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / math.pow(x, (-1.0 / n))) / (x * n)
	tmp = 0
	if (1.0 / n) <= -4e-103:
		tmp = t_0
	elif (1.0 / n) <= 1e-71:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 2e-6:
		tmp = t_0
	else:
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / (x ^ Float64(-1.0 / n))) / Float64(x * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-103)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-71)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = t_0;
	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) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / (x ^ (-1.0 / n))) / (x * n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-103)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e-71)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 2e-6)
		tmp = t_0;
	else
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-103], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-71], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], t$95$0, 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] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999983e-103 or 9.9999999999999992e-72 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 65.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 84.4%

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

   5. Simplified 84.4%

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

   if -3.99999999999999983e-103 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999992e-72

   1. Initial program 27.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 84.9%

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

   7. Applied egg-rr 69.3%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 84.9%

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

   10. Simplified 84.9%

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

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.4%

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

   3. Taylor expanded in 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 79.0%

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

   5. Simplified 79.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-71}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-71}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 (pow x (/ -1.0 n))) (* x n))))
   (if (<= (/ 1.0 n) -4e-103)
     t_0
     (if (<= (/ 1.0 n) 1e-71)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (if (<= (/ 1.0 n) 2e-6)
         t_0
         (if (<= (/ 1.0 n) 5e+119)
           (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
           (* x (+ (/ 1.0 n) (* x (+ (/ 0.5 (* n n)) (/ -0.5 n)))))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / pow(x, (-1.0 / n))) / (x * n);
	double tmp;
	if ((1.0 / n) <= -4e-103) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-71) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+119) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (x ** ((-1.0d0) / n))) / (x * n)
    if ((1.0d0 / n) <= (-4d-103)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d-71) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 2d-6) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d+119) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = x * ((1.0d0 / n) + (x * ((0.5d0 / (n * n)) + ((-0.5d0) / n))))
    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))) / (x * n);
	double tmp;
	if ((1.0 / n) <= -4e-103) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-71) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+119) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / math.pow(x, (-1.0 / n))) / (x * n)
	tmp = 0
	if (1.0 / n) <= -4e-103:
		tmp = t_0
	elif (1.0 / n) <= 1e-71:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 2e-6:
		tmp = t_0
	elif (1.0 / n) <= 5e+119:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / (x ^ Float64(-1.0 / n))) / Float64(x * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-103)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-71)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+119)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / (x ^ (-1.0 / n))) / (x * n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-103)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e-71)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 2e-6)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e+119)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-103], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-71], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+119], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.99999999999999983e-103 or 9.9999999999999992e-72 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 65.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 84.4%

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

   5. Simplified 84.4%

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

   if -3.99999999999999983e-103 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999992e-72

   1. Initial program 27.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 84.9%

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

   7. Applied egg-rr 69.3%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 84.9%

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

   10. Simplified 84.9%

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

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e119

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   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 71.0%

