2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 85.4%

Time: 31.6s

Alternatives: 17

Speedup: 2.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% 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: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ t_2 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{e^{t_1}}\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (exp (/ (log1p x) n)) t_0))
        (t_2 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_2 -5e-6)
     (* 2.0 (log (sqrt (exp t_1))))
     (if (<= t_2 0.0) (/ (- (log (/ x (+ x 1.0)))) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = exp((log1p(x) / n)) - t_0;
	double t_2 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_2 <= -5e-6) {
		tmp = 2.0 * log(sqrt(exp(t_1)));
	} else if (t_2 <= 0.0) {
		tmp = -log((x / (x + 1.0))) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.exp((Math.log1p(x) / n)) - t_0;
	double t_2 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_2 <= -5e-6) {
		tmp = 2.0 * Math.log(Math.sqrt(Math.exp(t_1)));
	} else if (t_2 <= 0.0) {
		tmp = -Math.log((x / (x + 1.0))) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.exp((math.log1p(x) / n)) - t_0
	t_2 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_2 <= -5e-6:
		tmp = 2.0 * math.log(math.sqrt(math.exp(t_1)))
	elif t_2 <= 0.0:
		tmp = -math.log((x / (x + 1.0))) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(exp(Float64(log1p(x) / n)) - t_0)
	t_2 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_2 <= -5e-6)
		tmp = Float64(2.0 * log(sqrt(exp(t_1))));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / 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[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-6], N[(2.0 * N[Log[N[Sqrt[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[((-N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < -5.00000000000000041e-6

   1. Initial program 99.1%

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

      1. add-log-exp 99.0%

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

      2. add-sqr-sqrt 99.0%

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

      3. log-prod 99.1%

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

      4. add-exp-log 99.1%

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

      5. log-pow 99.1%

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

      6. +-commutative 99.1%

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

      7. log1p-udef 99.1%

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

      8. *-commutative 99.1%

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

      9. un-div-inv 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. count-2 99.1%

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

   5. Simplified 99.1%

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

   if -5.00000000000000041e-6 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 0.0

   1. Initial program 43.2%

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

   2. Taylor expanded in n around inf 80.7%

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

      1. +-rgt-identity 80.7%

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

      2. +-rgt-identity 80.7%

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

      3. log1p-def 80.7%

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

   4. Simplified 80.7%

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

      1. log1p-udef 80.7%

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

      2. +-commutative 80.7%

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

      3. diff-log 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. clear-num 80.8%

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

      2. log-rec 80.8%

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

   8. Applied egg-rr 80.8%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 54.7%

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

   2. Taylor expanded in n around 0 54.7%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 85.7%

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

Alternative 2: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-6}:\\ \;\;\;\;1 - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -5e-6)
     (- 1.0 (cbrt (pow x (/ 3.0 n))))
     (if (<= t_1 0.0)
       (/ (- (log (/ x (+ x 1.0)))) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -5e-6) {
		tmp = 1.0 - cbrt(pow(x, (3.0 / n)));
	} else if (t_1 <= 0.0) {
		tmp = -log((x / (x + 1.0))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -5e-6) {
		tmp = 1.0 - Math.cbrt(Math.pow(x, (3.0 / n)));
	} else if (t_1 <= 0.0) {
		tmp = -Math.log((x / (x + 1.0))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -5e-6)
		tmp = Float64(1.0 - cbrt((x ^ Float64(3.0 / n))));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	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[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-6], N[(1.0 - N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[((-N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < -5.00000000000000041e-6

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 99.1%

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

      1. add-cbrt-cube 99.1%

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

      2. pow3 99.1%

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

      3. pow-pow 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. associate-*l/ 99.1%

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

      2. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   if -5.00000000000000041e-6 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 0.0

   1. Initial program 43.2%

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

   2. Taylor expanded in n around inf 80.7%

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

      1. +-rgt-identity 80.7%

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

      2. +-rgt-identity 80.7%

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

      3. log1p-def 80.7%

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

   4. Simplified 80.7%

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

      1. log1p-udef 80.7%

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

      2. +-commutative 80.7%

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

      3. diff-log 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. clear-num 80.8%

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

      2. log-rec 80.8%

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

   8. Applied egg-rr 80.8%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 54.7%

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

   2. Taylor expanded in n around 0 54.7%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;1 - \sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-35}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x \cdot n\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log (/ x (+ x 1.0)))) n)))
   (if (<= (/ 1.0 n) -5e-35)
     (/ (exp (/ (log x) n)) (* x n))
     (if (<= (/ 1.0 n) 4e-177)
       t_0
       (if (<= (/ 1.0 n) 4e-132)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 2e-5)
           t_0
           (if (<= (/ 1.0 n) 2e+137)
             (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
             (sqrt (pow (* x n) -2.0)))))))))

C

double code(double x, double n) {
	double t_0 = -log((x / (x + 1.0))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-35) {
		tmp = exp((log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 4e-177) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-132) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+137) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = sqrt(pow((x * n), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log((x / (x + 1.0d0))) / n
    if ((1.0d0 / n) <= (-5d-35)) then
        tmp = exp((log(x) / n)) / (x * n)
    else if ((1.0d0 / n) <= 4d-177) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-132) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 2d-5) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d+137) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else
        tmp = sqrt(((x * n) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log((x / (x + 1.0))) / n;
	double tmp;
	if ((1.0 / n) <= -5e-35) {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 4e-177) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-132) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+137) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.sqrt(Math.pow((x * n), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log((x / (x + 1.0))) / n
	tmp = 0
	if (1.0 / n) <= -5e-35:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	elif (1.0 / n) <= 4e-177:
		tmp = t_0
	elif (1.0 / n) <= 4e-132:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 2e-5:
		tmp = t_0
	elif (1.0 / n) <= 2e+137:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.sqrt(math.pow((x * n), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-35)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 4e-177)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-132)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e+137)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = sqrt((Float64(x * n) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log((x / (x + 1.0))) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-35)
		tmp = exp((log(x) / n)) / (x * n);
	elseif ((1.0 / n) <= 4e-177)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-132)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 2e-5)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e+137)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	else
		tmp = sqrt(((x * n) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-35], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-177], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-132], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+137], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[N[(x * n), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999964e-35

   1. Initial program 94.1%

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

   2. Taylor expanded in x around inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. log-rec 92.9%

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

      3. mul-1-neg 92.9%

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

      4. distribute-neg-frac 92.9%

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

      5. mul-1-neg 92.9%

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

      6. remove-double-neg 92.9%

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

      7. *-commutative 92.9%

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

   4. Simplified 92.9%

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

   if -4.99999999999999964e-35 < (/.f64 1 n) < 3.99999999999999981e-177 or 3.9999999999999999e-132 < (/.f64 1 n) < 2.00000000000000016e-5

