2nthrt (problem 3.4.6)

Percentage Accurate: 53.9% → 85.5%

Time: 41.5s

Alternatives: 16

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: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.0005)
   (/ (/ 1.0 (pow x (/ -1.0 n))) (* n x))
   (if (<= (/ 1.0 n) 2e-22)
     (/
      (-
       (* (/ 0.5 n) (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
       (log (/ x (+ x 1.0))))
      n)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0005) {
		tmp = (1.0 / pow(x, (-1.0 / n))) / (n * x);
	} else if ((1.0 / n) <= 2e-22) {
		tmp = (((0.5 / n) * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) - log((x / (x + 1.0)))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0005) {
		tmp = (1.0 / Math.pow(x, (-1.0 / n))) / (n * x);
	} else if ((1.0 / n) <= 2e-22) {
		tmp = (((0.5 / n) * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) - Math.log((x / (x + 1.0)))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.0005:
		tmp = (1.0 / math.pow(x, (-1.0 / n))) / (n * x)
	elif (1.0 / n) <= 2e-22:
		tmp = (((0.5 / n) * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) - math.log((x / (x + 1.0)))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.0005)
		tmp = Float64(Float64(1.0 / (x ^ Float64(-1.0 / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-22)
		tmp = Float64(Float64(Float64(Float64(0.5 / n) * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) - log(Float64(x / Float64(x + 1.0)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.0005], N[(N[(1.0 / N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-22], N[(N[(N[(N[(0.5 / n), $MachinePrecision] * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], 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) < -5.0000000000000001e-4

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. /-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. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if -5.0000000000000001e-4 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-22

   1. Initial program 31.1%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 79.2%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 79.6%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   if 2.0000000000000001e-22 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 64.3%

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

      1. 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 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.0005)
   (/ (/ 1.0 (pow x (/ -1.0 n))) (* n x))
   (if (<= (/ 1.0 n) 2e-22)
     (/ (log (/ x (+ x 1.0))) (- 0.0 n))
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0005) {
		tmp = (1.0 / pow(x, (-1.0 / n))) / (n * x);
	} else if ((1.0 / n) <= 2e-22) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0005) {
		tmp = (1.0 / Math.pow(x, (-1.0 / n))) / (n * x);
	} else if ((1.0 / n) <= 2e-22) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.0005:
		tmp = (1.0 / math.pow(x, (-1.0 / n))) / (n * x)
	elif (1.0 / n) <= 2e-22:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.0005)
		tmp = Float64(Float64(1.0 / (x ^ Float64(-1.0 / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-22)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.0005], N[(N[(1.0 / N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-22], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], 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) < -5.0000000000000001e-4

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. /-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. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if -5.0000000000000001e-4 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-22

   1. Initial program 31.1%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 79.2%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 79.6%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 79.3%

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

   10. Simplified 79.3%

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

   if 2.0000000000000001e-22 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 64.3%

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

      1. 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 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{-1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{t\_0} + 1\right) - x \cdot \frac{-1 - x \cdot \frac{0.5}{n}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ -1.0 n))))
   (if (<= (/ 1.0 n) -0.0005)
     (/ (/ 1.0 t_0) (* n x))
     (if (<= (/ 1.0 n) 2e-22)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (- (+ (/ -1.0 t_0) 1.0) (* x (/ (- -1.0 (* x (/ 0.5 n))) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (-1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.0005) {
		tmp = (1.0 / t_0) / (n * x);
	} else if ((1.0 / n) <= 2e-22) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((-1.0 / t_0) + 1.0) - (x * ((-1.0 - (x * (0.5 / n))) / 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 = x ** ((-1.0d0) / n)
    if ((1.0d0 / n) <= (-0.0005d0)) then
        tmp = (1.0d0 / t_0) / (n * x)
    else if ((1.0d0 / n) <= 2d-22) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else
        tmp = (((-1.0d0) / t_0) + 1.0d0) - (x * (((-1.0d0) - (x * (0.5d0 / n))) / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (-1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.0005) {
		tmp = (1.0 / t_0) / (n * x);
	} else if ((1.0 / n) <= 2e-22) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((-1.0 / t_0) + 1.0) - (x * ((-1.0 - (x * (0.5 / n))) / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (-1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.0005:
		tmp = (1.0 / t_0) / (n * x)
	elif (1.0 / n) <= 2e-22:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	else:
		tmp = ((-1.0 / t_0) + 1.0) - (x * ((-1.0 - (x * (0.5 / n))) / n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(-1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.0005)
		tmp = Float64(Float64(1.0 / t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-22)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	else
		tmp = Float64(Float64(Float64(-1.0 / t_0) + 1.0) - Float64(x * Float64(Float64(-1.0 - Float64(x * Float64(0.5 / n))) / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (-1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -0.0005)
		tmp = (1.0 / t_0) / (n * x);
	elseif ((1.0 / n) <= 2e-22)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	else
		tmp = ((-1.0 / t_0) + 1.0) - (x * ((-1.0 - (x * (0.5 / n))) / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.0005], N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-22], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] - N[(x * N[(N[(-1.0 - N[(x * N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000001e-4