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

   5. Simplified 71.0%

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

   if 4.9999999999999999e119 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf

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

   5. Simplified 6.2%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 82.6%

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

   8. Simplified 82.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-71}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{1}{x \cdot n} + \left(\frac{-0.5}{n} + \frac{\log \left(\frac{1}{x}\right)}{n \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.6e-162)
   (- 0.0 (/ (log x) n))
   (if (<= x 7.5e-39)
     (*
      (* x x)
      (+ (/ 1.0 (* x n)) (+ (/ -0.5 n) (/ (log (/ 1.0 x)) (* n (* x x))))))
     (/ (/ 1.0 (pow x (/ -1.0 n))) (* x n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.6e-162) {
		tmp = 0.0 - (log(x) / n);
	} else if (x <= 7.5e-39) {
		tmp = (x * x) * ((1.0 / (x * n)) + ((-0.5 / n) + (log((1.0 / x)) / (n * (x * x)))));
	} else {
		tmp = (1.0 / pow(x, (-1.0 / n))) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.6d-162) then
        tmp = 0.0d0 - (log(x) / n)
    else if (x <= 7.5d-39) then
        tmp = (x * x) * ((1.0d0 / (x * n)) + (((-0.5d0) / n) + (log((1.0d0 / x)) / (n * (x * x)))))
    else
        tmp = (1.0d0 / (x ** ((-1.0d0) / n))) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.6e-162) {
		tmp = 0.0 - (Math.log(x) / n);
	} else if (x <= 7.5e-39) {
		tmp = (x * x) * ((1.0 / (x * n)) + ((-0.5 / n) + (Math.log((1.0 / x)) / (n * (x * x)))));
	} else {
		tmp = (1.0 / Math.pow(x, (-1.0 / n))) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.6e-162:
		tmp = 0.0 - (math.log(x) / n)
	elif x <= 7.5e-39:
		tmp = (x * x) * ((1.0 / (x * n)) + ((-0.5 / n) + (math.log((1.0 / x)) / (n * (x * x)))))
	else:
		tmp = (1.0 / math.pow(x, (-1.0 / n))) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.6e-162)
		tmp = Float64(0.0 - Float64(log(x) / n));
	elseif (x <= 7.5e-39)
		tmp = Float64(Float64(x * x) * Float64(Float64(1.0 / Float64(x * n)) + Float64(Float64(-0.5 / n) + Float64(log(Float64(1.0 / x)) / Float64(n * Float64(x * x))))));
	else
		tmp = Float64(Float64(1.0 / (x ^ Float64(-1.0 / n))) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.6e-162)
		tmp = 0.0 - (log(x) / n);
	elseif (x <= 7.5e-39)
		tmp = (x * x) * ((1.0 / (x * n)) + ((-0.5 / n) + (log((1.0 / x)) / (n * (x * x)))));
	else
		tmp = (1.0 / (x ^ (-1.0 / n))) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.6e-162], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e-39], N[(N[(x * x), $MachinePrecision] * N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / n), $MachinePrecision] + N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.59999999999999988e-162

   1. Initial program 34.8%

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

   3. Taylor expanded in 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 34.8%

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

   6. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. log-lowering-log.f64 63.4%

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

   8. Simplified 63.4%

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

   if 1.59999999999999988e-162 < x < 7.49999999999999971e-39

   1. Initial program 21.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   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 35.1%

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

   5. Simplified 35.1%

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

   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 62.8%

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

   8. Simplified 62.8%

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

   9. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. associate--l+ N/A

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

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

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

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

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

       7. *-commutative N/A

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

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

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

       9. sub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. associate-*r/ N/A

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

       18. metadata-eval N/A

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

       19. distribute-neg-frac N/A

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

       20. metadata-eval N/A

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

   11. Simplified 86.2%

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

   if 7.49999999999999971e-39 < x

   1. Initial program 60.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{1}{x \cdot n} + \left(\frac{-0.5}{n} + \frac{\log \left(\frac{1}{x}\right)}{n \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 1.8e-9)
   (/ (log (/ x (+ x 1.0))) (- 0.0 n))
   (if (<= (/ 1.0 n) 5e+119)
     (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
     (* x (+ (/ 1.0 n) (* x (+ (/ 0.5 (* n n)) (/ -0.5 n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 1.8e-9) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 5e+119) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 1.8d-9) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 5d+119) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = x * ((1.0d0 / n) + (x * ((0.5d0 / (n * n)) + ((-0.5d0) / n))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 1.8e-9) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 5e+119) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 1.8e-9:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 5e+119:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 1.8e-9)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 5e+119)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 1.8e-9)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 5e+119)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.8e-9], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+119], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < 1.8e-9

   1. Initial program 46.6%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 76.2%

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

   7. Applied egg-rr 60.5%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 68.5%

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

   10. Simplified 68.5%

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

   if 1.8e-9 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e119

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   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 61.0%

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

   5. Simplified 61.0%

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

   if 4.9999999999999999e119 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf

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

   5. Simplified 6.2%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 82.6%

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

   8. Simplified 82.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+119}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 2e-9)
   (/ (log (/ x (+ x 1.0))) (- 0.0 n))
   (if (<= (/ 1.0 n) 5e+119)
     (- 1.0 (pow x (/ 1.0 n)))
     (* x (+ (/ 1.0 n) (* x (+ (/ 0.5 (* n n)) (/ -0.5 n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-9) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 5e+119) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 2d-9) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 5d+119) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = x * ((1.0d0 / n) + (x * ((0.5d0 / (n * n)) + ((-0.5d0) / n))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-9) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 5e+119) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 2e-9:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 5e+119:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 2e-9)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 5e+119)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 2e-9)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 5e+119)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-9], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+119], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < 2.00000000000000012e-9

   1. Initial program 46.5%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 75.9%

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

   7. Applied egg-rr 60.2%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 68.2%

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

   10. Simplified 68.2%

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

   if 2.00000000000000012e-9 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e119