   1. Initial program 29.7%

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

   2. Taylor expanded in n around inf 81.2%

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

      1. +-rgt-identity 81.2%

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

      2. +-rgt-identity 81.2%

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

      3. log1p-def 81.2%

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

   4. Simplified 81.2%

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

      1. log1p-udef 81.2%

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

      2. +-commutative 81.2%

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

      3. diff-log 81.3%

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

   6. Applied egg-rr 81.3%

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

      1. clear-num 81.3%

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

      2. log-rec 81.3%

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

   8. Applied egg-rr 81.3%

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

   if 3.99999999999999981e-177 < (/.f64 1 n) < 3.9999999999999999e-132

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf 45.4%

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

      1. +-rgt-identity 45.4%

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

      2. +-rgt-identity 45.4%

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

      3. log1p-def 45.4%

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

   4. Simplified 45.4%

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

      1. log1p-udef 45.4%

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

      2. +-commutative 45.4%

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

      3. diff-log 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-/r* 84.1%

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

   9. Simplified 84.1%

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

   if 2.00000000000000016e-5 < (/.f64 1 n) < 2.0000000000000001e137

   1. Initial program 76.1%

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

   2. Taylor expanded in x around 0 77.7%

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

   if 2.0000000000000001e137 < (/.f64 1 n)

   1. Initial program 21.3%

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

   2. Taylor expanded in n around inf 5.7%

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

      1. +-rgt-identity 5.7%

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

      2. +-rgt-identity 5.7%

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

      3. log1p-def 5.7%

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

   4. Simplified 5.7%

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

   5. Taylor expanded in x around inf 48.8%

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

      1. *-commutative 48.8%

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

   7. Simplified 48.8%

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

      1. add-sqr-sqrt 48.8%

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

      2. sqrt-unprod 77.6%

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

      3. inv-pow 77.6%

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

      4. inv-pow 77.6%

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

      5. pow-prod-up 77.6%

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

      6. metadata-eval 77.6%

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

   9. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-35}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x \cdot n\right)}^{-2}}\\ \end{array} \]

Alternative 4: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x \cdot n\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log (/ x (+ x 1.0)))) n)) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-13)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_1)
     (if (<= (/ 1.0 n) 4e-177)
       t_0
       (if (<= (/ 1.0 n) 4e-132)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 2e-5)
           t_0
           (if (<= (/ 1.0 n) 2e+137)
             (- (+ 1.0 (/ x n)) t_1)
             (sqrt (pow (* x n) -2.0)))))))))

C

double code(double x, double n) {
	double t_0 = -log((x / (x + 1.0))) / n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-13) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_1;
	} else if ((1.0 / n) <= 4e-177) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-132) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+137) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = sqrt(pow((x * n), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -log((x / (x + 1.0d0))) / n
    t_1 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-13)) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - t_1
    else if ((1.0d0 / n) <= 4d-177) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-132) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 2d-5) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d+137) then
        tmp = (1.0d0 + (x / n)) - t_1
    else
        tmp = sqrt(((x * n) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log((x / (x + 1.0))) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-13) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - t_1;
	} else if ((1.0 / n) <= 4e-177) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-132) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+137) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = Math.sqrt(Math.pow((x * n), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log((x / (x + 1.0))) / n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-13:
		tmp = math.pow((x + 1.0), (1.0 / n)) - t_1
	elif (1.0 / n) <= 4e-177:
		tmp = t_0
	elif (1.0 / n) <= 4e-132:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 2e-5:
		tmp = t_0
	elif (1.0 / n) <= 2e+137:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = math.sqrt(math.pow((x * n), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-13)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_1);
	elseif (Float64(1.0 / n) <= 4e-177)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-132)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e+137)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = sqrt((Float64(x * n) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log((x / (x + 1.0))) / n;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-13)
		tmp = ((x + 1.0) ^ (1.0 / n)) - t_1;
	elseif ((1.0 / n) <= 4e-177)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-132)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 2e-5)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e+137)
		tmp = (1.0 + (x / n)) - t_1;
	else
		tmp = sqrt(((x * n) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-13], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-177], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-132], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+137], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[Sqrt[N[Power[N[(x * n), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -1e-13

   1. Initial program 98.3%

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

   if -1e-13 < (/.f64 1 n) < 3.99999999999999981e-177 or 3.9999999999999999e-132 < (/.f64 1 n) < 2.00000000000000016e-5

   1. Initial program 29.2%

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

   2. Taylor expanded in n around inf 80.3%

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

      1. +-rgt-identity 80.3%

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

      2. +-rgt-identity 80.3%

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

      3. log1p-def 80.3%

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

   4. Simplified 80.3%

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

      1. log1p-udef 80.3%

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

      2. +-commutative 80.3%

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

      3. diff-log 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. clear-num 80.4%

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

      2. log-rec 80.4%

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

   8. Applied egg-rr 80.4%

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

   if 3.99999999999999981e-177 < (/.f64 1 n) < 3.9999999999999999e-132

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf 45.4%

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

      1. +-rgt-identity 45.4%

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

      2. +-rgt-identity 45.4%

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

      3. log1p-def 45.4%

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

   4. Simplified 45.4%

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

      1. log1p-udef 45.4%

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

      2. +-commutative 45.4%

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

      3. diff-log 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-/r* 84.1%

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

   9. Simplified 84.1%

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

   if 2.00000000000000016e-5 < (/.f64 1 n) < 2.0000000000000001e137

   1. Initial program 76.1%

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

   2. Taylor expanded in x around 0 77.7%

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

   if 2.0000000000000001e137 < (/.f64 1 n)

   1. Initial program 21.3%

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

   2. Taylor expanded in n around inf 5.7%

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

      1. +-rgt-identity 5.7%

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

      2. +-rgt-identity 5.7%

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

      3. log1p-def 5.7%

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

   4. Simplified 5.7%

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

   5. Taylor expanded in x around inf 48.8%

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

      1. *-commutative 48.8%

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

   7. Simplified 48.8%

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

      1. add-sqr-sqrt 48.8%

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

      2. sqrt-unprod 77.6%

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

      3. inv-pow 77.6%

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

      4. inv-pow 77.6%

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

      5. pow-prod-up 77.6%

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

      6. metadata-eval 77.6%

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

   9. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-13}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x \cdot n\right)}^{-2}}\\ \end{array} \]

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;n \leq -1.02:\\ \;\;\;\;\frac{-\log t_0}{n}\\ \mathbf{elif}\;n \leq 5.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(t_0 + -1\right)}{n}\\ \mathbf{elif}\;n \leq 90000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= n -1.02)
     (/ (- (log t_0)) n)
     (if (<= n 5.1e-141)
       (/ (- (log1p (+ t_0 -1.0))) n)
       (if (<= n 90000.0)
         (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
         (if (<= n 4.5e+131)
           (/ (- (log1p x) (log x)) n)
           (if (<= n 2.1e+168)
             (/ (/ 1.0 n) x)
             (- (/ (log1p x) n) (/ (log x) n)))))))))

