2nthrt (problem 3.4.6)

Percentage Accurate: 52.8% → 84.4%

Time: 44.8s

Alternatives: 18

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: 52.8% 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: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-18}:\\ \;\;\;\;\frac{n \cdot x - n \cdot \log x}{n \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (- -1.0 (/ -1.0 n))) n)))
   (if (<= (/ 1.0 n) -5e-88)
     t_0
     (if (<= (/ 1.0 n) 5e-101)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 2e-33)
         (/
          (+
           (/ 0.3333333333333333 (pow x 3.0))
           (- (/ 1.0 x) (/ 0.5 (pow x 2.0))))
          n)
         (if (<= (/ 1.0 n) 1e-18)
           (/ (- (* n x) (* n (log x))) (* n n))
           (if (<= (/ 1.0 n) 200000.0)
             t_0
             (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-101) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e-33) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 1e-18) {
		tmp = ((n * x) - (n * log(x))) / (n * n);
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_0;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-101) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e-33) {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / Math.pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 1e-18) {
		tmp = ((n * x) - (n * Math.log(x))) / (n * n);
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (-1.0 - (-1.0 / n))) / n
	tmp = 0
	if (1.0 / n) <= -5e-88:
		tmp = t_0
	elif (1.0 / n) <= 5e-101:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 2e-33:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / math.pow(x, 2.0)))) / n
	elif (1.0 / n) <= 1e-18:
		tmp = ((n * x) - (n * math.log(x))) / (n * n)
	elif (1.0 / n) <= 200000.0:
		tmp = t_0
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-88)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-101)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e-33)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0)))) / n);
	elseif (Float64(1.0 / n) <= 1e-18)
		tmp = Float64(Float64(Float64(n * x) - Float64(n * log(x))) / Float64(n * n));
	elseif (Float64(1.0 / n) <= 200000.0)
		tmp = t_0;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-88], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-101], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-33], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-18], N[(N[(N[(n * x), $MachinePrecision] - N[(n * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 200000.0], t$95$0, N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5.00000000000000009e-88 or 1.0000000000000001e-18 < (/.f64 1 n) < 2e5

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf 89.4%

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

      1. log-rec 89.4%

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

      2. mul-1-neg 89.4%

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

      3. associate-*r/ 89.4%

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

      4. associate-*r* 89.4%

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

      5. metadata-eval 89.4%

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

      6. *-commutative 89.4%

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

      7. associate-/l* 89.5%

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

      8. exp-to-pow 89.5%

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

      9. *-commutative 89.5%

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

   5. Simplified 89.5%

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

      1. add0 89.5%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + 0} \]

      2. associate-/r* 90.5%

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} + 0 \]

      3. pow1 90.5%

         \[\leadsto \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n} + 0 \]

      4. pow-div 90.1%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n} + 0 \]

   7. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n} + 0} \]
   8. Step-by-step derivation

      1. add0 90.1%

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

      2. sub-neg 90.1%

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

      3. metadata-eval 90.1%

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

   9. Simplified 90.1%

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

   if -5.00000000000000009e-88 < (/.f64 1 n) < 5.0000000000000001e-101

   1. Initial program 38.2%

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

   3. Taylor expanded in n around inf 90.1%

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

      1. log1p-define 90.1%

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

   5. Simplified 90.1%

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

      1. log1p-undefine 90.1%

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

      2. diff-log 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. clear-num 90.2%

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

      2. log-rec 90.3%

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

   9. Applied egg-rr 90.3%

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

   if 5.0000000000000001e-101 < (/.f64 1 n) < 2.0000000000000001e-33

   1. Initial program 4.6%

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

   3. Taylor expanded in n around inf 39.1%

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

      1. log1p-define 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in x around inf 71.4%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   7. Step-by-step derivation

      1. associate--l+ 71.4%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. associate-*r/ 71.4%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. metadata-eval 71.4%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      4. associate-*r/ 71.4%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{n} \]

      5. metadata-eval 71.4%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{n} \]

   8. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}}{n} \]

   if 2.0000000000000001e-33 < (/.f64 1 n) < 1.0000000000000001e-18

   1. Initial program 8.2%

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

   3. Taylor expanded in n around inf 99.6%

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

      1. log1p-define 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 99.6%

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

      1. associate-*r/ 99.6%

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

      2. log-pow 99.6%

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

      3. inv-pow 99.6%

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

      4. neg-log 99.6%

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

      5. frac-2neg 99.6%

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

      6. frac-add 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 2e5 < (/.f64 1 n)

   1. Initial program 49.8%

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

   3. Taylor expanded in n around 0 49.8%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-18}:\\ \;\;\;\;\frac{n \cdot x - n \cdot \log x}{n \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \mathsf{fma}\left(t\_0, \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{{x}^{2}}, \frac{\frac{t\_0}{n}}{x}\right)\\ \mathbf{if}\;n \leq -3.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;n \leq -0.56:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \mathbf{elif}\;n \leq 8 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1
         (fma
          t_0
          (/ (+ (/ 0.5 (pow n 2.0)) (/ -0.5 n)) (pow x 2.0))
          (/ (/ t_0 n) x))))
   (if (<= n -3.2e+80)
     (/ (log (/ x (+ x 1.0))) (- n))
     (if (<= n -0.56)
       t_1
       (if (<= n 5.8e-6)
         (- (exp (/ (log1p x) n)) t_0)
         (if (<= n 8e+96) t_1 (/ 1.0 (/ n (- (log1p x) (log x))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = fma(t_0, (((0.5 / pow(n, 2.0)) + (-0.5 / n)) / pow(x, 2.0)), ((t_0 / n) / x));
	double tmp;
	if (n <= -3.2e+80) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if (n <= -0.56) {
		tmp = t_1;
	} else if (n <= 5.8e-6) {
		tmp = exp((log1p(x) / n)) - t_0;
	} else if (n <= 8e+96) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (n / (log1p(x) - log(x)));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = fma(t_0, Float64(Float64(Float64(0.5 / (n ^ 2.0)) + Float64(-0.5 / n)) / (x ^ 2.0)), Float64(Float64(t_0 / n) / x))
	tmp = 0.0
	if (n <= -3.2e+80)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (n <= -0.56)
		tmp = t_1;
	elseif (n <= 5.8e-6)
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	elseif (n <= 8e+96)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(n / Float64(log1p(x) - log(x))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.2e+80], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[n, -0.56], t$95$1, If[LessEqual[n, 5.8e-6], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[n, 8e+96], t$95$1, N[(1.0 / N[(n / N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.1999999999999999e80

   1. Initial program 32.3%

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

   3. Taylor expanded in n around inf 89.2%

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

      1. log1p-define 89.2%

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

   5. Simplified 89.2%

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

      1. log1p-undefine 89.2%

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

      2. diff-log 89.6%

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

   7. Applied egg-rr 89.6%

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

      1. clear-num 89.6%

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

      2. log-rec 89.7%

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

   9. Applied egg-rr 89.7%

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

   if -3.1999999999999999e80 < n < -0.56000000000000005 or 5.8000000000000004e-6 < n < 8.0000000000000004e96