   1. Initial program 73.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 65.2%

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

   if 4.9999999999999999e119 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf

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

   5. Simplified 6.2%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 82.6%

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

   8. Simplified 82.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+119}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+119}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 2e-9)
   (/ (log (/ (+ x 1.0) x)) n)
   (if (<= (/ 1.0 n) 5e+119)
     (- 1.0 (pow x (/ 1.0 n)))
     (* x (+ (/ 1.0 n) (* x (+ (/ 0.5 (* n n)) (/ -0.5 n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-9) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+119) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 2d-9) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+119) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = x * ((1.0d0 / n) + (x * ((0.5d0 / (n * n)) + ((-0.5d0) / n))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-9) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+119) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 2e-9:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+119:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 2e-9)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+119)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 2e-9)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+119)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-9], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+119], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < 2.00000000000000012e-9

   1. Initial program 46.5%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 75.9%

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

   7. Applied egg-rr 60.2%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 68.2%

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

   10. Simplified 68.2%

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

       1. clear-num N/A

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

       2. log-rec N/A

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

       3. diff-log N/A

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

       4. frac-2neg N/A

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

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

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

       6. diff-log N/A

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

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

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

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

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

       9. +-commutative N/A

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

       10. +-lowering-+.f64 68.2%

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

   12. Applied egg-rr 68.2%

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

   if 2.00000000000000012e-9 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e119

   1. Initial program 73.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 65.2%

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

   if 4.9999999999999999e119 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf

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

   5. Simplified 6.2%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 82.6%

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

   8. Simplified 82.6%

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

Alternative 12: 59.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+219}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{3}{n} + \frac{\frac{1}{x \cdot n} + \frac{-1.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.85)
   (/ (- x (log x)) n)
   (if (<= x 2.4e+219)
     (*
      0.3333333333333333
      (/ (+ (/ 3.0 n) (/ (+ (/ 1.0 (* x n)) (/ -1.5 n)) x)) x))
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.85) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.4e+219) {
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.85d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.4d+219) then
        tmp = 0.3333333333333333d0 * (((3.0d0 / n) + (((1.0d0 / (x * n)) + ((-1.5d0) / n)) / x)) / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.85) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.4e+219) {
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.85:
		tmp = (x - math.log(x)) / n
	elif x <= 2.4e+219:
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.85)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.4e+219)
		tmp = Float64(0.3333333333333333 * Float64(Float64(Float64(3.0 / n) + Float64(Float64(Float64(1.0 / Float64(x * n)) + Float64(-1.5 / n)) / x)) / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.85)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.4e+219)
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.85], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.4e+219], N[(0.3333333333333333 * N[(N[(N[(3.0 / n), $MachinePrecision] + N[(N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-1.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.849999999999999978

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   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 35.7%

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

   5. Simplified 35.7%

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

   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 59.0%

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

   8. Simplified 59.0%

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

   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 58.9%

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

   11. Simplified 58.9%

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

   if 0.849999999999999978 < x < 2.4e219

   1. Initial program 51.3%

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

      1. sub-neg N/A

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

      2. flip3-+ N/A

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

      3. div-inv N/A

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

      4. cube-neg N/A

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

      5. sub-neg N/A

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

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

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

   4. Applied egg-rr 19.5%

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

   5. Taylor expanded in n around inf

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

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

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

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

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

      3. distribute-lft-out-- N/A

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

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

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

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

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

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), \log x\right)\right), n\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(x\right), \log x\right)\right), n\right)\right) \]

      8. log-lowering-log.f64 52.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(x\right), \mathsf{log.f64}\left(x\right)\right)\right), n\right)\right) \]

   7. Simplified 52.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{3 \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]

   8. Taylor expanded in x around inf

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

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

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

   10. Simplified 67.0%

       \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\frac{3}{n} + \frac{\frac{1}{x \cdot n} + \frac{-1.5}{n}}{x}}{x}} \]

   if 2.4e219 < x

   1. Initial program 96.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 55.8%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 96.1%

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

      1. metadata-eval 96.1%

         \[\leadsto 0 \]