C

double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (n <= -1.02) {
		tmp = -log(t_0) / n;
	} else if (n <= 5.1e-141) {
		tmp = -log1p((t_0 + -1.0)) / n;
	} else if (n <= 90000.0) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else if (n <= 4.5e+131) {
		tmp = (log1p(x) - log(x)) / n;
	} else if (n <= 2.1e+168) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = (log1p(x) / n) - (log(x) / n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (n <= -1.02) {
		tmp = -Math.log(t_0) / n;
	} else if (n <= 5.1e-141) {
		tmp = -Math.log1p((t_0 + -1.0)) / n;
	} else if (n <= 90000.0) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else if (n <= 4.5e+131) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if (n <= 2.1e+168) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = (Math.log1p(x) / n) - (Math.log(x) / n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / (x + 1.0)
	tmp = 0
	if n <= -1.02:
		tmp = -math.log(t_0) / n
	elif n <= 5.1e-141:
		tmp = -math.log1p((t_0 + -1.0)) / n
	elif n <= 90000.0:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	elif n <= 4.5e+131:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif n <= 2.1e+168:
		tmp = (1.0 / n) / x
	else:
		tmp = (math.log1p(x) / n) - (math.log(x) / n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (n <= -1.02)
		tmp = Float64(Float64(-log(t_0)) / n);
	elseif (n <= 5.1e-141)
		tmp = Float64(Float64(-log1p(Float64(t_0 + -1.0))) / n);
	elseif (n <= 90000.0)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (n <= 4.5e+131)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (n <= 2.1e+168)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(Float64(log1p(x) / n) - Float64(log(x) / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.02], N[((-N[Log[t$95$0], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[n, 5.1e-141], N[((-N[Log[1 + N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[n, 90000.0], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.5e+131], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 2.1e+168], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -1.02

   1. Initial program 32.2%

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

   2. Taylor expanded in n around inf 71.1%

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

      1. +-rgt-identity 71.1%

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

      2. +-rgt-identity 71.1%

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

      3. log1p-def 71.1%

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

   4. Simplified 71.1%

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

      1. log1p-udef 71.1%

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

      2. +-commutative 71.1%

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

      3. diff-log 71.2%

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

   6. Applied egg-rr 71.2%

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

      1. clear-num 71.2%

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

      2. log-rec 71.3%

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

   8. Applied egg-rr 71.3%

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

   if -1.02 < n < 5.09999999999999977e-141

   1. Initial program 87.8%

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

   2. Taylor expanded in n around inf 48.2%

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

      1. +-rgt-identity 48.2%

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

      2. +-rgt-identity 48.2%

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

      3. log1p-def 48.2%

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

   4. Simplified 48.2%

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

      1. log1p-udef 48.2%

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

      2. +-commutative 48.2%

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

      3. diff-log 48.2%

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

   6. Applied egg-rr 48.2%

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

      1. clear-num 48.2%

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

      2. log-rec 48.2%

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

   8. Applied egg-rr 48.2%

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

      1. add-sqr-sqrt 44.0%

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

      2. sqrt-unprod 44.3%

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

      3. sqr-neg 44.3%

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

      4. sqrt-unprod 44.3%

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

      5. add-sqr-sqrt 44.3%

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

      6. neg-log 44.3%

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

      7. clear-num 44.3%

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

      8. diff-log 44.3%

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

      9. +-commutative 44.3%

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

      10. log1p-udef 44.3%

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

      11. add-sqr-sqrt 44.3%

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

      12. unpow2 44.3%

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

      13. log1p-expm1-u 44.3%

          \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(x\right) - \log x}\right)}^{2}\right)\right)}}{n} \]

      14. unpow2 44.3%

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

      15. add-sqr-sqrt 44.3%

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

   10. Applied egg-rr 95.4%

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

   if 5.09999999999999977e-141 < n < 9e4

   1. Initial program 76.1%

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

   2. Taylor expanded in x around 0 77.7%

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

   if 9e4 < n < 4.5000000000000002e131

   1. Initial program 15.2%

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

   2. Taylor expanded in n around inf 71.8%

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

      1. +-rgt-identity 71.8%

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

      2. +-rgt-identity 71.8%

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

      3. log1p-def 71.8%

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

   4. Simplified 71.8%

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

   if 4.5000000000000002e131 < n < 2.10000000000000003e168

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf 45.4%

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

      1. +-rgt-identity 45.4%

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

      2. +-rgt-identity 45.4%

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

      3. log1p-def 45.4%

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

   4. Simplified 45.4%

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

      1. log1p-udef 45.4%

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

      2. +-commutative 45.4%

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

      3. diff-log 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-/r* 84.1%

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

   9. Simplified 84.1%

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

   if 2.10000000000000003e168 < n

   1. Initial program 36.2%

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

   2. Taylor expanded in n around inf 93.4%

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

      1. +-rgt-identity 93.4%

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

      2. +-rgt-identity 93.4%

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

      3. log1p-def 93.4%

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

   4. Simplified 93.4%

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

      1. div-sub 93.5%

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

   6. Applied egg-rr 93.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.02:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq 5.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(\frac{x}{x + 1} + -1\right)}{n}\\ \mathbf{elif}\;n \leq 90000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \end{array} \]

Alternative 6: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{-\log t_0}{n}\\ \mathbf{if}\;n \leq -0.98:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(t_0 + -1\right)}{n}\\ \mathbf{elif}\;n \leq 90000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (/ (- (log t_0)) n)))
   (if (<= n -0.98)
     t_1
     (if (<= n 2.3e-139)
       (/ (- (log1p (+ t_0 -1.0))) n)
       (if (<= n 90000.0)
         (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
         (if (<= n 4.5e+131)
           (/ (- (log1p x) (log x)) n)
           (if (<= n 2.1e+168) (/ (/ 1.0 n) x) t_1)))))))

C

double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = -log(t_0) / n;
	double tmp;
	if (n <= -0.98) {
		tmp = t_1;
	} else if (n <= 2.3e-139) {
		tmp = -log1p((t_0 + -1.0)) / n;
	} else if (n <= 90000.0) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else if (n <= 4.5e+131) {
		tmp = (log1p(x) - log(x)) / n;
	} else if (n <= 2.1e+168) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = -Math.log(t_0) / n;
	double tmp;
	if (n <= -0.98) {
		tmp = t_1;
	} else if (n <= 2.3e-139) {
		tmp = -Math.log1p((t_0 + -1.0)) / n;
	} else if (n <= 90000.0) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else if (n <= 4.5e+131) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if (n <= 2.1e+168) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / (x + 1.0)
	t_1 = -math.log(t_0) / n
	tmp = 0
	if n <= -0.98:
		tmp = t_1
	elif n <= 2.3e-139:
		tmp = -math.log1p((t_0 + -1.0)) / n
	elif n <= 90000.0:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	elif n <= 4.5e+131:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif n <= 2.1e+168:
		tmp = (1.0 / n) / x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = Float64(Float64(-log(t_0)) / n)
	tmp = 0.0
	if (n <= -0.98)
		tmp = t_1;
	elseif (n <= 2.3e-139)
		tmp = Float64(Float64(-log1p(Float64(t_0 + -1.0))) / n);
	elseif (n <= 90000.0)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (n <= 4.5e+131)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (n <= 2.1e+168)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = t_1;
	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[Log[t$95$0], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[n, -0.98], t$95$1, If[LessEqual[n, 2.3e-139], N[((-N[Log[1 + N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[n, 90000.0], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.5e+131], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 2.1e+168], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -0.97999999999999998 or 2.10000000000000003e168 < n