   1. Initial program 12.6%

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

   3. Taylor expanded in x around inf 66.2%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 66.2%

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

      2. associate-/l* 66.2%

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

      3. fma-define 66.2%

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

   5. Simplified 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{{x}^{2}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\right)} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 66.3%

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

      2. times-frac 68.0%

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

   7. Applied egg-rr 68.0%

      \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{{x}^{2}}, \color{blue}{\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}}\right) \]
   8. Step-by-step derivation

      1. associate-*l/ 68.1%

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

      2. *-un-lft-identity 68.1%

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

   9. Applied egg-rr 68.1%

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

   if -0.56000000000000005 < n < 5.8000000000000004e-6

   1. Initial program 84.6%

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

   3. Taylor expanded in n around 0 84.6%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if 8.0000000000000004e96 < n

   1. Initial program 43.5%

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

   3. Taylor expanded in n around inf 89.6%

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

      1. log1p-define 89.6%

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

   5. Simplified 89.6%

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

      1. clear-num 89.6%

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

      2. inv-pow 89.6%

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

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 89.6%

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

   9. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;n \leq -0.56:\\ \;\;\;\;\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{{x}^{2}}, \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\right)\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{{x}^{2}}, \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{if}\;n \leq -3.2 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -7.6:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{\frac{-0.5}{n}}{{x}^{2}}, \frac{t\_0}{n} \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ x (+ x 1.0))) (- n))))
   (if (<= n -3.2e+80)
     t_1
     (if (<= n -7.6)
       (fma t_0 (/ (/ -0.5 n) (pow x 2.0)) (* (/ t_0 n) (/ 1.0 x)))
       (if (<= n 2.2e-6)
         (- (exp (/ (log1p x) n)) t_0)
         (if (<= n 2.5e+72) (/ (pow x (- -1.0 (/ -1.0 n))) n) t_1))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log((x / (x + 1.0))) / -n;
	double tmp;
	if (n <= -3.2e+80) {
		tmp = t_1;
	} else if (n <= -7.6) {
		tmp = fma(t_0, ((-0.5 / n) / pow(x, 2.0)), ((t_0 / n) * (1.0 / x)));
	} else if (n <= 2.2e-6) {
		tmp = exp((log1p(x) / n)) - t_0;
	} else if (n <= 2.5e+72) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n))
	tmp = 0.0
	if (n <= -3.2e+80)
		tmp = t_1;
	elseif (n <= -7.6)
		tmp = fma(t_0, Float64(Float64(-0.5 / n) / (x ^ 2.0)), Float64(Float64(t_0 / n) * Float64(1.0 / x)));
	elseif (n <= 2.2e-6)
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	elseif (n <= 2.5e+72)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[n, -3.2e+80], t$95$1, If[LessEqual[n, -7.6], N[(t$95$0 * N[(N[(-0.5 / n), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.2e-6], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[n, 2.5e+72], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.1999999999999999e80 or 2.49999999999999996e72 < n

   1. Initial program 35.7%

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

   3. Taylor expanded in n around inf 87.3%

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

      1. log1p-define 87.3%

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

   5. Simplified 87.3%

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

      1. log1p-undefine 87.3%

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

      2. diff-log 87.5%

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

   7. Applied egg-rr 87.5%

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

      1. clear-num 87.5%

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

      2. log-rec 87.6%

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

   9. Applied egg-rr 87.6%

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

   if -3.1999999999999999e80 < n < -7.5999999999999996

   1. Initial program 13.4%

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

   3. Taylor expanded in x around inf 72.9%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 72.9%

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

      2. associate-/l* 72.9%

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

      3. fma-define 72.9%

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

   5. Simplified 72.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{{x}^{2}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\right)} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 72.9%

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

      2. times-frac 75.1%

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

   7. Applied egg-rr 75.1%

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

   8. Taylor expanded in n around inf 75.1%

      \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}}, \frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right) \]
   9. Step-by-step derivation

      1. associate-/r* 75.1%

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

   10. Simplified 75.1%

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

   if -7.5999999999999996 < n < 2.2000000000000001e-6

   1. Initial program 84.6%

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

   3. Taylor expanded in n around 0 84.6%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if 2.2000000000000001e-6 < n < 2.49999999999999996e72

   1. Initial program 14.3%

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

   3. Taylor expanded in x around inf 61.5%

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

      1. log-rec 61.5%

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

      2. mul-1-neg 61.5%

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

      3. associate-*r/ 61.5%

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

      4. associate-*r* 61.5%

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

      5. metadata-eval 61.5%

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

      6. *-commutative 61.5%

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

      7. associate-/l* 61.5%

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

      8. exp-to-pow 61.7%

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

      9. *-commutative 61.7%

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

   5. Simplified 61.7%

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

      1. add0 61.7%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + 0} \]

      2. associate-/r* 63.6%

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} + 0 \]

      3. pow1 63.6%

         \[\leadsto \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n} + 0 \]

      4. pow-div 62.8%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n} + 0 \]

   7. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n} + 0} \]
   8. Step-by-step derivation

      1. add0 62.8%

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

      2. sub-neg 62.8%

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

      3. metadata-eval 62.8%

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

   9. Simplified 62.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;n \leq -7.6:\\ \;\;\;\;\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{\frac{-0.5}{n}}{{x}^{2}}, \frac{{x}^{\left(\frac{1}{n}\right)}}{n} \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (- -1.0 (/ -1.0 n))) n)))
   (if (<= (/ 1.0 n) -5e-88)
     t_0
     (if (<= (/ 1.0 n) 5e-101)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 2e-39)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 1e-18)
           (/ (log (/ (+ x 1.0) x)) n)
           (if (<= (/ 1.0 n) 200000.0)
             t_0
             (if (<= (/ 1.0 n) 2e+78)
               (- 1.0 (pow x (/ 1.0 n)))
               (log1p (expm1 (/ 1.0 (* n x))))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-101) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e-39) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 1e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+78) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = log1p(expm1((1.0 / (n * x))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-101) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e-39) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 1e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+78) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p(Math.expm1((1.0 / (n * x))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (-1.0 - (-1.0 / n))) / n
	tmp = 0
	if (1.0 / n) <= -5e-88:
		tmp = t_0
	elif (1.0 / n) <= 5e-101:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 2e-39:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 1e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 200000.0:
		tmp = t_0
	elif (1.0 / n) <= 2e+78:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p(math.expm1((1.0 / (n * x))))
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-88)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-101)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e-39)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 1e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 200000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e+78)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = log1p(expm1(Float64(1.0 / Float64(n * x))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-88], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-101], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-39], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 200000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+78], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.00000000000000009e-88 or 1.0000000000000001e-18 < (/.f64 1 n) < 2e5

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf 89.4%

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

      1. log-rec 89.4%

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

      2. mul-1-neg 89.4%

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

      3. associate-*r/ 89.4%

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

      4. associate-*r* 89.4%

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

      5. metadata-eval 89.4%

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

      6. *-commutative 89.4%

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

      7. associate-/l* 89.5%

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

      8. exp-to-pow 89.5%

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

      9. *-commutative 89.5%

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

   5. Simplified 89.5%

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

      1. add0 89.5%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + 0} \]