   10. Applied egg-rr 96.1%

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

Alternative 13: 59.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+218}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{3}{n} + \frac{\frac{1}{x \cdot n} + \frac{-1.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.6)
   (- 0.0 (/ (log x) n))
   (if (<= x 8e+218)
     (*
      0.3333333333333333
      (/ (+ (/ 3.0 n) (/ (+ (/ 1.0 (* x n)) (/ -1.5 n)) x)) x))
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.6) {
		tmp = 0.0 - (log(x) / n);
	} else if (x <= 8e+218) {
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = 0.0d0 - (log(x) / n)
    else if (x <= 8d+218) then
        tmp = 0.3333333333333333d0 * (((3.0d0 / n) + (((1.0d0 / (x * n)) + ((-1.5d0) / n)) / x)) / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.6) {
		tmp = 0.0 - (Math.log(x) / n);
	} else if (x <= 8e+218) {
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.6:
		tmp = 0.0 - (math.log(x) / n)
	elif x <= 8e+218:
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(0.0 - Float64(log(x) / n));
	elseif (x <= 8e+218)
		tmp = Float64(0.3333333333333333 * Float64(Float64(Float64(3.0 / n) + Float64(Float64(Float64(1.0 / Float64(x * n)) + Float64(-1.5 / n)) / x)) / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.6)
		tmp = 0.0 - (log(x) / n);
	elseif (x <= 8e+218)
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.6], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e+218], N[(0.3333333333333333 * N[(N[(N[(3.0 / n), $MachinePrecision] + N[(N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-1.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.599999999999999978

   1. Initial program 33.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 31.9%

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

   6. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. log-lowering-log.f64 58.8%

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

   8. Simplified 58.8%

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

   if 0.599999999999999978 < x < 8.00000000000000066e218

   1. Initial program 51.3%

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

      1. sub-neg N/A

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

      2. flip3-+ N/A

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

      3. div-inv N/A

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

      4. cube-neg N/A

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

      5. sub-neg N/A

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

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

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

   4. Applied egg-rr 19.5%

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

   5. Taylor expanded in n around inf

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

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

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

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

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

      3. distribute-lft-out-- N/A

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

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

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

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

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

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), \log x\right)\right), n\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(x\right), \log x\right)\right), n\right)\right) \]

      8. log-lowering-log.f64 52.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(x\right), \mathsf{log.f64}\left(x\right)\right)\right), n\right)\right) \]

   7. Simplified 52.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{3 \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]

   8. Taylor expanded in x around inf

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

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

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

   10. Simplified 67.0%

       \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\frac{3}{n} + \frac{\frac{1}{x \cdot n} + \frac{-1.5}{n}}{x}}{x}} \]

   if 8.00000000000000066e218 < x

   1. Initial program 96.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 55.8%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 96.1%

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

      1. metadata-eval 96.1%

         \[\leadsto 0 \]

   10. Applied egg-rr 96.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+218}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{3}{n} + \frac{\frac{1}{x \cdot n} + \frac{-1.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.6% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{n}}{x}}{x}}{x}\\ \mathbf{elif}\;n \leq -1.36 \cdot 10^{-301}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{3}{n} + \frac{\frac{1}{x \cdot n} + \frac{-1.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -1.05e-6)
   (/ (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ (/ 0.3333333333333333 n) x)) x)) x)
   (if (<= n -1.36e-301)
     0.0
     (*
      0.3333333333333333
      (/ (+ (/ 3.0 n) (/ (+ (/ 1.0 (* x n)) (/ -1.5 n)) x)) x)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -1.05e-6) {
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / n) / x)) / x)) / x;
	} else if (n <= -1.36e-301) {
		tmp = 0.0;
	} else {
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.05d-6)) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + ((0.3333333333333333d0 / n) / x)) / x)) / x
    else if (n <= (-1.36d-301)) then
        tmp = 0.0d0
    else
        tmp = 0.3333333333333333d0 * (((3.0d0 / n) + (((1.0d0 / (x * n)) + ((-1.5d0) / n)) / x)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -1.05e-6) {
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / n) / x)) / x)) / x;
	} else if (n <= -1.36e-301) {
		tmp = 0.0;
	} else {
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -1.05e-6:
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / n) / x)) / x)) / x
	elif n <= -1.36e-301:
		tmp = 0.0
	else:
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -1.05e-6)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(Float64(0.3333333333333333 / n) / x)) / x)) / x);
	elseif (n <= -1.36e-301)
		tmp = 0.0;
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(Float64(3.0 / n) + Float64(Float64(Float64(1.0 / Float64(x * n)) + Float64(-1.5 / n)) / x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -1.05e-6)
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / n) / x)) / x)) / x;
	elseif (n <= -1.36e-301)
		tmp = 0.0;
	else
		tmp = 0.3333333333333333 * (((3.0 / n) + (((1.0 / (x * n)) + (-1.5 / n)) / x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -1.05e-6], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(0.3333333333333333 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -1.36e-301], 0.0, N[(0.3333333333333333 * N[(N[(N[(3.0 / n), $MachinePrecision] + N[(N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-1.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.0499999999999999e-6