   1. Initial program 33.7%

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

   2. Taylor expanded in n around inf 79.5%

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

      1. +-rgt-identity 79.5%

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

      2. +-rgt-identity 79.5%

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

      3. log1p-def 79.5%

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

   4. Simplified 79.5%

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

      1. log1p-udef 79.5%

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

      2. +-commutative 79.5%

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

      3. diff-log 79.5%

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

   6. Applied egg-rr 79.5%

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

      1. clear-num 79.5%

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

      2. log-rec 79.6%

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

   8. Applied egg-rr 79.6%

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

   if -0.97999999999999998 < n < 2.30000000000000012e-139

   1. Initial program 87.8%

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

   2. Taylor expanded in n around inf 48.2%

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

      1. +-rgt-identity 48.2%

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

      2. +-rgt-identity 48.2%

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

      3. log1p-def 48.2%

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

   4. Simplified 48.2%

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

      1. log1p-udef 48.2%

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

      2. +-commutative 48.2%

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

      3. diff-log 48.2%

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

   6. Applied egg-rr 48.2%

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

      1. clear-num 48.2%

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

      2. log-rec 48.2%

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

   8. Applied egg-rr 48.2%

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

      1. add-sqr-sqrt 44.0%

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

      2. sqrt-unprod 44.3%

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

      3. sqr-neg 44.3%

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

      4. sqrt-unprod 44.3%

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

      5. add-sqr-sqrt 44.3%

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

      6. neg-log 44.3%

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

      7. clear-num 44.3%

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

      8. diff-log 44.3%

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

      9. +-commutative 44.3%

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

      10. log1p-udef 44.3%

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

      11. add-sqr-sqrt 44.3%

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

      12. unpow2 44.3%

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

      13. log1p-expm1-u 44.3%

          \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(x\right) - \log x}\right)}^{2}\right)\right)}}{n} \]

      14. unpow2 44.3%

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

      15. add-sqr-sqrt 44.3%

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

   10. Applied egg-rr 95.4%

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

   if 2.30000000000000012e-139 < n < 9e4

   1. Initial program 76.1%

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

   2. Taylor expanded in x around 0 77.7%

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

   if 9e4 < n < 4.5000000000000002e131

   1. Initial program 15.2%

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

   2. Taylor expanded in n around inf 71.8%

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

      1. +-rgt-identity 71.8%

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

      2. +-rgt-identity 71.8%

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

      3. log1p-def 71.8%

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

   4. Simplified 71.8%

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

   if 4.5000000000000002e131 < n < 2.10000000000000003e168

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf 45.4%

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

      1. +-rgt-identity 45.4%

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

      2. +-rgt-identity 45.4%

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

      3. log1p-def 45.4%

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

   4. Simplified 45.4%

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

      1. log1p-udef 45.4%

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

      2. +-commutative 45.4%

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

      3. diff-log 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-/r* 84.1%

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

   9. Simplified 84.1%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.98:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(\frac{x}{x + 1} + -1\right)}{n}\\ \mathbf{elif}\;n \leq 90000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \end{array} \]

Alternative 7: 66.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (- (log (/ x (+ x 1.0)))) n)))
   (if (<= (/ 1.0 n) -2e+25)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= (/ 1.0 n) -1e-8)
       (- 1.0 t_0)
       (if (<= (/ 1.0 n) 4e-177)
         t_1
         (if (<= (/ 1.0 n) 4e-132)
           (/ (/ 1.0 n) x)
           (if (<= (/ 1.0 n) 2e-5)
             t_1
             (if (<= (/ 1.0 n) 5e+218)
               (- (+ 1.0 (/ x n)) t_0)
               (/ 1.0 (* x n))))))))))

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 ((1.0 / n) <= -2e+25) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= -1e-8) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 4e-177) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-132) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = -log((x / (x + 1.0d0))) / n
    if ((1.0d0 / n) <= (-2d+25)) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= (-1d-8)) then
        tmp = 1.0d0 - t_0
    else if ((1.0d0 / n) <= 4d-177) then
        tmp = t_1
    else if ((1.0d0 / n) <= 4d-132) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 2d-5) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+218) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = -Math.log((x / (x + 1.0))) / n;
	double tmp;
	if ((1.0 / n) <= -2e+25) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= -1e-8) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 4e-177) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-132) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = -math.log((x / (x + 1.0))) / n
	tmp = 0
	if (1.0 / n) <= -2e+25:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= -1e-8:
		tmp = 1.0 - t_0
	elif (1.0 / n) <= 4e-177:
		tmp = t_1
	elif (1.0 / n) <= 4e-132:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 2e-5:
		tmp = t_1
	elif (1.0 / n) <= 5e+218:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+25)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= -1e-8)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= 4e-177)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e-132)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+218)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = -log((x / (x + 1.0))) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -2e+25)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= -1e-8)
		tmp = 1.0 - t_0;
	elseif ((1.0 / n) <= 4e-177)
		tmp = t_1;
	elseif ((1.0 / n) <= 4e-132)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 2e-5)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+218)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (x * n);
	end
	tmp_2 = 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[(1.0 / n), $MachinePrecision], -2e+25], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-8], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-177], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-132], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+218], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -2.00000000000000018e25

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf 61.1%

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

      1. +-rgt-identity 61.1%

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

      2. +-rgt-identity 61.1%

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

      3. log1p-def 61.1%

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

   4. Simplified 61.1%

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

      1. log1p-udef 61.1%

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

      2. +-commutative 61.1%

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

      3. diff-log 61.1%

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

   6. Applied egg-rr 61.1%

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

   if -2.00000000000000018e25 < (/.f64 1 n) < -1e-8

   1. Initial program 97.1%

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

   2. Taylor expanded in x around 0 81.0%

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

   if -1e-8 < (/.f64 1 n) < 3.99999999999999981e-177 or 3.9999999999999999e-132 < (/.f64 1 n) < 2.00000000000000016e-5

   1. Initial program 29.1%

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

   2. Taylor expanded in n around inf 79.8%

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

      1. +-rgt-identity 79.8%

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

      2. +-rgt-identity 79.8%

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

      3. log1p-def 79.8%

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

   4. Simplified 79.8%

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

      1. log1p-udef 79.8%

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

      2. +-commutative 79.8%

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

      3. diff-log 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. clear-num 79.8%

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

      2. log-rec 79.9%

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

   8. Applied egg-rr 79.9%

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

   if 3.99999999999999981e-177 < (/.f64 1 n) < 3.9999999999999999e-132

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf 45.4%

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

      1. +-rgt-identity 45.4%

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

      2. +-rgt-identity 45.4%

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

      3. log1p-def 45.4%

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

   4. Simplified 45.4%

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

      1. log1p-udef 45.4%

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

      2. +-commutative 45.4%

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

      3. diff-log 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-/r* 84.1%

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

   9. Simplified 84.1%

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

   if 2.00000000000000016e-5 < (/.f64 1 n) < 4.99999999999999983e218

   1. Initial program 64.4%

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

   2. Taylor expanded in x around 0 65.9%

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

   if 4.99999999999999983e218 < (/.f64 1 n)