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. /-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. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if -5.0000000000000001e-4 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-22

   1. Initial program 31.1%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 79.2%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 79.6%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 79.3%

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

   10. Simplified 79.3%

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

   if 2.0000000000000001e-22 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 64.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 65.6%

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

   6. Taylor expanded in n around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

      8. *-lowering-*.f64 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in n around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. /-lowering-/.f64 76.4%

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

   11. Simplified 76.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{{x}^{\left(\frac{-1}{n}\right)}} + 1\right) - x \cdot \frac{-1 - x \cdot \frac{0.5}{n}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{-1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{t\_0} + 1\right) + x \cdot \frac{x \cdot 0.5}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ -1.0 n))))
   (if (<= (/ 1.0 n) -0.0005)
     (/ (/ 1.0 t_0) (* n x))
     (if (<= (/ 1.0 n) 2e-22)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (+ (+ (/ -1.0 t_0) 1.0) (* x (/ (* x 0.5) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (-1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.0005) {
		tmp = (1.0 / t_0) / (n * x);
	} else if ((1.0 / n) <= 2e-22) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((-1.0 / t_0) + 1.0) + (x * ((x * 0.5) / (n * 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 = x ** ((-1.0d0) / n)
    if ((1.0d0 / n) <= (-0.0005d0)) then
        tmp = (1.0d0 / t_0) / (n * x)
    else if ((1.0d0 / n) <= 2d-22) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else
        tmp = (((-1.0d0) / t_0) + 1.0d0) + (x * ((x * 0.5d0) / (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (-1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.0005) {
		tmp = (1.0 / t_0) / (n * x);
	} else if ((1.0 / n) <= 2e-22) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = ((-1.0 / t_0) + 1.0) + (x * ((x * 0.5) / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (-1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.0005:
		tmp = (1.0 / t_0) / (n * x)
	elif (1.0 / n) <= 2e-22:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	else:
		tmp = ((-1.0 / t_0) + 1.0) + (x * ((x * 0.5) / (n * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(-1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.0005)
		tmp = Float64(Float64(1.0 / t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-22)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	else
		tmp = Float64(Float64(Float64(-1.0 / t_0) + 1.0) + Float64(x * Float64(Float64(x * 0.5) / Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (-1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -0.0005)
		tmp = (1.0 / t_0) / (n * x);
	elseif ((1.0 / n) <= 2e-22)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	else
		tmp = ((-1.0 / t_0) + 1.0) + (x * ((x * 0.5) / (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.0005], N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-22], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(x * N[(N[(x * 0.5), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000001e-4

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. /-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. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if -5.0000000000000001e-4 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-22

   1. Initial program 31.1%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 79.2%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 79.6%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 79.3%

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

   10. Simplified 79.3%

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

   if 2.0000000000000001e-22 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 64.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 65.6%

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

   6. Taylor expanded in n around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 65.5%

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

   8. Simplified 65.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{{x}^{\left(\frac{-1}{n}\right)}} + 1\right) + x \cdot \frac{x \cdot 0.5}{n \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.0005)
   (/ (/ 1.0 (pow x (/ -1.0 n))) (* n x))
   (if (<= (/ 1.0 n) 2e-22)
     (/ (log (/ x (+ x 1.0))) (- 0.0 n))
     (if (<= (/ 1.0 n) 4e+244)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (/ 1.0 (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0005) {
		tmp = (1.0 / pow(x, (-1.0 / n))) / (n * x);
	} else if ((1.0 / n) <= 2e-22) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 4e+244) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-0.0005d0)) then
        tmp = (1.0d0 / (x ** ((-1.0d0) / n))) / (n * x)
    else if ((1.0d0 / n) <= 2d-22) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 4d+244) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0005) {
		tmp = (1.0 / Math.pow(x, (-1.0 / n))) / (n * x);
	} else if ((1.0 / n) <= 2e-22) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 4e+244) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.0005:
		tmp = (1.0 / math.pow(x, (-1.0 / n))) / (n * x)
	elif (1.0 / n) <= 2e-22:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 4e+244:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.0005)
		tmp = Float64(Float64(1.0 / (x ^ Float64(-1.0 / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-22)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 4e+244)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.0005)
		tmp = (1.0 / (x ^ (-1.0 / n))) / (n * x);
	elseif ((1.0 / n) <= 2e-22)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 4e+244)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.0005], N[(N[(1.0 / N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-22], 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], 4e+244], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000001e-4