   1. Initial program 30.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 76.9%

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

   7. Applied egg-rr 61.4%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 76.3%

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

   10. Simplified 76.3%

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

   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 46.9%

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

   if -1.0499999999999999e-6 < n < -1.36e-301

   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 44.6%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 57.8%

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

      1. metadata-eval 57.8%

         \[\leadsto 0 \]

   10. Applied egg-rr 57.8%

       \[\leadsto \color{blue}{0} \]

   if -1.36e-301 < n

   1. Initial program 24.9%

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

      1. sub-neg N/A

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

      2. flip3-+ N/A

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

      3. div-inv N/A

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

      4. cube-neg N/A

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

      5. sub-neg N/A

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

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

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

   4. Applied egg-rr 21.1%

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

   5. Taylor expanded in n around inf

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

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

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

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

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

      3. distribute-lft-out-- N/A

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

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

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

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

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

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), \log x\right)\right), n\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(x\right), \log x\right)\right), n\right)\right) \]

      8. log-lowering-log.f64 48.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(x\right), \mathsf{log.f64}\left(x\right)\right)\right), n\right)\right) \]

   7. Simplified 48.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{3 \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]

   8. Taylor expanded in x around inf

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

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

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

   10. Simplified 47.9%

       \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\frac{3}{n} + \frac{\frac{1}{x \cdot n} + \frac{-1.5}{n}}{x}}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{n}}{x}}{x}}{x}\\ \mathbf{elif}\;n \leq -1.36 \cdot 10^{-301}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{3}{n} + \frac{\frac{1}{x \cdot n} + \frac{-1.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{n}}{x}}{x}}{x}\\ \mathbf{if}\;n \leq -1.05 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.36 \cdot 10^{-301}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0
         (/
          (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ (/ 0.3333333333333333 n) x)) x))
          x)))
   (if (<= n -1.05e-6) t_0 (if (<= n -1.36e-301) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / n) / x)) / x)) / x;
	double tmp;
	if (n <= -1.05e-6) {
		tmp = t_0;
	} else if (n <= -1.36e-301) {
		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) + ((((-0.5d0) / n) + ((0.3333333333333333d0 / n) / x)) / x)) / x
    if (n <= (-1.05d-6)) then
        tmp = t_0
    else if (n <= (-1.36d-301)) 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) + (((-0.5 / n) + ((0.3333333333333333 / n) / x)) / x)) / x;
	double tmp;
	if (n <= -1.05e-6) {
		tmp = t_0;
	} else if (n <= -1.36e-301) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / n) / x)) / x)) / x
	tmp = 0
	if n <= -1.05e-6:
		tmp = t_0
	elif n <= -1.36e-301:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(Float64(0.3333333333333333 / n) / x)) / x)) / x)
	tmp = 0.0
	if (n <= -1.05e-6)
		tmp = t_0;
	elseif (n <= -1.36e-301)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / n) / x)) / x)) / x;
	tmp = 0.0;
	if (n <= -1.05e-6)
		tmp = t_0;
	elseif (n <= -1.36e-301)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(0.3333333333333333 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[n, -1.05e-6], t$95$0, If[LessEqual[n, -1.36e-301], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.0499999999999999e-6 or -1.36e-301 < n

   1. Initial program 27.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 71.2%

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

   7. Applied egg-rr 60.5%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 60.9%

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

   10. Simplified 60.9%

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

   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 47.5%

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

   if -1.0499999999999999e-6 < n < -1.36e-301

   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 44.6%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 57.8%

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

      1. metadata-eval 57.8%

         \[\leadsto 0 \]