   1. Initial program 20.5%

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

   2. Taylor expanded in n around inf 6.4%

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

      1. +-rgt-identity 6.4%

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

      2. +-rgt-identity 6.4%

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

      3. log1p-def 6.4%

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

   4. Simplified 6.4%

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

   5. Taylor expanded in x around inf 75.8%

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

      1. *-commutative 75.8%

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

   7. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \]

Alternative 8: 66.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

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 (<= (/ 1.0 n) -2e+25)
     t_1
     (if (<= (/ 1.0 n) -1e-8)
       t_0
       (if (<= (/ 1.0 n) 4e-177)
         t_1
         (if (<= (/ 1.0 n) 4e-132)
           (/ (/ 1.0 n) x)
           (if (<= (/ 1.0 n) 5e-5)
             t_1
             (if (<= (/ 1.0 n) 5e+218) t_0 (/ 1.0 (* x n))))))))))

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 ((1.0 / n) <= -2e+25) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-177) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-132) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: 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 ((1.0d0 / n) <= (-2d+25)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-1d-8)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-177) then
        tmp = t_1
    else if ((1.0d0 / n) <= 4d-132) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 5d-5) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+218) then
        tmp = t_0
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -2e+25) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-177) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-132) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	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 (1.0 / n) <= -2e+25:
		tmp = t_1
	elif (1.0 / n) <= -1e-8:
		tmp = t_0
	elif (1.0 / n) <= 4e-177:
		tmp = t_1
	elif (1.0 / n) <= 4e-132:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 5e-5:
		tmp = t_1
	elif (1.0 / n) <= 5e+218:
		tmp = t_0
	else:
		tmp = 1.0 / (x * n)
	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 (Float64(1.0 / n) <= -2e+25)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -1e-8)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-177)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e-132)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-5)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+218)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(x * n));
	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 ((1.0 / n) <= -2e+25)
		tmp = t_1;
	elseif ((1.0 / n) <= -1e-8)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-177)
		tmp = t_1;
	elseif ((1.0 / n) <= 4e-132)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 5e-5)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+218)
		tmp = t_0;
	else
		tmp = 1.0 / (x * n);
	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[(1.0 / n), $MachinePrecision], -2e+25], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-8], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-177], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-132], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-5], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+218], t$95$0, N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2.00000000000000018e25 or -1e-8 < (/.f64 1 n) < 3.99999999999999981e-177 or 3.9999999999999999e-132 < (/.f64 1 n) < 5.00000000000000024e-5

   1. Initial program 51.1%

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

   2. Taylor expanded in n around inf 73.6%

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

      1. +-rgt-identity 73.6%

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

      2. +-rgt-identity 73.6%

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

      3. log1p-def 73.6%

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

   4. Simplified 73.6%

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

      1. log1p-udef 73.6%

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

      2. +-commutative 73.6%

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

      3. diff-log 73.6%

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

   6. Applied egg-rr 73.6%

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

   if -2.00000000000000018e25 < (/.f64 1 n) < -1e-8 or 5.00000000000000024e-5 < (/.f64 1 n) < 4.99999999999999983e218

   1. Initial program 77.8%

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

   2. Taylor expanded in x around 0 72.1%

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

   if 3.99999999999999981e-177 < (/.f64 1 n) < 3.9999999999999999e-132

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf 45.4%

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

      1. +-rgt-identity 45.4%

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

      2. +-rgt-identity 45.4%

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

      3. log1p-def 45.4%

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

   4. Simplified 45.4%

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

      1. log1p-udef 45.4%

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

      2. +-commutative 45.4%

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

      3. diff-log 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-/r* 84.1%

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

   9. Simplified 84.1%

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

   if 4.99999999999999983e218 < (/.f64 1 n)

   1. Initial program 20.5%

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

   2. Taylor expanded in n around inf 6.4%

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

      1. +-rgt-identity 6.4%

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

      2. +-rgt-identity 6.4%

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

      3. log1p-def 6.4%

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

   4. Simplified 6.4%

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

   5. Taylor expanded in x around inf 75.8%

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

      1. *-commutative 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-177}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \]

Alternative 9: 66.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

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 (<= (/ 1.0 n) -2e+25)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= (/ 1.0 n) -1e-8)
       t_0
       (if (<= (/ 1.0 n) 4e-177)
         t_1
         (if (<= (/ 1.0 n) 4e-132)
           (/ (/ 1.0 n) x)
           (if (<= (/ 1.0 n) 5e-5)
             t_1
             (if (<= (/ 1.0 n) 5e+218) t_0 (/ 1.0 (* x n))))))))))

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 ((1.0 / n) <= -2e+25) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= -1e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-177) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-132) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: 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 ((1.0d0 / n) <= (-2d+25)) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= (-1d-8)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-177) then
        tmp = t_1
    else if ((1.0d0 / n) <= 4d-132) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 5d-5) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+218) then
        tmp = t_0
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = -Math.log((x / (x + 1.0))) / n;
	double tmp;
	if ((1.0 / n) <= -2e+25) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= -1e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-177) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-132) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 5e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+218) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (x * n);
	}
	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 (1.0 / n) <= -2e+25:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= -1e-8:
		tmp = t_0
	elif (1.0 / n) <= 4e-177:
		tmp = t_1
	elif (1.0 / n) <= 4e-132:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 5e-5:
		tmp = t_1
	elif (1.0 / n) <= 5e+218:
		tmp = t_0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+25)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= -1e-8)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-177)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e-132)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-5)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+218)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(x * n));
	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 ((1.0 / n) <= -2e+25)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= -1e-8)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-177)
		tmp = t_1;
	elseif ((1.0 / n) <= 4e-132)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 5e-5)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+218)
		tmp = t_0;
	else
		tmp = 1.0 / (x * n);
	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[(1.0 / n), $MachinePrecision], -2e+25], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-8], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-177], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-132], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-5], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+218], t$95$0, N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2.00000000000000018e25

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf 61.1%

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

      1. +-rgt-identity 61.1%

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

      2. +-rgt-identity 61.1%

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

      3. log1p-def 61.1%

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

   4. Simplified 61.1%

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

      1. log1p-udef 61.1%

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

      2. +-commutative 61.1%

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

      3. diff-log 61.1%

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

   6. Applied egg-rr 61.1%

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

   if -2.00000000000000018e25 < (/.f64 1 n) < -1e-8 or 5.00000000000000024e-5 < (/.f64 1 n) < 4.99999999999999983e218

   1. Initial program 77.8%

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

   2. Taylor expanded in x around 0 72.1%

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

   if -1e-8 < (/.f64 1 n) < 3.99999999999999981e-177 or 3.9999999999999999e-132 < (/.f64 1 n) < 5.00000000000000024e-5