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. /-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. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

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

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if -5.0000000000000001e-4 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-22

   1. Initial program 31.1%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 79.2%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 79.6%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 79.3%

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

   10. Simplified 79.3%

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

   if 2.0000000000000001e-22 < (/.f64 #s(literal 1 binary64) n) < 4.0000000000000003e244

   1. Initial program 76.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   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 70.8%

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

   5. Simplified 70.8%

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

   if 4.0000000000000003e244 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

      1. flip3-- N/A

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

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

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

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

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

      4. pow-pow N/A

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

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

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

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

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. pow-pow N/A

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

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

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

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

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

   7. Simplified 0.0%

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

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   9. 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 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0005:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+167}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e+167)
     (- 1.0 t_0)
     (if (<= (/ 1.0 n) 2e-22)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (if (<= (/ 1.0 n) 4e+244) (- (+ (/ x n) 1.0) t_0) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+167) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 2e-22) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 4e+244) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d+167)) then
        tmp = 1.0d0 - t_0
    else if ((1.0d0 / n) <= 2d-22) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 4d+244) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+167) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 2e-22) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 4e+244) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e+167:
		tmp = 1.0 - t_0
	elif (1.0 / n) <= 2e-22:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 4e+244:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+167)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= 2e-22)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 4e+244)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e+167)
		tmp = 1.0 - t_0;
	elseif ((1.0 / n) <= 2e-22)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 4e+244)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+167], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-22], 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], 4e+244], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e167

   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 70.2%

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

   if -1e167 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-22

   1. Initial program 47.1%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 75.7%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 76.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 76.0%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 75.1%

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

   10. Simplified 75.1%

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

   if 2.0000000000000001e-22 < (/.f64 #s(literal 1 binary64) n) < 4.0000000000000003e244

   1. Initial program 76.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   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 70.8%

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

   5. Simplified 70.8%

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

   if 4.0000000000000003e244 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

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

      1. flip3-- N/A

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

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

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

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

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

      4. pow-pow N/A

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

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

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

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

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. pow-pow N/A

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

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

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

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

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

   7. Simplified 0.0%

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

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   9. 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 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+167}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -1e+167)
     t_0
     (if (<= (/ 1.0 n) 2e-22)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (if (<= (/ 1.0 n) 5e+234) t_0 (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-22) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 5e+234) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-1d+167)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-22) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 5d+234) then
        tmp = t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-22) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 5e+234) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e+167:
		tmp = t_0
	elif (1.0 / n) <= 2e-22:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 5e+234:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+167)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-22)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 5e+234)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -1e+167)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-22)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 5e+234)
		tmp = t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+167], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-22], 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+234], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e167 or 2.0000000000000001e-22 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000003e234

   1. Initial program 90.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 70.9%

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

   if -1e167 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-22

   1. Initial program 47.1%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 75.7%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 76.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 76.0%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 75.1%

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

   10. Simplified 75.1%

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

   if 5.0000000000000003e234 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.0%

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

      1. flip3-- N/A

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

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

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

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

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

      4. pow-pow N/A

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

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

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

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

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. pow-pow N/A

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

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

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

   4. Applied egg-rr 17.0%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

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

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

   7. Simplified 0.0%

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

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   9. 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 86.2%

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

   10. Simplified 86.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+167}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -1e+167)
     t_0
     (if (<= (/ 1.0 n) 2e-22)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+234) t_0 (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-22) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+234) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-1d+167)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-22) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+234) then
        tmp = t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-22) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+234) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e+167:
		tmp = t_0
	elif (1.0 / n) <= 2e-22:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+234:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+167)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-22)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+234)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -1e+167)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-22)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+234)
		tmp = t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+167], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-22], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+234], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e167 or 2.0000000000000001e-22 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000003e234

   1. Initial program 90.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 70.9%

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

   if -1e167 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-22

   1. Initial program 47.1%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 75.7%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 76.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 76.0%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 75.1%