      2. associate-/r* 90.5%

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} + 0 \]

      3. pow1 90.5%

         \[\leadsto \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n} + 0 \]

      4. pow-div 90.1%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n} + 0 \]

   7. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n} + 0} \]
   8. Step-by-step derivation

      1. add0 90.1%

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

      2. sub-neg 90.1%

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

      3. metadata-eval 90.1%

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

   9. Simplified 90.1%

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

   if -5.00000000000000009e-88 < (/.f64 1 n) < 5.0000000000000001e-101

   1. Initial program 38.2%

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

   3. Taylor expanded in n around inf 90.1%

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

      1. log1p-define 90.1%

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

   5. Simplified 90.1%

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

      1. log1p-undefine 90.1%

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

      2. diff-log 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. clear-num 90.2%

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

      2. log-rec 90.3%

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

   9. Applied egg-rr 90.3%

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

   if 5.0000000000000001e-101 < (/.f64 1 n) < 1.99999999999999986e-39

   1. Initial program 4.7%

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

   3. Taylor expanded in n around inf 37.3%

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

      1. log1p-define 37.3%

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

   5. Simplified 37.3%

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

   6. Taylor expanded in x around inf 72.9%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   7. Step-by-step derivation

      1. associate-*r/ 72.9%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 72.9%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

   8. Simplified 72.9%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{{x}^{2}}}}{n} \]

   if 1.99999999999999986e-39 < (/.f64 1 n) < 1.0000000000000001e-18

   1. Initial program 6.9%

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

   3. Taylor expanded in n around inf 83.9%

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

      1. log1p-define 83.9%

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

   5. Simplified 83.9%

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

      1. log1p-undefine 83.9%

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

      2. diff-log 83.9%

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

   7. Applied egg-rr 83.9%

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

   if 2e5 < (/.f64 1 n) < 2.00000000000000002e78

   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 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if 2.00000000000000002e78 < (/.f64 1 n)

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf 5.9%

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

      1. log1p-define 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around inf 35.4%

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

      1. *-commutative 35.4%

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

   8. Simplified 35.4%

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

      1. log1p-expm1-u 79.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]

   10. Applied egg-rr 79.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-18}:\\ \;\;\;\;\frac{n \cdot x - n \cdot \log x}{n \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (- -1.0 (/ -1.0 n))) n)))
   (if (<= (/ 1.0 n) -5e-88)
     t_0
     (if (<= (/ 1.0 n) 5e-101)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 2e-33)
         (/
          (+
           (/ 0.3333333333333333 (pow x 3.0))
           (- (/ 1.0 x) (/ 0.5 (pow x 2.0))))
          n)
         (if (<= (/ 1.0 n) 1e-18)
           (/ (- (* n x) (* n (log x))) (* n n))
           (if (<= (/ 1.0 n) 200000.0)
             t_0
             (if (<= (/ 1.0 n) 2e+78)
               (- 1.0 (pow x (/ 1.0 n)))
               (log1p (expm1 (/ 1.0 (* n x))))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-101) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e-33) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 1e-18) {
		tmp = ((n * x) - (n * log(x))) / (n * n);
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+78) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = log1p(expm1((1.0 / (n * x))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-101) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 2e-33) {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / Math.pow(x, 2.0)))) / n;
	} else if ((1.0 / n) <= 1e-18) {
		tmp = ((n * x) - (n * Math.log(x))) / (n * n);
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+78) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p(Math.expm1((1.0 / (n * x))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (-1.0 - (-1.0 / n))) / n
	tmp = 0
	if (1.0 / n) <= -5e-88:
		tmp = t_0
	elif (1.0 / n) <= 5e-101:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 2e-33:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / math.pow(x, 2.0)))) / n
	elif (1.0 / n) <= 1e-18:
		tmp = ((n * x) - (n * math.log(x))) / (n * n)
	elif (1.0 / n) <= 200000.0:
		tmp = t_0
	elif (1.0 / n) <= 2e+78:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p(math.expm1((1.0 / (n * x))))
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-88)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-101)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e-33)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0)))) / n);
	elseif (Float64(1.0 / n) <= 1e-18)
		tmp = Float64(Float64(Float64(n * x) - Float64(n * log(x))) / Float64(n * n));
	elseif (Float64(1.0 / n) <= 200000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e+78)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = log1p(expm1(Float64(1.0 / Float64(n * x))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-88], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-101], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-33], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-18], N[(N[(N[(n * x), $MachinePrecision] - N[(n * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 200000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+78], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.00000000000000009e-88 or 1.0000000000000001e-18 < (/.f64 1 n) < 2e5

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf 89.4%

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

      1. log-rec 89.4%

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

      2. mul-1-neg 89.4%

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

      3. associate-*r/ 89.4%

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

      4. associate-*r* 89.4%

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

      5. metadata-eval 89.4%

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

      6. *-commutative 89.4%

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

      7. associate-/l* 89.5%

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

      8. exp-to-pow 89.5%

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

      9. *-commutative 89.5%

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

   5. Simplified 89.5%

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

      1. add0 89.5%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + 0} \]

      2. associate-/r* 90.5%

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} + 0 \]

      3. pow1 90.5%

         \[\leadsto \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n} + 0 \]

      4. pow-div 90.1%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n} + 0 \]

   7. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n} + 0} \]
   8. Step-by-step derivation

      1. add0 90.1%

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

      2. sub-neg 90.1%

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

      3. metadata-eval 90.1%

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

   9. Simplified 90.1%

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

   if -5.00000000000000009e-88 < (/.f64 1 n) < 5.0000000000000001e-101

   1. Initial program 38.2%

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

   3. Taylor expanded in n around inf 90.1%

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

      1. log1p-define 90.1%

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

   5. Simplified 90.1%

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

      1. log1p-undefine 90.1%

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

      2. diff-log 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. clear-num 90.2%

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

      2. log-rec 90.3%

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

   9. Applied egg-rr 90.3%

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

   if 5.0000000000000001e-101 < (/.f64 1 n) < 2.0000000000000001e-33

   1. Initial program 4.6%

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

   3. Taylor expanded in n around inf 39.1%

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

      1. log1p-define 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in x around inf 71.4%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   7. Step-by-step derivation

      1. associate--l+ 71.4%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. associate-*r/ 71.4%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. metadata-eval 71.4%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      4. associate-*r/ 71.4%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{n} \]

      5. metadata-eval 71.4%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{n} \]

   8. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}}{n} \]

   if 2.0000000000000001e-33 < (/.f64 1 n) < 1.0000000000000001e-18

   1. Initial program 8.2%

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

   3. Taylor expanded in n around inf 99.6%

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

      1. log1p-define 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 99.6%

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

      1. associate-*r/ 99.6%

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

      2. log-pow 99.6%

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

      3. inv-pow 99.6%

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

      4. neg-log 99.6%

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

      5. frac-2neg 99.6%

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

      6. frac-add 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 2e5 < (/.f64 1 n) < 2.00000000000000002e78

   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 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if 2.00000000000000002e78 < (/.f64 1 n)