   1. Initial program 28.9%

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

   2. Taylor expanded in n around inf 79.2%

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

      1. +-rgt-identity 79.2%

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

      2. +-rgt-identity 79.2%

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

      3. log1p-def 79.2%

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

   4. Simplified 79.2%

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

      1. log1p-udef 79.2%

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

      2. +-commutative 79.2%

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

      3. diff-log 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. clear-num 79.3%

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

      2. log-rec 79.3%

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

   8. Applied egg-rr 79.3%

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

   if 3.99999999999999981e-177 < (/.f64 1 n) < 3.9999999999999999e-132

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf 45.4%

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

      1. +-rgt-identity 45.4%

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

      2. +-rgt-identity 45.4%

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

      3. log1p-def 45.4%

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

   4. Simplified 45.4%

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

      1. log1p-udef 45.4%

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

      2. +-commutative 45.4%

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

      3. diff-log 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-/r* 84.1%

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

   9. Simplified 84.1%

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

   if 4.99999999999999983e218 < (/.f64 1 n)

   1. Initial program 20.5%

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

   2. Taylor expanded in n around inf 6.4%

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

      1. +-rgt-identity 6.4%

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

      2. +-rgt-identity 6.4%

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

      3. log1p-def 6.4%

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

   4. Simplified 6.4%

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

   5. Taylor expanded in x around inf 75.8%

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

      1. *-commutative 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \]

Alternative 10: 79.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{-\log t_0}{n}\\ \mathbf{if}\;n \leq -1.28:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-140}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(t_0 + -1\right)}{n}\\ \mathbf{elif}\;n \leq 90000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (/ (- (log t_0)) n)))
   (if (<= n -1.28)
     t_1
     (if (<= n 1.55e-140)
       (/ (- (log1p (+ t_0 -1.0))) n)
       (if (<= n 90000.0)
         (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
         (if (<= n 3.9e+131)
           (/ (log (/ (+ x 1.0) x)) n)
           (if (<= n 2.1e+168) (/ (/ 1.0 n) x) t_1)))))))

C

double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = -log(t_0) / n;
	double tmp;
	if (n <= -1.28) {
		tmp = t_1;
	} else if (n <= 1.55e-140) {
		tmp = -log1p((t_0 + -1.0)) / n;
	} else if (n <= 90000.0) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else if (n <= 3.9e+131) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if (n <= 2.1e+168) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = -Math.log(t_0) / n;
	double tmp;
	if (n <= -1.28) {
		tmp = t_1;
	} else if (n <= 1.55e-140) {
		tmp = -Math.log1p((t_0 + -1.0)) / n;
	} else if (n <= 90000.0) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else if (n <= 3.9e+131) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if (n <= 2.1e+168) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / (x + 1.0)
	t_1 = -math.log(t_0) / n
	tmp = 0
	if n <= -1.28:
		tmp = t_1
	elif n <= 1.55e-140:
		tmp = -math.log1p((t_0 + -1.0)) / n
	elif n <= 90000.0:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	elif n <= 3.9e+131:
		tmp = math.log(((x + 1.0) / x)) / n
	elif n <= 2.1e+168:
		tmp = (1.0 / n) / x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = Float64(Float64(-log(t_0)) / n)
	tmp = 0.0
	if (n <= -1.28)
		tmp = t_1;
	elseif (n <= 1.55e-140)
		tmp = Float64(Float64(-log1p(Float64(t_0 + -1.0))) / n);
	elseif (n <= 90000.0)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (n <= 3.9e+131)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (n <= 2.1e+168)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = t_1;
	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[Log[t$95$0], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[n, -1.28], t$95$1, If[LessEqual[n, 1.55e-140], N[((-N[Log[1 + N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[n, 90000.0], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.9e+131], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 2.1e+168], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.28000000000000003 or 2.10000000000000003e168 < n

   1. Initial program 33.7%

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

   2. Taylor expanded in n around inf 79.5%

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

      1. +-rgt-identity 79.5%

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

      2. +-rgt-identity 79.5%

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

      3. log1p-def 79.5%

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

   4. Simplified 79.5%

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

      1. log1p-udef 79.5%

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

      2. +-commutative 79.5%

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

      3. diff-log 79.5%

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

   6. Applied egg-rr 79.5%

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

      1. clear-num 79.5%

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

      2. log-rec 79.6%

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

   8. Applied egg-rr 79.6%

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

   if -1.28000000000000003 < n < 1.55e-140

   1. Initial program 87.8%

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

   2. Taylor expanded in n around inf 48.2%

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

      1. +-rgt-identity 48.2%

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

      2. +-rgt-identity 48.2%

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

      3. log1p-def 48.2%

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

   4. Simplified 48.2%

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

      1. log1p-udef 48.2%

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

      2. +-commutative 48.2%

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

      3. diff-log 48.2%

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

   6. Applied egg-rr 48.2%

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

      1. clear-num 48.2%

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

      2. log-rec 48.2%

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

   8. Applied egg-rr 48.2%

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

      1. add-sqr-sqrt 44.0%

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

      2. sqrt-unprod 44.3%

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

      3. sqr-neg 44.3%

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

      4. sqrt-unprod 44.3%

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

      5. add-sqr-sqrt 44.3%

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

      6. neg-log 44.3%

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

      7. clear-num 44.3%

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

      8. diff-log 44.3%

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

      9. +-commutative 44.3%

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

      10. log1p-udef 44.3%

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

      11. add-sqr-sqrt 44.3%

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

      12. unpow2 44.3%

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

      13. log1p-expm1-u 44.3%

          \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt{\mathsf{log1p}\left(x\right) - \log x}\right)}^{2}\right)\right)}}{n} \]

      14. unpow2 44.3%

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

      15. add-sqr-sqrt 44.3%

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

   10. Applied egg-rr 95.4%

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

   if 1.55e-140 < n < 9e4

   1. Initial program 76.1%

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

   2. Taylor expanded in x around 0 77.7%

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

   if 9e4 < n < 3.9e131

   1. Initial program 15.2%

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

   2. Taylor expanded in n around inf 71.8%

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

      1. +-rgt-identity 71.8%

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

      2. +-rgt-identity 71.8%

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

      3. log1p-def 71.8%

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

   4. Simplified 71.8%

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

      1. log1p-udef 71.8%

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

      2. +-commutative 71.8%

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

      3. diff-log 71.8%

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

   6. Applied egg-rr 71.8%

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

   if 3.9e131 < n < 2.10000000000000003e168

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf 45.4%

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

      1. +-rgt-identity 45.4%

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

      2. +-rgt-identity 45.4%

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

      3. log1p-def 45.4%

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

   4. Simplified 45.4%

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

      1. log1p-udef 45.4%

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

      2. +-commutative 45.4%

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

      3. diff-log 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. associate-/r* 84.1%

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

   9. Simplified 84.1%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.28:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-140}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(\frac{x}{x + 1} + -1\right)}{n}\\ \mathbf{elif}\;n \leq 90000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \end{array} \]