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

   10. Simplified 75.1%

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

       1. distribute-frac-neg2 N/A

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

       2. distribute-frac-neg N/A

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

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

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

       4. neg-log N/A

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

       5. clear-num N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. +-lowering-+.f64 75.1%

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

   12. Applied egg-rr 75.1%

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

   if 5.0000000000000003e234 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.0%

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

      1. flip3-- N/A

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

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

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

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

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

      4. pow-pow N/A

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

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

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

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

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. pow-pow N/A

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

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

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

   4. Applied egg-rr 17.0%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

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

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

   7. Simplified 0.0%

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

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   9. 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 86.2%

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

   10. Simplified 86.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+167}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.26 \cdot 10^{-266}:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-223}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.26e-266)
   (- 0.0 (/ (log x) n))
   (if (<= x 6.5e-223)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.88)
       (/ (- x (log x)) n)
       (if (<= x 6.5e+45)
         (/
          (/ (+ (/ (- (/ (+ 0.3333333333333333 (/ -0.25 x)) x) 0.5) x) 1.0) x)
          n)
         0.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.26e-266) {
		tmp = 0.0 - (log(x) / n);
	} else if (x <= 6.5e-223) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 6.5e+45) {
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.26d-266) then
        tmp = 0.0d0 - (log(x) / n)
    else if (x <= 6.5d-223) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 6.5d+45) then
        tmp = ((((((0.3333333333333333d0 + ((-0.25d0) / x)) / x) - 0.5d0) / x) + 1.0d0) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.26e-266) {
		tmp = 0.0 - (Math.log(x) / n);
	} else if (x <= 6.5e-223) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 6.5e+45) {
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.26e-266:
		tmp = 0.0 - (math.log(x) / n)
	elif x <= 6.5e-223:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 6.5e+45:
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.26e-266)
		tmp = Float64(0.0 - Float64(log(x) / n));
	elseif (x <= 6.5e-223)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 6.5e+45)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.26e-266)
		tmp = 0.0 - (log(x) / n);
	elseif (x <= 6.5e-223)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 6.5e+45)
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.26e-266], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-223], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 6.5e+45], N[(N[(N[(N[(N[(N[(N[(0.3333333333333333 + N[(-0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.26000000000000008e-266

   1. Initial program 40.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 40.0%

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

   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 66.1%

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

   8. Simplified 66.1%

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

   if 1.26000000000000008e-266 < x < 6.4999999999999996e-223

   1. Initial program 68.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 6.4999999999999996e-223 < x < 0.880000000000000004

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 29.4%

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

   6. Taylor expanded in n around inf

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

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. sub-neg N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. log-lowering-log.f64 60.0%

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

   8. Simplified 60.0%

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

   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 59.7%

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

   11. Simplified 59.7%

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

   if 0.880000000000000004 < x < 6.50000000000000034e45

   1. Initial program 21.0%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 28.1%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 32.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 32.2%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in x around -inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{x}\right)}, \mathsf{neg.f64}\left(n\right)\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{x}\right)\right), \mathsf{neg.f64}\left(\color{blue}{n}\right)\right) \]

       2. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{\mathsf{neg}\left(x\right)}\right), \mathsf{neg.f64}\left(\color{blue}{n}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{-1 \cdot x}\right), \mathsf{neg.f64}\left(n\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}\right), \left(-1 \cdot x\right)\right), \mathsf{neg.f64}\left(\color{blue}{n}\right)\right) \]

   13. Simplified 85.9%

       \[\leadsto \frac{\color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{0 - x}}}{-n} \]

   if 6.50000000000000034e45 < x

   1. Initial program 78.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 42.3%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 78.3%

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

      1. metadata-eval 78.3%

         \[\leadsto 0 \]

   10. Applied egg-rr 78.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.26 \cdot 10^{-266}:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-223}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.88)
   (/ (- x (log x)) n)
   (if (<= x 1.95e+42)
     (/ (/ (+ (/ (- (/ (+ 0.3333333333333333 (/ -0.25 x)) x) 0.5) x) 1.0) x) n)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.95e+42) {
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.95d+42) then
        tmp = ((((((0.3333333333333333d0 + ((-0.25d0) / x)) / x) - 0.5d0) / x) + 1.0d0) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.95e+42) {
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 1.95e+42:
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.95e+42)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.95e+42)
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.95e+42], N[(N[(N[(N[(N[(N[(N[(0.3333333333333333 + N[(-0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.880000000000000004