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf 5.9%

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

      1. log1p-define 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around inf 35.4%

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

      1. *-commutative 35.4%

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

   8. Simplified 35.4%

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

      1. log1p-expm1-u 79.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]

   10. Applied egg-rr 79.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-18}:\\ \;\;\;\;\frac{n \cdot x - n \cdot \log x}{n \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\log t\_0}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-1 + t\_0\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (/ (pow x (- -1.0 (/ -1.0 n))) n)))
   (if (<= (/ 1.0 n) -5e-88)
     t_1
     (if (<= (/ 1.0 n) 5e-101)
       (/ (log t_0) (- n))
       (if (<= (/ 1.0 n) 2e-39)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 1e-18)
           (/ (log (/ (+ x 1.0) x)) n)
           (if (<= (/ 1.0 n) 200000.0)
             t_1
             (if (<= (/ 1.0 n) 2e+78)
               (- 1.0 (pow x (/ 1.0 n)))
               (/ (log1p (+ -1.0 t_0)) (- n))))))))))

C

double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-101) {
		tmp = log(t_0) / -n;
	} else if ((1.0 / n) <= 2e-39) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 1e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+78) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = log1p((-1.0 + t_0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-101) {
		tmp = Math.log(t_0) / -n;
	} else if ((1.0 / n) <= 2e-39) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 1e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+78) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p((-1.0 + t_0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / (x + 1.0)
	t_1 = math.pow(x, (-1.0 - (-1.0 / n))) / n
	tmp = 0
	if (1.0 / n) <= -5e-88:
		tmp = t_1
	elif (1.0 / n) <= 5e-101:
		tmp = math.log(t_0) / -n
	elif (1.0 / n) <= 2e-39:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 1e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 200000.0:
		tmp = t_1
	elif (1.0 / n) <= 2e+78:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p((-1.0 + t_0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-88)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-101)
		tmp = Float64(log(t_0) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2e-39)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 1e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 200000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+78)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log1p(Float64(-1.0 + t_0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-88], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-101], N[(N[Log[t$95$0], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-39], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 200000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+78], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(-1.0 + t$95$0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.00000000000000009e-88 or 1.0000000000000001e-18 < (/.f64 1 n) < 2e5

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf 89.4%

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

      1. log-rec 89.4%

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

      2. mul-1-neg 89.4%

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

      3. associate-*r/ 89.4%

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

      4. associate-*r* 89.4%

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

      5. metadata-eval 89.4%

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

      6. *-commutative 89.4%

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

      7. associate-/l* 89.5%

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

      8. exp-to-pow 89.5%

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

      9. *-commutative 89.5%

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

   5. Simplified 89.5%

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

      1. add0 89.5%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + 0} \]

      2. associate-/r* 90.5%

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} + 0 \]

      3. pow1 90.5%

         \[\leadsto \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n} + 0 \]

      4. pow-div 90.1%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n} + 0 \]

   7. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n} + 0} \]
   8. Step-by-step derivation

      1. add0 90.1%

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

      2. sub-neg 90.1%

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

      3. metadata-eval 90.1%

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

   9. Simplified 90.1%

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

   if -5.00000000000000009e-88 < (/.f64 1 n) < 5.0000000000000001e-101

   1. Initial program 38.2%

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

   3. Taylor expanded in n around inf 90.1%

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

      1. log1p-define 90.1%

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

   5. Simplified 90.1%

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

      1. log1p-undefine 90.1%

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

      2. diff-log 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. clear-num 90.2%

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

      2. log-rec 90.3%

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

   9. Applied egg-rr 90.3%

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

   if 5.0000000000000001e-101 < (/.f64 1 n) < 1.99999999999999986e-39

   1. Initial program 4.7%

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

   3. Taylor expanded in n around inf 37.3%

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

      1. log1p-define 37.3%

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

   5. Simplified 37.3%

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

   6. Taylor expanded in x around inf 72.9%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   7. Step-by-step derivation

      1. associate-*r/ 72.9%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 72.9%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

   8. Simplified 72.9%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{{x}^{2}}}}{n} \]

   if 1.99999999999999986e-39 < (/.f64 1 n) < 1.0000000000000001e-18

   1. Initial program 6.9%

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

   3. Taylor expanded in n around inf 83.9%

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

      1. log1p-define 83.9%

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

   5. Simplified 83.9%

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

      1. log1p-undefine 83.9%

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

      2. diff-log 83.9%

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

   7. Applied egg-rr 83.9%

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

   if 2e5 < (/.f64 1 n) < 2.00000000000000002e78

   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 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if 2.00000000000000002e78 < (/.f64 1 n)

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf 5.9%

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

      1. log1p-define 5.9%

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

   5. Simplified 5.9%

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

      1. log1p-undefine 5.9%

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

      2. diff-log 5.9%

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

   7. Applied egg-rr 5.9%

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

      1. clear-num 5.9%

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

      2. log-rec 5.9%

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

   9. Applied egg-rr 5.9%

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

       1. log1p-expm1-u 74.5%

          \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{x}{1 + x}\right)\right)\right)}}{n} \]

       2. expm1-undefine 74.5%

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

       3. add-exp-log 74.5%

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

       4. +-commutative 74.5%

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

   11. Applied egg-rr 74.5%

       \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\frac{x}{x + 1} - 1\right)}}{n} \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-1 + \frac{x}{x + 1}\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -1400:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -3.2 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -6.7 \cdot 10^{-276}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 1900000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (log (/ x (+ x 1.0))) (- n))))
   (if (<= n -2.8e+87)
     t_1
     (if (<= n -1400.0)
       (/ (/ 1.0 n) x)
       (if (<= n -3.2e-154)
         t_0
         (if (<= n -6.7e-276)
           (/ 0.0 n)
           (if (<= n 9.5e-224)
             (/ 1.0 (* n x))
             (if (<= n 1900000000.0) t_0 t_1))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = log((x / (x + 1.0))) / -n;
	double tmp;
	if (n <= -2.8e+87) {
		tmp = t_1;
	} else if (n <= -1400.0) {
		tmp = (1.0 / n) / x;
	} else if (n <= -3.2e-154) {
		tmp = t_0;
	} else if (n <= -6.7e-276) {
		tmp = 0.0 / n;
	} else if (n <= 9.5e-224) {
		tmp = 1.0 / (n * x);
	} else if (n <= 1900000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = log((x / (x + 1.0d0))) / -n
    if (n <= (-2.8d+87)) then
        tmp = t_1
    else if (n <= (-1400.0d0)) then
        tmp = (1.0d0 / n) / x
    else if (n <= (-3.2d-154)) then
        tmp = t_0
    else if (n <= (-6.7d-276)) then
        tmp = 0.0d0 / n
    else if (n <= 9.5d-224) then
        tmp = 1.0d0 / (n * x)
    else if (n <= 1900000000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = Math.log((x / (x + 1.0))) / -n;
	double tmp;
	if (n <= -2.8e+87) {
		tmp = t_1;
	} else if (n <= -1400.0) {
		tmp = (1.0 / n) / x;
	} else if (n <= -3.2e-154) {
		tmp = t_0;
	} else if (n <= -6.7e-276) {
		tmp = 0.0 / n;
	} else if (n <= 9.5e-224) {
		tmp = 1.0 / (n * x);
	} else if (n <= 1900000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = math.log((x / (x + 1.0))) / -n
	tmp = 0
	if n <= -2.8e+87:
		tmp = t_1
	elif n <= -1400.0:
		tmp = (1.0 / n) / x
	elif n <= -3.2e-154:
		tmp = t_0
	elif n <= -6.7e-276:
		tmp = 0.0 / n
	elif n <= 9.5e-224:
		tmp = 1.0 / (n * x)
	elif n <= 1900000000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n))
	tmp = 0.0
	if (n <= -2.8e+87)
		tmp = t_1;
	elseif (n <= -1400.0)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (n <= -3.2e-154)
		tmp = t_0;
	elseif (n <= -6.7e-276)
		tmp = Float64(0.0 / n);
	elseif (n <= 9.5e-224)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= 1900000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = log((x / (x + 1.0))) / -n;
	tmp = 0.0;
	if (n <= -2.8e+87)
		tmp = t_1;
	elseif (n <= -1400.0)
		tmp = (1.0 / n) / x;
	elseif (n <= -3.2e-154)
		tmp = t_0;
	elseif (n <= -6.7e-276)
		tmp = 0.0 / n;
	elseif (n <= 9.5e-224)
		tmp = 1.0 / (n * x);
	elseif (n <= 1900000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[n, -2.8e+87], t$95$1, If[LessEqual[n, -1400.0], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -3.2e-154], t$95$0, If[LessEqual[n, -6.7e-276], N[(0.0 / n), $MachinePrecision], If[LessEqual[n, 9.5e-224], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1900000000.0], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -2.80000000000000015e87 or 1.9e9 < n