Alternative 11: 59.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-268}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-200}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.55e-268)
   (/ (- (log x)) n)
   (if (<= x 2.45e-200)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 1.0)
       (/ (- x (log x)) n)
       (if (<= x 4.7e+120)
         (/ (/ 1.0 x) n)
         (* 0.3333333333333333 (/ 0.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.55e-268) {
		tmp = -log(x) / n;
	} else if (x <= 2.45e-200) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.7e+120) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.3333333333333333 * (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 <= 1.55d-268) then
        tmp = -log(x) / n
    else if (x <= 2.45d-200) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4.7d+120) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.3333333333333333d0 * (0.0d0 / n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.55e-268) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.45e-200) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.7e+120) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.3333333333333333 * (0.0 / n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.55e-268:
		tmp = -math.log(x) / n
	elif x <= 2.45e-200:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 4.7e+120:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.3333333333333333 * (0.0 / n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.55e-268)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.45e-200)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.7e+120)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(0.3333333333333333 * Float64(0.0 / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.55e-268)
		tmp = -log(x) / n;
	elseif (x <= 2.45e-200)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.7e+120)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.3333333333333333 * (0.0 / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.55e-268], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.45e-200], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.7e+120], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(0.3333333333333333 * N[(0.0 / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.5499999999999999e-268

   1. Initial program 19.0%

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

   2. Taylor expanded in n around inf 87.1%

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

      1. +-rgt-identity 87.1%

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

      2. +-rgt-identity 87.1%

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

      3. log1p-def 87.1%

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

   4. Simplified 87.1%

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

   5. Taylor expanded in x around 0 87.1%

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

      1. neg-mul-1 87.1%

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

   7. Simplified 87.1%

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

   if 1.5499999999999999e-268 < x < 2.45e-200

   1. Initial program 60.4%

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

   2. Taylor expanded in x around 0 60.4%

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

   if 2.45e-200 < x < 1

   1. Initial program 37.7%

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

   2. Taylor expanded in n around inf 57.6%

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

      1. +-rgt-identity 57.6%

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

      2. +-rgt-identity 57.6%

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

      3. log1p-def 57.6%

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

   4. Simplified 57.6%

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

      1. log1p-udef 57.6%

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

      2. +-commutative 57.6%

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

      3. diff-log 57.6%

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

   6. Applied egg-rr 57.6%

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

   7. Taylor expanded in x around 0 57.3%

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

      1. neg-mul-1 57.3%

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

      2. unsub-neg 57.3%

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

   9. Simplified 57.3%

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

   if 1 < x < 4.69999999999999993e120

   1. Initial program 35.7%

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

   2. Taylor expanded in n around inf 41.1%

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

      1. +-rgt-identity 41.1%

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

      2. +-rgt-identity 41.1%

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

      3. log1p-def 41.1%

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

   4. Simplified 41.1%

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

   5. Taylor expanded in x around inf 62.5%

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

   if 4.69999999999999993e120 < x

   1. Initial program 82.9%

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

      1. flip3-- 47.4%

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

      2. div-inv 47.4%

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

      3. add-exp-log 47.4%

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

      4. log-pow 47.4%

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

      5. +-commutative 47.4%

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

      6. log1p-udef 47.4%

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

      7. *-commutative 47.4%

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

      8. un-div-inv 47.4%

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

      9. pow-pow 47.2%

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

   3. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      2. metadata-eval 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. +-commutative 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} \]

      4. pow-sqr 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      5. sqr-neg 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)}\right)} \]

      6. +-commutative 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left(\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      7. associate-+r+ 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}}} \]

   5. Simplified 47.3%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + {x}^{\left(\frac{2}{n}\right)}\right) + {\left(x \cdot \left(1 + x\right)\right)}^{\left(\frac{1}{n}\right)}}} \]

   6. Taylor expanded in n around inf 82.9%

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

      1. cancel-sign-sub-inv 82.9%

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

      2. log1p-def 82.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\color{blue}{\mathsf{log1p}\left(x\right)} + 2 \cdot \log \left(1 + x\right)\right) + \left(-3\right) \cdot \log x}{n} \]

      3. log1p-def 82.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\mathsf{log1p}\left(x\right) + 2 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right) + \left(-3\right) \cdot \log x}{n} \]

      4. distribute-rgt1-in 82.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\left(2 + 1\right) \cdot \mathsf{log1p}\left(x\right)} + \left(-3\right) \cdot \log x}{n} \]

      5. metadata-eval 82.9%

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

      6. metadata-eval 82.9%

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

      7. *-commutative 82.9%

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

   8. Simplified 82.9%

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

   9. Taylor expanded in x around inf 82.9%

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

       1. log-rec 82.9%

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

       2. log-rec 82.9%

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

       3. distribute-rgt-out 82.9%

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

       4. metadata-eval 82.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\left(-\log x\right) \cdot \color{blue}{0}}{n} \]

       5. mul0-rgt 82.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]

   11. Simplified 82.9%

       \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-268}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-200}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \end{array} \]

Alternative 12: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (- x (log x)) n)
   (if (<= x 3.55e+120) (/ (/ 1.0 x) n) (* 0.3333333333333333 (/ 0.0 n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 3.55e+120) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.3333333333333333 * (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 <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 3.55d+120) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.3333333333333333d0 * (0.0d0 / n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 3.55e+120) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.3333333333333333 * (0.0 / n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 3.55e+120:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.3333333333333333 * (0.0 / n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 3.55e+120)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(0.3333333333333333 * Float64(0.0 / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 3.55e+120)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.3333333333333333 * (0.0 / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3.55e+120], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(0.3333333333333333 * N[(0.0 / n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 41.0%

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

   2. Taylor expanded in n around inf 56.2%

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

      1. +-rgt-identity 56.2%

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

      2. +-rgt-identity 56.2%

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

      3. log1p-def 56.2%

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

   4. Simplified 56.2%

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

      1. log1p-udef 56.2%

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

      2. +-commutative 56.2%

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

      3. diff-log 56.2%

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

   6. Applied egg-rr 56.2%

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

   7. Taylor expanded in x around 0 56.0%

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

      1. neg-mul-1 56.0%

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

      2. unsub-neg 56.0%

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

   9. Simplified 56.0%

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

   if 1 < x < 3.5500000000000001e120

   1. Initial program 35.7%

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

   2. Taylor expanded in n around inf 41.1%

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

      1. +-rgt-identity 41.1%

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

      2. +-rgt-identity 41.1%

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

      3. log1p-def 41.1%

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

   4. Simplified 41.1%

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

   5. Taylor expanded in x around inf 62.5%

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

   if 3.5500000000000001e120 < x

   1. Initial program 82.9%

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

      1. flip3-- 47.4%

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

      2. div-inv 47.4%

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

      3. add-exp-log 47.4%

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

      4. log-pow 47.4%

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

      5. +-commutative 47.4%

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

      6. log1p-udef 47.4%

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

      7. *-commutative 47.4%

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

      8. un-div-inv 47.4%

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

      9. pow-pow 47.2%

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

   3. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      2. metadata-eval 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. +-commutative 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} \]

      4. pow-sqr 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      5. sqr-neg 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)}\right)} \]

      6. +-commutative 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left(\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      7. associate-+r+ 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}}} \]