   10. Applied egg-rr 57.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{n}}{x}}{x}}{x}\\ \mathbf{elif}\;n \leq -1.36 \cdot 10^{-301}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{n}}{x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.6% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.22 \cdot 10^{-301}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ (+ 1.0 (/ (- (/ 0.3333333333333333 x) 0.5) x)) x) n)))
   (if (<= n -1.15e-6) t_0 (if (<= n -1.22e-301) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = ((1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / x) / n;
	double tmp;
	if (n <= -1.15e-6) {
		tmp = t_0;
	} else if (n <= -1.22e-301) {
		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 + (((0.3333333333333333d0 / x) - 0.5d0) / x)) / x) / n
    if (n <= (-1.15d-6)) then
        tmp = t_0
    else if (n <= (-1.22d-301)) 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 + (((0.3333333333333333 / x) - 0.5) / x)) / x) / n;
	double tmp;
	if (n <= -1.15e-6) {
		tmp = t_0;
	} else if (n <= -1.22e-301) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / x) / n
	tmp = 0
	if n <= -1.15e-6:
		tmp = t_0
	elif n <= -1.22e-301:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) / x) / n)
	tmp = 0.0
	if (n <= -1.15e-6)
		tmp = t_0;
	elseif (n <= -1.22e-301)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / x) / n;
	tmp = 0.0;
	if (n <= -1.15e-6)
		tmp = t_0;
	elseif (n <= -1.22e-301)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(1.0 + N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -1.15e-6], t$95$0, If[LessEqual[n, -1.22e-301], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.15e-6 or -1.2199999999999999e-301 < n

   1. Initial program 27.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 71.2%

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

   7. Applied egg-rr 60.5%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 60.9%

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

   10. Simplified 60.9%

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

   11. Taylor expanded in x around inf

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

       1. +-commutative N/A

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

       2. associate--r+ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. unpow2 N/A

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

       6. associate-/r* N/A

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

       7. metadata-eval N/A

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

       8. associate-*r/ N/A

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

       9. div-sub N/A

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

       10. sub-neg N/A

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

       11. +-commutative N/A

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

       12. neg-mul-1 N/A

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

       13. metadata-eval N/A

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

       14. distribute-lft-in N/A

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

       15. metadata-eval N/A

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

       16. sub-neg N/A

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

       17. associate-*r/ N/A

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

   13. Simplified 47.4%

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

   if -1.15e-6 < n < -1.2199999999999999e-301

   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 44.6%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 57.8%

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

      1. metadata-eval 57.8%

         \[\leadsto 0 \]

   10. Applied egg-rr 57.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{elif}\;n \leq -1.22 \cdot 10^{-301}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.8% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{3}{x}}{n}\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{-301}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -1.15e-6)
   (* 0.3333333333333333 (/ (/ 3.0 x) n))
   (if (<= n -1.3e-301) 0.0 (/ 1.0 (* x n)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -1.15e-6) {
		tmp = 0.3333333333333333 * ((3.0 / x) / n);
	} else if (n <= -1.3e-301) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.15d-6)) then
        tmp = 0.3333333333333333d0 * ((3.0d0 / x) / n)
    else if (n <= (-1.3d-301)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -1.15e-6) {
		tmp = 0.3333333333333333 * ((3.0 / x) / n);
	} else if (n <= -1.3e-301) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -1.15e-6:
		tmp = 0.3333333333333333 * ((3.0 / x) / n)
	elif n <= -1.3e-301:
		tmp = 0.0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -1.15e-6)
		tmp = Float64(0.3333333333333333 * Float64(Float64(3.0 / x) / n));
	elseif (n <= -1.3e-301)
		tmp = 0.0;
	else
		tmp = Float64(1.0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -1.15e-6)
		tmp = 0.3333333333333333 * ((3.0 / x) / n);
	elseif (n <= -1.3e-301)
		tmp = 0.0;
	else
		tmp = 1.0 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -1.15e-6], N[(0.3333333333333333 * N[(N[(3.0 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.3e-301], 0.0, N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.15e-6

   1. Initial program 30.2%

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

      1. sub-neg N/A

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

      2. flip3-+ N/A

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

      3. div-inv N/A

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

      4. cube-neg N/A

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

      5. sub-neg N/A

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

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

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

   4. Applied egg-rr 29.0%

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

   5. Taylor expanded in n around inf

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

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

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

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

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

      3. distribute-lft-out-- N/A

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

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

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

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

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

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), \log x\right)\right), n\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(x\right), \log x\right)\right), n\right)\right) \]

      8. log-lowering-log.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(3, \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(x\right), \mathsf{log.f64}\left(x\right)\right)\right), n\right)\right) \]

   7. Simplified 75.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{3 \cdot \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 46.5%