   1. Initial program 42.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 29.7%

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

   6. Taylor expanded in n around inf

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

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. sub-neg N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. log-lowering-log.f64 55.8%

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

   8. Simplified 55.8%

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

   9. Taylor expanded in x around 0

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

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

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

       2. log-lowering-log.f64 55.6%

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

   11. Simplified 55.6%

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

   if 0.880000000000000004 < x < 1.94999999999999985e42

   1. Initial program 21.0%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 28.1%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 32.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 32.2%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in x around -inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{x}\right)}, \mathsf{neg.f64}\left(n\right)\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{x}\right)\right), \mathsf{neg.f64}\left(\color{blue}{n}\right)\right) \]

       2. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{\mathsf{neg}\left(x\right)}\right), \mathsf{neg.f64}\left(\color{blue}{n}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{-1 \cdot x}\right), \mathsf{neg.f64}\left(n\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}\right), \left(-1 \cdot x\right)\right), \mathsf{neg.f64}\left(\color{blue}{n}\right)\right) \]

   13. Simplified 85.9%

       \[\leadsto \frac{\color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{0 - x}}}{-n} \]

   if 1.94999999999999985e42 < x

   1. Initial program 78.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 42.3%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 78.3%

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

      1. metadata-eval 78.3%

         \[\leadsto 0 \]

   10. Applied egg-rr 78.3%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.7)
   (- 0.0 (/ (log x) n))
   (if (<= x 1.8e+44)
     (/ (/ (+ (/ (- (/ (+ 0.3333333333333333 (/ -0.25 x)) x) 0.5) x) 1.0) x) n)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = 0.0 - (log(x) / n);
	} else if (x <= 1.8e+44) {
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = 0.0d0 - (log(x) / n)
    else if (x <= 1.8d+44) then
        tmp = ((((((0.3333333333333333d0 + ((-0.25d0) / x)) / x) - 0.5d0) / x) + 1.0d0) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = 0.0 - (Math.log(x) / n);
	} else if (x <= 1.8e+44) {
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.7:
		tmp = 0.0 - (math.log(x) / n)
	elif x <= 1.8e+44:
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(0.0 - Float64(log(x) / n));
	elseif (x <= 1.8e+44)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = 0.0 - (log(x) / n);
	elseif (x <= 1.8e+44)
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.7], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e+44], N[(N[(N[(N[(N[(N[(N[(0.3333333333333333 + N[(-0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.69999999999999996

   1. Initial program 42.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 41.0%

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

   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 55.0%

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

   8. Simplified 55.0%

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

   if 0.69999999999999996 < x < 1.8e44

   1. Initial program 21.0%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 28.1%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 32.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 32.2%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 32.2%

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

   10. Simplified 32.2%

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

   11. Taylor expanded in x around -inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{x}\right)}, \mathsf{neg.f64}\left(n\right)\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{x}\right)\right), \mathsf{neg.f64}\left(\color{blue}{n}\right)\right) \]

       2. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{\mathsf{neg}\left(x\right)}\right), \mathsf{neg.f64}\left(\color{blue}{n}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}}{-1 \cdot x}\right), \mathsf{neg.f64}\left(n\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{3} - \frac{1}{4} \cdot \frac{1}{x}}{x}}{x}\right), \left(-1 \cdot x\right)\right), \mathsf{neg.f64}\left(\color{blue}{n}\right)\right) \]

   13. Simplified 85.9%

       \[\leadsto \frac{\color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{0 - x}}}{-n} \]

   if 1.8e44 < x

   1. Initial program 78.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 42.3%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 78.3%

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

      1. metadata-eval 78.3%

         \[\leadsto 0 \]

   10. Applied egg-rr 78.3%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.5e+45)
   (/ (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x)) x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.5e+45) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 6.5d+45) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6.5e+45) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6.5e+45:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.5e+45)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6.5e+45)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.5e+45], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.50000000000000034e45

   1. Initial program 40.0%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 63.9%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 64.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 64.3%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 53.8%

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

   10. Simplified 53.8%

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

   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 33.2%

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

   if 6.50000000000000034e45 < x

   1. Initial program 78.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 42.3%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 78.3%

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

      1. metadata-eval 78.3%

         \[\leadsto 0 \]

   10. Applied egg-rr 78.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.5% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8.5e+44)
   (/ (/ (+ 1.0 (/ (- (/ 0.3333333333333333 x) 0.5) x)) x) n)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8.5e+44) {
		tmp = ((1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 8.5d+44) then
        tmp = ((1.0d0 + (((0.3333333333333333d0 / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8.5e+44) {
		tmp = ((1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8.5e+44:
		tmp = ((1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8.5e+44)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 8.5e+44)
		tmp = ((1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8.5e+44], N[(N[(N[(1.0 + N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.5e44