   1. Initial program 32.9%

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

   3. Taylor expanded in n around inf 82.2%

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

      1. log1p-define 82.2%

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

   5. Simplified 82.2%

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

      1. log1p-undefine 82.2%

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

      2. diff-log 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. clear-num 82.5%

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

      2. log-rec 82.6%

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

   9. Applied egg-rr 82.6%

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

   if -2.80000000000000015e87 < n < -1400

   1. Initial program 12.8%

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

   3. Taylor expanded in n around inf 36.9%

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

      1. log1p-define 36.9%

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

   5. Simplified 36.9%

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

      1. log1p-undefine 36.9%

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

      2. diff-log 37.2%

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

   7. Applied egg-rr 37.2%

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

      1. clear-num 37.2%

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

      2. log-rec 37.1%

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

   9. Applied egg-rr 37.1%

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

   10. Taylor expanded in x around inf 64.2%

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

       1. associate-/r* 64.7%

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

   12. Simplified 64.7%

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

   if -1400 < n < -3.20000000000000005e-154 or 9.5000000000000003e-224 < n < 1.9e9

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0 58.2%

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

      1. *-rgt-identity 58.2%

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

      2. associate-/l* 58.2%

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

      3. exp-to-pow 58.2%

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

   5. Simplified 58.2%

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

   if -3.20000000000000005e-154 < n < -6.69999999999999983e-276

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. log1p-define 60.5%

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

   5. Simplified 60.5%

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

      1. log1p-undefine 60.5%

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

      2. diff-log 60.5%

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

   7. Applied egg-rr 60.5%

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

   8. Taylor expanded in x around inf 67.2%

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

   if -6.69999999999999983e-276 < n < 9.5000000000000003e-224

   1. Initial program 63.7%

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

   3. Taylor expanded in n around inf 32.7%

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

      1. log1p-define 32.7%

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

   5. Simplified 32.7%

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

   6. Taylor expanded in x around inf 69.6%

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

      1. *-commutative 69.6%

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

   8. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;n \leq -1400:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -3.2 \cdot 10^{-154}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq -6.7 \cdot 10^{-276}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 1900000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (- -1.0 (/ -1.0 n))) n)))
   (if (<= (/ 1.0 n) -5e-88)
     t_0
     (if (<= (/ 1.0 n) 2e-73)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 200000.0)
         t_0
         (if (<= (/ 1.0 n) 2e+233)
           (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
           (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-73) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+233) {
		tmp = (1.0 + (x / n)) - 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) :: t_0
    real(8) :: tmp
    t_0 = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    if ((1.0d0 / n) <= (-5d-88)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-73) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 200000.0d0) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d+233) then
        tmp = (1.0d0 + (x / n)) - (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 t_0 = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-73) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+233) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (-1.0 - (-1.0 / n))) / n
	tmp = 0
	if (1.0 / n) <= -5e-88:
		tmp = t_0
	elif (1.0 / n) <= 2e-73:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 200000.0:
		tmp = t_0
	elif (1.0 / n) <= 2e+233:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-88)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-73)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 200000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e+233)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x ^ (-1.0 - (-1.0 / n))) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-88)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-73)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 200000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e+233)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-88], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-73], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 200000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+233], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $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 1 n) < -5.00000000000000009e-88 or 1.99999999999999999e-73 < (/.f64 1 n) < 2e5

   1. Initial program 62.1%

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

   3. Taylor expanded in x around inf 85.3%

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

      1. log-rec 85.3%

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

      2. mul-1-neg 85.3%

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

      3. associate-*r/ 85.3%

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

      4. associate-*r* 85.3%

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

      5. metadata-eval 85.3%

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

      6. *-commutative 85.3%

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

      7. associate-/l* 85.4%

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

      8. exp-to-pow 85.4%

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

      9. *-commutative 85.4%

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

   5. Simplified 85.4%

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

      1. add0 85.4%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + 0} \]

      2. associate-/r* 86.3%

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} + 0 \]

      3. pow1 86.3%

         \[\leadsto \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n} + 0 \]

      4. pow-div 85.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n} + 0 \]

   7. Applied egg-rr 85.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n} + 0} \]
   8. Step-by-step derivation

      1. add0 85.9%

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

      2. sub-neg 85.9%

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

      3. metadata-eval 85.9%

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

   9. Simplified 85.9%

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

   if -5.00000000000000009e-88 < (/.f64 1 n) < 1.99999999999999999e-73

   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 n around inf 87.9%

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

      1. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.1%

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

   7. Applied egg-rr 88.1%

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

      1. clear-num 88.1%

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

      2. log-rec 88.1%

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

   9. Applied egg-rr 88.1%

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

   if 2e5 < (/.f64 1 n) < 1.99999999999999995e233

   1. Initial program 62.6%

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

   3. Taylor expanded in x around 0 60.6%

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

   if 1.99999999999999995e233 < (/.f64 1 n)

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf 8.7%

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

      1. log1p-define 8.7%

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

   5. Simplified 8.7%

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

   6. Taylor expanded in x around inf 85.1%

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

      1. *-commutative 85.1%

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

   8. Simplified 85.1%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\log t\_0}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-1 + t\_0\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (/ (pow x (- -1.0 (/ -1.0 n))) n)))
   (if (<= (/ 1.0 n) -5e-88)
     t_1
     (if (<= (/ 1.0 n) 2e-73)
       (/ (log t_0) (- n))
       (if (<= (/ 1.0 n) 200000.0)
         t_1
         (if (<= (/ 1.0 n) 2e+78)
           (- 1.0 (pow x (/ 1.0 n)))
           (/ (log1p (+ -1.0 t_0)) (- n))))))))