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

   10. Simplified 46.5%

       \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\frac{3}{x}}}{n} \]

   if -1.15e-6 < n < -1.2999999999999999e-301

   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 44.6%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 57.8%

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

      1. metadata-eval 57.8%

         \[\leadsto 0 \]

   10. Applied egg-rr 57.8%

       \[\leadsto \color{blue}{0} \]

   if -1.2999999999999999e-301 < n

   1. Initial program 24.9%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 66.7%

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

   7. Applied egg-rr 59.8%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 48.6%

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

   10. Simplified 48.6%

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

   11. Taylor expanded in x around inf

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 45.8%

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

   13. Simplified 45.8%

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

Alternative 18: 46.8% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -1.36 \cdot 10^{-301}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -1.15e-6)
   (/ (/ 1.0 x) n)
   (if (<= n -1.36e-301) 0.0 (/ 1.0 (* x n)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -1.15e-6) {
		tmp = (1.0 / x) / n;
	} else if (n <= -1.36e-301) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.15d-6)) then
        tmp = (1.0d0 / x) / n
    else if (n <= (-1.36d-301)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -1.15e-6) {
		tmp = (1.0 / x) / n;
	} else if (n <= -1.36e-301) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -1.15e-6:
		tmp = (1.0 / x) / n
	elif n <= -1.36e-301:
		tmp = 0.0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -1.15e-6)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (n <= -1.36e-301)
		tmp = 0.0;
	else
		tmp = Float64(1.0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -1.15e-6)
		tmp = (1.0 / x) / n;
	elseif (n <= -1.36e-301)
		tmp = 0.0;
	else
		tmp = 1.0 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -1.15e-6], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -1.36e-301], 0.0, N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.15e-6

   1. Initial program 30.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 76.9%

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

   7. Applied egg-rr 61.4%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 76.3%

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

   10. Simplified 76.3%

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

   11. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 46.5%

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

   13. Simplified 46.5%

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

   if -1.15e-6 < n < -1.36e-301

   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 44.6%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 57.8%

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

      1. metadata-eval 57.8%

         \[\leadsto 0 \]

   10. Applied egg-rr 57.8%

       \[\leadsto \color{blue}{0} \]

   if -1.36e-301 < n

   1. Initial program 24.9%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 66.7%

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

   7. Applied egg-rr 59.8%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 48.6%

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

   10. Simplified 48.6%

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

   11. Taylor expanded in x around inf

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 45.8%

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

   13. Simplified 45.8%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -1.36 \cdot 10^{-301}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot n}\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.36 \cdot 10^{-301}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x n))))
   (if (<= n -1.15e-6) t_0 (if (<= n -1.36e-301) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (x * n);
	double tmp;
	if (n <= -1.15e-6) {
		tmp = t_0;
	} else if (n <= -1.36e-301) {
		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 / (x * n)
    if (n <= (-1.15d-6)) then
        tmp = t_0
    else if (n <= (-1.36d-301)) 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 / (x * n);
	double tmp;
	if (n <= -1.15e-6) {
		tmp = t_0;
	} else if (n <= -1.36e-301) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (x * n)
	tmp = 0
	if n <= -1.15e-6:
		tmp = t_0
	elif n <= -1.36e-301:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(x * n))
	tmp = 0.0
	if (n <= -1.15e-6)
		tmp = t_0;
	elseif (n <= -1.36e-301)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (x * n);
	tmp = 0.0;
	if (n <= -1.15e-6)
		tmp = t_0;
	elseif (n <= -1.36e-301)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.15e-6], t$95$0, If[LessEqual[n, -1.36e-301], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.15e-6 or -1.36e-301 < n

   1. Initial program 27.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 71.2%

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

   7. Applied egg-rr 60.5%

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

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      9. neg-lowering-neg.f64 60.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 60.9%

       \[\leadsto \color{blue}{\frac{\log \left(\frac{x}{1 + x}\right)}{-n}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(n \cdot x\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{n}\right)\right) \]

       3. *-lowering-*.f64 46.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   13. Simplified 46.1%

       \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if -1.15e-6 < n < -1.36e-301

   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 44.6%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 57.8%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 57.8%

         \[\leadsto 0 \]

   10. Applied egg-rr 57.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 30.9% 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 45.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 30.3%

   \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

6. Taylor expanded in n around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
7. Step-by-step derivation

8. Simplified 28.2%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 28.2%

      \[\leadsto 0 \]

10. Applied egg-rr 28.2%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))