   1. Initial program 40.0%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 63.9%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 64.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 64.3%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 53.8%

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

   10. Simplified 53.8%

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

   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. /-lowering-/.f64 N/A

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

   13. Simplified 33.1%

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

   if 8.5e44 < x

   1. Initial program 78.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 42.3%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 78.3%

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

      1. metadata-eval 78.3%

         \[\leadsto 0 \]

   10. Applied egg-rr 78.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;n \leq -7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -9 \cdot 10^{-168}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 n) x)))
   (if (<= n -7.0) t_0 (if (<= n -9e-168) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double tmp;
	if (n <= -7.0) {
		tmp = t_0;
	} else if (n <= -9e-168) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / n) / x
    if (n <= (-7.0d0)) then
        tmp = t_0
    else if (n <= (-9d-168)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double tmp;
	if (n <= -7.0) {
		tmp = t_0;
	} else if (n <= -9e-168) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / n) / x
	tmp = 0
	if n <= -7.0:
		tmp = t_0
	elif n <= -9e-168:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / n) / x)
	tmp = 0.0
	if (n <= -7.0)
		tmp = t_0;
	elseif (n <= -9e-168)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / n) / x;
	tmp = 0.0;
	if (n <= -7.0)
		tmp = t_0;
	elseif (n <= -9e-168)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[n, -7.0], t$95$0, If[LessEqual[n, -9e-168], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -7 or -9.0000000000000002e-168 < n

   1. Initial program 44.3%

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

   3. Taylor expanded in n around inf

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

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

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

   5. Simplified 70.2%

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

      1. diff-log N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1 + x}{x}\right)\right), n\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{x + 1}{x}\right)\right), n\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \log \left(\frac{1}{\frac{x}{x + 1}}\right)\right), n\right) \]

      4. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{x}{x + 1}\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{x + 1}\right)\right)\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right)\right)\right), n\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right)\right), n\right) \]

      9. +-lowering-+.f64 70.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, n\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, x\right)\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 70.5%

      \[\leadsto \frac{\frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \color{blue}{\left(-\log \left(\frac{x}{1 + x}\right)\right)}}{n} \]

   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 63.1%

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

   10. Simplified 63.1%

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

   11. Taylor expanded in x around inf

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

       1. associate-/r* N/A

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

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

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

       3. /-lowering-/.f64 47.5%

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

   13. Simplified 47.5%

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

   if -7 < n < -9.0000000000000002e-168

   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 38.2%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 64.4%

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

      1. metadata-eval 64.4%

         \[\leadsto 0 \]

   10. Applied egg-rr 64.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 45.0% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;n \leq -2.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-167}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))))
   (if (<= n -2.9) t_0 (if (<= n -1.35e-167) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -2.9) {
		tmp = t_0;
	} else if (n <= -1.35e-167) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    if (n <= (-2.9d0)) then
        tmp = t_0
    else if (n <= (-1.35d-167)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -2.9) {
		tmp = t_0;
	} else if (n <= -1.35e-167) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	tmp = 0
	if n <= -2.9:
		tmp = t_0
	elif n <= -1.35e-167:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (n <= -2.9)
		tmp = t_0;
	elseif (n <= -1.35e-167)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	tmp = 0.0;
	if (n <= -2.9)
		tmp = t_0;
	elseif (n <= -1.35e-167)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.9], t$95$0, If[LessEqual[n, -1.35e-167], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.89999999999999991 or -1.35e-167 < n

   1. Initial program 44.3%

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

      1. flip3-- N/A

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

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

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

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

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

      4. pow-pow N/A

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

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

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

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

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. pow-pow N/A

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

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

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

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

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

   4. Applied egg-rr 31.4%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

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

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

   7. Simplified 35.8%

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

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   9. 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 46.5%

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

   10. Simplified 46.5%

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

   if -2.89999999999999991 < n < -1.35e-167

   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 38.2%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 64.4%

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

      1. metadata-eval 64.4%

         \[\leadsto 0 \]

   10. Applied egg-rr 64.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.9:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-167}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 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 54.5%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 39.5%

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

6. Taylor expanded in n around inf

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

8. Simplified 33.1%

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

   1. metadata-eval 33.1%

      \[\leadsto 0 \]

10. Applied egg-rr 33.1%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024140 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))