C

double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-73) {
		tmp = log(t_0) / -n;
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+78) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = log1p((-1.0 + t_0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-73) {
		tmp = Math.log(t_0) / -n;
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+78) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p((-1.0 + t_0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / (x + 1.0)
	t_1 = math.pow(x, (-1.0 - (-1.0 / n))) / n
	tmp = 0
	if (1.0 / n) <= -5e-88:
		tmp = t_1
	elif (1.0 / n) <= 2e-73:
		tmp = math.log(t_0) / -n
	elif (1.0 / n) <= 200000.0:
		tmp = t_1
	elif (1.0 / n) <= 2e+78:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p((-1.0 + t_0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-88)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-73)
		tmp = Float64(log(t_0) / Float64(-n));
	elseif (Float64(1.0 / n) <= 200000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+78)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log1p(Float64(-1.0 + t_0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-88], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-73], N[(N[Log[t$95$0], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 200000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+78], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(-1.0 + t$95$0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000009e-88 or 1.99999999999999999e-73 < (/.f64 1 n) < 2e5

   1. Initial program 62.1%

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

   3. Taylor expanded in x around inf 85.3%

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

      1. log-rec 85.3%

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

      2. mul-1-neg 85.3%

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

      3. associate-*r/ 85.3%

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

      4. associate-*r* 85.3%

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

      5. metadata-eval 85.3%

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

      6. *-commutative 85.3%

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

      7. associate-/l* 85.4%

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

      8. exp-to-pow 85.4%

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

      9. *-commutative 85.4%

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

   5. Simplified 85.4%

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

      1. add0 85.4%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + 0} \]

      2. associate-/r* 86.3%

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} + 0 \]

      3. pow1 86.3%

         \[\leadsto \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n} + 0 \]

      4. pow-div 85.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n} + 0 \]

   7. Applied egg-rr 85.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n} + 0} \]
   8. Step-by-step derivation

      1. add0 85.9%

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

      2. sub-neg 85.9%

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

      3. metadata-eval 85.9%

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

   9. Simplified 85.9%

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

   if -5.00000000000000009e-88 < (/.f64 1 n) < 1.99999999999999999e-73

   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 n around inf 87.9%

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

      1. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.1%

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

   7. Applied egg-rr 88.1%

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

      1. clear-num 88.1%

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

      2. log-rec 88.1%

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

   9. Applied egg-rr 88.1%

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

   if 2e5 < (/.f64 1 n) < 2.00000000000000002e78

   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 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if 2.00000000000000002e78 < (/.f64 1 n)

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf 5.9%

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

      1. log1p-define 5.9%

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

   5. Simplified 5.9%

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

      1. log1p-undefine 5.9%

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

      2. diff-log 5.9%

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

   7. Applied egg-rr 5.9%

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

      1. clear-num 5.9%

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

      2. log-rec 5.9%

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

   9. Applied egg-rr 5.9%

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

       1. log1p-expm1-u 74.5%

          \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{x}{1 + x}\right)\right)\right)}}{n} \]

       2. expm1-undefine 74.5%

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

       3. add-exp-log 74.5%

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

       4. +-commutative 74.5%

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

   11. Applied egg-rr 74.5%

       \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\frac{x}{x + 1} - 1\right)}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-1 + \frac{x}{x + 1}\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -1.6 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -1400:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -2.6 \cdot 10^{-146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -3.7 \cdot 10^{-276}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-218}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 1800000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= n -1.6e+87)
     t_1
     (if (<= n -1400.0)
       (/ (/ 1.0 n) x)
       (if (<= n -2.6e-146)
         t_0
         (if (<= n -3.7e-276)
           (/ 0.0 n)
           (if (<= n 2.4e-218)
             (/ 1.0 (* n x))
             (if (<= n 1800000000.0) t_0 t_1))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if (n <= -1.6e+87) {
		tmp = t_1;
	} else if (n <= -1400.0) {
		tmp = (1.0 / n) / x;
	} else if (n <= -2.6e-146) {
		tmp = t_0;
	} else if (n <= -3.7e-276) {
		tmp = 0.0 / n;
	} else if (n <= 2.4e-218) {
		tmp = 1.0 / (n * x);
	} else if (n <= 1800000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = log(((x + 1.0d0) / x)) / n
    if (n <= (-1.6d+87)) then
        tmp = t_1
    else if (n <= (-1400.0d0)) then
        tmp = (1.0d0 / n) / x
    else if (n <= (-2.6d-146)) then
        tmp = t_0
    else if (n <= (-3.7d-276)) then
        tmp = 0.0d0 / n
    else if (n <= 2.4d-218) then
        tmp = 1.0d0 / (n * x)
    else if (n <= 1800000000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if (n <= -1.6e+87) {
		tmp = t_1;
	} else if (n <= -1400.0) {
		tmp = (1.0 / n) / x;
	} else if (n <= -2.6e-146) {
		tmp = t_0;
	} else if (n <= -3.7e-276) {
		tmp = 0.0 / n;
	} else if (n <= 2.4e-218) {
		tmp = 1.0 / (n * x);
	} else if (n <= 1800000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if n <= -1.6e+87:
		tmp = t_1
	elif n <= -1400.0:
		tmp = (1.0 / n) / x
	elif n <= -2.6e-146:
		tmp = t_0
	elif n <= -3.7e-276:
		tmp = 0.0 / n
	elif n <= 2.4e-218:
		tmp = 1.0 / (n * x)
	elif n <= 1800000000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (n <= -1.6e+87)
		tmp = t_1;
	elseif (n <= -1400.0)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (n <= -2.6e-146)
		tmp = t_0;
	elseif (n <= -3.7e-276)
		tmp = Float64(0.0 / n);
	elseif (n <= 2.4e-218)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= 1800000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if (n <= -1.6e+87)
		tmp = t_1;
	elseif (n <= -1400.0)
		tmp = (1.0 / n) / x;
	elseif (n <= -2.6e-146)
		tmp = t_0;
	elseif (n <= -3.7e-276)
		tmp = 0.0 / n;
	elseif (n <= 2.4e-218)
		tmp = 1.0 / (n * x);
	elseif (n <= 1800000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -1.6e+87], t$95$1, If[LessEqual[n, -1400.0], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -2.6e-146], t$95$0, If[LessEqual[n, -3.7e-276], N[(0.0 / n), $MachinePrecision], If[LessEqual[n, 2.4e-218], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1800000000.0], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.6e87 or 1.8e9 < n

   1. Initial program 32.9%

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

   3. Taylor expanded in n around inf 82.2%

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

      1. log1p-define 82.2%

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

   5. Simplified 82.2%

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

      1. log1p-undefine 82.2%

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

      2. diff-log 82.5%

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

   7. Applied egg-rr 82.5%

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

   if -1.6e87 < n < -1400

   1. Initial program 12.8%

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

   3. Taylor expanded in n around inf 36.9%

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

      1. log1p-define 36.9%

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

   5. Simplified 36.9%

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

      1. log1p-undefine 36.9%

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

      2. diff-log 37.2%

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

   7. Applied egg-rr 37.2%

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

      1. clear-num 37.2%

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

      2. log-rec 37.1%

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

   9. Applied egg-rr 37.1%

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

   10. Taylor expanded in x around inf 64.2%

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

       1. associate-/r* 64.7%

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

   12. Simplified 64.7%

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

   if -1400 < n < -2.59999999999999987e-146 or 2.4000000000000001e-218 < n < 1.8e9

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0 58.2%

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

      1. *-rgt-identity 58.2%

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

      2. associate-/l* 58.2%

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

      3. exp-to-pow 58.2%

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

   5. Simplified 58.2%

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

   if -2.59999999999999987e-146 < n < -3.7e-276

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. log1p-define 60.5%

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

   5. Simplified 60.5%

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

      1. log1p-undefine 60.5%

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

      2. diff-log 60.5%

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

   7. Applied egg-rr 60.5%

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

   8. Taylor expanded in x around inf 67.2%

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

   if -3.7e-276 < n < 2.4000000000000001e-218

   1. Initial program 63.7%

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

   3. Taylor expanded in n around inf 32.7%

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

      1. log1p-define 32.7%

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

   5. Simplified 32.7%

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

   6. Taylor expanded in x around inf 69.6%

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

      1. *-commutative 69.6%

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

   8. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -1400:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -2.6 \cdot 10^{-146}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq -3.7 \cdot 10^{-276}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-218}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 1800000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+211}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (- -1.0 (/ -1.0 n))) n)))
   (if (<= (/ 1.0 n) -5e-88)
     t_0
     (if (<= (/ 1.0 n) 2e-73)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 200000.0)
         t_0
         (if (<= (/ 1.0 n) 5e+211)
           (- 1.0 (pow x (/ 1.0 n)))
           (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-73) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+211) {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    if ((1.0d0 / n) <= (-5d-88)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-73) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 200000.0d0) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d+211) then
        tmp = 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 t_0 = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-73) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+211) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (-1.0 - (-1.0 / n))) / n
	tmp = 0
	if (1.0 / n) <= -5e-88:
		tmp = t_0
	elif (1.0 / n) <= 2e-73:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 200000.0:
		tmp = t_0
	elif (1.0 / n) <= 5e+211:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-88)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-73)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 200000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+211)
		tmp = Float64(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)
	t_0 = (x ^ (-1.0 - (-1.0 / n))) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-88)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-73)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 200000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e+211)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-88], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-73], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 200000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+211], N[(1.0 - 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 1 n) < -5.00000000000000009e-88 or 1.99999999999999999e-73 < (/.f64 1 n) < 2e5

   1. Initial program 62.1%

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

   3. Taylor expanded in x around inf 85.3%

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

      1. log-rec 85.3%

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

      2. mul-1-neg 85.3%

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

      3. associate-*r/ 85.3%

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

      4. associate-*r* 85.3%

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

      5. metadata-eval 85.3%

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

      6. *-commutative 85.3%

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

      7. associate-/l* 85.4%

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

      8. exp-to-pow 85.4%

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

      9. *-commutative 85.4%

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

   5. Simplified 85.4%

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

      1. add0 85.4%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + 0} \]

      2. associate-/r* 86.3%

         \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} + 0 \]

      3. pow1 86.3%

         \[\leadsto \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n} + 0 \]

      4. pow-div 85.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n} + 0 \]

   7. Applied egg-rr 85.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n} + 0} \]
   8. Step-by-step derivation

      1. add0 85.9%

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

      2. sub-neg 85.9%

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

      3. metadata-eval 85.9%

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

   9. Simplified 85.9%

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

   if -5.00000000000000009e-88 < (/.f64 1 n) < 1.99999999999999999e-73

   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 n around inf 87.9%

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

      1. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.1%

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

   7. Applied egg-rr 88.1%

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

      1. clear-num 88.1%

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

      2. log-rec 88.1%

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

   9. Applied egg-rr 88.1%

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

   if 2e5 < (/.f64 1 n) < 4.9999999999999995e211

   1. Initial program 64.2%

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

   3. Taylor expanded in x around 0 59.4%

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

      1. *-rgt-identity 59.4%

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

      2. associate-/l* 59.4%

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

      3. exp-to-pow 59.4%

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

   5. Simplified 59.4%

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

   if 4.9999999999999995e211 < (/.f64 1 n)

   1. Initial program 13.8%

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

   3. Taylor expanded in n around inf 7.8%

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

      1. log1p-define 7.8%

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

   5. Simplified 7.8%

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

   6. Taylor expanded in x around inf 76.7%

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

      1. *-commutative 76.7%

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

   8. Simplified 76.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 200000:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+211}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-230}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+182}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.5e-247)
   (/ (log x) (- n))
   (if (<= x 2.5e-230)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 2.05e-7)
       (/ (- x (log x)) n)
       (if (<= x 9e+182) (/ (/ 1.0 n) x) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.5e-247) {
		tmp = log(x) / -n;
	} else if (x <= 2.5e-230) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 2.05e-7) {
		tmp = (x - log(x)) / n;
	} else if (x <= 9e+182) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.5d-247) then
        tmp = log(x) / -n
    else if (x <= 2.5d-230) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 2.05d-7) then
        tmp = (x - log(x)) / n
    else if (x <= 9d+182) then
        tmp = (1.0d0 / n) / x
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.5e-247) {
		tmp = Math.log(x) / -n;
	} else if (x <= 2.5e-230) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 2.05e-7) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 9e+182) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.5e-247:
		tmp = math.log(x) / -n
	elif x <= 2.5e-230:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 2.05e-7:
		tmp = (x - math.log(x)) / n
	elif x <= 9e+182:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.5e-247)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 2.5e-230)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 2.05e-7)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 9e+182)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.5e-247)
		tmp = log(x) / -n;
	elseif (x <= 2.5e-230)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 2.05e-7)
		tmp = (x - log(x)) / n;
	elseif (x <= 9e+182)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.5e-247], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 2.5e-230], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05e-7], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 9e+182], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.49999999999999989e-247

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf 81.4%

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

      1. log1p-define 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in x around 0 81.4%

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

      1. neg-mul-1 81.4%

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

   8. Simplified 81.4%

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

   if 2.49999999999999989e-247 < x < 2.50000000000000017e-230

   1. Initial program 84.8%

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

   3. Taylor expanded in x around 0 84.8%

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

      1. *-rgt-identity 84.8%

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

      2. associate-/l* 84.8%

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

      3. exp-to-pow 84.8%

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

   5. Simplified 84.8%

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

   if 2.50000000000000017e-230 < x < 2.05e-7

   1. Initial program 36.9%

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

   3. Taylor expanded in n around inf 54.9%

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

      1. log1p-define 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in x around 0 54.9%

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

      1. neg-mul-1 54.9%

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

      2. sub-neg 54.9%

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

   8. Simplified 54.9%

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

   if 2.05e-7 < x < 9.00000000000000058e182

   1. Initial program 48.8%

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

   3. Taylor expanded in n around inf 46.5%

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

      1. log1p-define 46.5%

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

   5. Simplified 46.5%

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

      1. log1p-undefine 46.5%

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

      2. diff-log 47.0%

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

   7. Applied egg-rr 47.0%

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

      1. clear-num 47.0%

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

      2. log-rec 47.2%

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

   9. Applied egg-rr 47.2%

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

   10. Taylor expanded in x around inf 63.4%

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

       1. associate-/r* 65.3%

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

   12. Simplified 65.3%

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

   if 9.00000000000000058e182 < x

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 83.9%

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

      1. log1p-define 83.9%

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

   5. Simplified 83.9%

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

      1. log1p-undefine 83.9%

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

      2. diff-log 83.9%

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

   7. Applied egg-rr 83.9%

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

   8. Taylor expanded in x around inf 83.9%

      \[\leadsto \frac{\log \color{blue}{1}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-230}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+182}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.05e-7)
   (/ (- x (log x)) n)
   (if (<= x 4.1e+185) (/ (/ 1.0 n) x) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.05e-7) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.1e+185) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.05d-7) then
        tmp = (x - log(x)) / n
    else if (x <= 4.1d+185) then
        tmp = (1.0d0 / n) / x
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.05e-7) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.1e+185) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.05e-7:
		tmp = (x - math.log(x)) / n
	elif x <= 4.1e+185:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.05e-7)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.1e+185)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.05e-7)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.1e+185)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.05e-7], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.1e+185], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.05e-7

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 59.0%

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

      1. log1p-define 59.0%

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

   5. Simplified 59.0%

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

   6. Taylor expanded in x around 0 59.0%

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

      1. neg-mul-1 59.0%

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

      2. sub-neg 59.0%

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

   8. Simplified 59.0%

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

   if 2.05e-7 < x < 4.1e185

   1. Initial program 48.8%

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

   3. Taylor expanded in n around inf 46.5%

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

      1. log1p-define 46.5%

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

   5. Simplified 46.5%

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

      1. log1p-undefine 46.5%

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

      2. diff-log 47.0%

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

   7. Applied egg-rr 47.0%

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

      1. clear-num 47.0%

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

      2. log-rec 47.2%

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

   9. Applied egg-rr 47.2%

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

   10. Taylor expanded in x around inf 63.4%

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

       1. associate-/r* 65.3%

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

   12. Simplified 65.3%

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

   if 4.1e185 < x

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 83.9%

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

      1. log1p-define 83.9%

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

   5. Simplified 83.9%

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

      1. log1p-undefine 83.9%

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

      2. diff-log 83.9%

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

   7. Applied egg-rr 83.9%

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

   8. Taylor expanded in x around inf 83.9%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.05e-7)
   (/ (log x) (- n))
   (if (<= x 7.8e+181) (/ (/ 1.0 n) x) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.05e-7) {
		tmp = log(x) / -n;
	} else if (x <= 7.8e+181) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.05d-7) then
        tmp = log(x) / -n
    else if (x <= 7.8d+181) then
        tmp = (1.0d0 / n) / x
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.05e-7) {
		tmp = Math.log(x) / -n;
	} else if (x <= 7.8e+181) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.05e-7:
		tmp = math.log(x) / -n
	elif x <= 7.8e+181:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.05e-7)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 7.8e+181)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.05e-7)
		tmp = log(x) / -n;
	elseif (x <= 7.8e+181)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.05e-7], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 7.8e+181], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.05e-7

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 59.0%

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

      1. log1p-define 59.0%

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

   5. Simplified 59.0%

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

   6. Taylor expanded in x around 0 58.4%

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

      1. neg-mul-1 58.4%

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

   8. Simplified 58.4%

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

   if 2.05e-7 < x < 7.8e181

   1. Initial program 48.8%

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

   3. Taylor expanded in n around inf 46.5%

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

      1. log1p-define 46.5%

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

   5. Simplified 46.5%

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

      1. log1p-undefine 46.5%

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

      2. diff-log 47.0%

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

   7. Applied egg-rr 47.0%

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

      1. clear-num 47.0%

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

      2. log-rec 47.2%

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

   9. Applied egg-rr 47.2%

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

   10. Taylor expanded in x around inf 63.4%

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

       1. associate-/r* 65.3%

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

   12. Simplified 65.3%

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

   if 7.8e181 < x

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 83.9%

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

      1. log1p-define 83.9%

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

   5. Simplified 83.9%

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

      1. log1p-undefine 83.9%

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

      2. diff-log 83.9%

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

   7. Applied egg-rr 83.9%

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

   8. Taylor expanded in x around inf 83.9%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.6% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e+14) (/ 0.0 n) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+14) {
		tmp = 0.0 / 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) <= (-2d+14)) then
        tmp = 0.0d0 / 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) <= -2e+14) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e+14:
		tmp = 0.0 / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e+14)
		tmp = 0.0 / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+14], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -2e14

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 49.6%

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

      1. log1p-define 49.6%

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

   5. Simplified 49.6%

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

      1. log1p-undefine 49.6%

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

      2. diff-log 49.6%

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

   7. Applied egg-rr 49.6%

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

   8. Taylor expanded in x around inf 50.8%

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

   if -2e14 < (/.f64 1 n)

   1. Initial program 32.7%

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

   3. Taylor expanded in n around inf 63.0%

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

      1. log1p-define 63.0%

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

   5. Simplified 63.0%

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

      1. log1p-undefine 63.0%

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

      2. diff-log 63.2%

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

   7. Applied egg-rr 63.2%

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

      1. clear-num 63.2%

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

      2. log-rec 63.3%

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

   9. Applied egg-rr 63.3%

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

   10. Taylor expanded in x around inf 46.0%

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

       1. associate-/r* 46.9%

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

   12. Simplified 46.9%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.4% accurate, 42.2× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

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

Fortran

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

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in n around inf 59.8%

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

   1. log1p-define 59.8%

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

5. Simplified 59.8%

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

6. Taylor expanded in x around inf 41.1%

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

   1. *-commutative 41.1%

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

8. Simplified 41.1%

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

9. Final simplification 41.1%

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

Alternative 17: 41.1% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in n around inf 59.8%

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

   1. log1p-define 59.8%

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

5. Simplified 59.8%

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

   1. log1p-undefine 59.8%

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

   2. diff-log 59.9%

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

7. Applied egg-rr 59.9%

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

   1. clear-num 59.9%

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

   2. log-rec 60.0%

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

9. Applied egg-rr 60.0%

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

10. Taylor expanded in x around inf 41.1%

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

    1. associate-/r* 41.8%

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

12. Simplified 41.8%

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

13. Final simplification 41.8%

    \[\leadsto \frac{\frac{1}{n}}{x} \]
14. Add Preprocessing

Alternative 18: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in x around 0 28.2%

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

4. Taylor expanded in x around inf 4.4%

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

5. Final simplification 4.4%

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

Reproduce

herbie shell --seed 2024044 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))