   5. Simplified 47.3%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + {x}^{\left(\frac{2}{n}\right)}\right) + {\left(x \cdot \left(1 + x\right)\right)}^{\left(\frac{1}{n}\right)}}} \]

   6. Taylor expanded in n around inf 82.9%

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

      1. cancel-sign-sub-inv 82.9%

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

      2. log1p-def 82.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\color{blue}{\mathsf{log1p}\left(x\right)} + 2 \cdot \log \left(1 + x\right)\right) + \left(-3\right) \cdot \log x}{n} \]

      3. log1p-def 82.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\mathsf{log1p}\left(x\right) + 2 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right) + \left(-3\right) \cdot \log x}{n} \]

      4. distribute-rgt1-in 82.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\left(2 + 1\right) \cdot \mathsf{log1p}\left(x\right)} + \left(-3\right) \cdot \log x}{n} \]

      5. metadata-eval 82.9%

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

      6. metadata-eval 82.9%

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

      7. *-commutative 82.9%

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

   8. Simplified 82.9%

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

   9. Taylor expanded in x around inf 82.9%

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

       1. log-rec 82.9%

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

       2. log-rec 82.9%

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

       3. distribute-rgt-out 82.9%

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

       4. metadata-eval 82.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\left(-\log x\right) \cdot \color{blue}{0}}{n} \]

       5. mul0-rgt 82.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]

   11. Simplified 82.9%

       \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \end{array} \]

Alternative 13: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55)
   (/ (- (log x)) n)
   (if (<= x 6.8e+120) (/ (/ 1.0 x) n) (* 0.3333333333333333 (/ 0.0 n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -log(x) / n;
	} else if (x <= 6.8e+120) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.3333333333333333 * (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 <= 0.55d0) then
        tmp = -log(x) / n
    else if (x <= 6.8d+120) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.3333333333333333d0 * (0.0d0 / n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -Math.log(x) / n;
	} else if (x <= 6.8e+120) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.3333333333333333 * (0.0 / n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -math.log(x) / n
	elif x <= 6.8e+120:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.3333333333333333 * (0.0 / n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 6.8e+120)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(0.3333333333333333 * Float64(0.0 / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -log(x) / n;
	elseif (x <= 6.8e+120)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.3333333333333333 * (0.0 / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 6.8e+120], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(0.3333333333333333 * N[(0.0 / n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.55000000000000004

   1. Initial program 41.0%

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

   2. Taylor expanded in n around inf 56.2%

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

      1. +-rgt-identity 56.2%

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

      2. +-rgt-identity 56.2%

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

      3. log1p-def 56.2%

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

   4. Simplified 56.2%

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

   5. Taylor expanded in x around 0 55.6%

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

      1. neg-mul-1 55.6%

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

   7. Simplified 55.6%

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

   if 0.55000000000000004 < x < 6.79999999999999998e120

   1. Initial program 35.7%

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

   2. Taylor expanded in n around inf 41.1%

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

      1. +-rgt-identity 41.1%

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

      2. +-rgt-identity 41.1%

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

      3. log1p-def 41.1%

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

   4. Simplified 41.1%

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

   5. Taylor expanded in x around inf 62.5%

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

   if 6.79999999999999998e120 < x

   1. Initial program 82.9%

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

      1. flip3-- 47.4%

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

      2. div-inv 47.4%

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

      3. add-exp-log 47.4%

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

      4. log-pow 47.4%

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

      5. +-commutative 47.4%

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

      6. log1p-udef 47.4%

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

      7. *-commutative 47.4%

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

      8. un-div-inv 47.4%

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

      9. pow-pow 47.2%

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

   3. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      2. metadata-eval 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. +-commutative 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} \]

      4. pow-sqr 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      5. sqr-neg 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)}\right)} \]

      6. +-commutative 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left(\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      7. associate-+r+ 47.3%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}}} \]

   5. Simplified 47.3%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + {x}^{\left(\frac{2}{n}\right)}\right) + {\left(x \cdot \left(1 + x\right)\right)}^{\left(\frac{1}{n}\right)}}} \]

   6. Taylor expanded in n around inf 82.9%

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

      1. cancel-sign-sub-inv 82.9%

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

      2. log1p-def 82.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\color{blue}{\mathsf{log1p}\left(x\right)} + 2 \cdot \log \left(1 + x\right)\right) + \left(-3\right) \cdot \log x}{n} \]

      3. log1p-def 82.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\mathsf{log1p}\left(x\right) + 2 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right) + \left(-3\right) \cdot \log x}{n} \]

      4. distribute-rgt1-in 82.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\left(2 + 1\right) \cdot \mathsf{log1p}\left(x\right)} + \left(-3\right) \cdot \log x}{n} \]

      5. metadata-eval 82.9%

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

      6. metadata-eval 82.9%

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

      7. *-commutative 82.9%

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

   8. Simplified 82.9%

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

   9. Taylor expanded in x around inf 82.9%

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

       1. log-rec 82.9%

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

       2. log-rec 82.9%

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

       3. distribute-rgt-out 82.9%

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

       4. metadata-eval 82.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\left(-\log x\right) \cdot \color{blue}{0}}{n} \]

       5. mul0-rgt 82.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]

   11. Simplified 82.9%

       \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \end{array} \]

Alternative 14: 46.6% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.4 \lor \neg \left(n \leq -3.2 \cdot 10^{-265}\right):\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -6.4) (not (<= n -3.2e-265)))
   (/ 1.0 (* x n))
   (* 0.3333333333333333 (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -6.4) || !(n <= -3.2e-265)) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.3333333333333333 * (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 ((n <= (-6.4d0)) .or. (.not. (n <= (-3.2d-265)))) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = 0.3333333333333333d0 * (0.0d0 / n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -6.4) || !(n <= -3.2e-265)) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.3333333333333333 * (0.0 / n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -6.4) or not (n <= -3.2e-265):
		tmp = 1.0 / (x * n)
	else:
		tmp = 0.3333333333333333 * (0.0 / n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -6.4) || !(n <= -3.2e-265))
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = Float64(0.3333333333333333 * Float64(0.0 / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -6.4) || ~((n <= -3.2e-265)))
		tmp = 1.0 / (x * n);
	else
		tmp = 0.3333333333333333 * (0.0 / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -6.4], N[Not[LessEqual[n, -3.2e-265]], $MachinePrecision]], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(0.0 / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6.4000000000000004 or -3.2e-265 < n

   1. Initial program 36.9%

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

   2. Taylor expanded in n around inf 63.5%

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

      1. +-rgt-identity 63.5%

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

      2. +-rgt-identity 63.5%

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

      3. log1p-def 63.5%

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

   4. Simplified 63.5%

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

   5. Taylor expanded in x around inf 45.4%

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

      1. *-commutative 45.4%

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

   7. Simplified 45.4%

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

   if -6.4000000000000004 < n < -3.2e-265

   1. Initial program 100.0%

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

      1. flip3-- 0.0%

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

      2. div-inv 0.0%

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

      3. add-exp-log 0.0%

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

      4. log-pow 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-udef 0.0%

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

      7. *-commutative 0.0%

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

      8. un-div-inv 0.0%

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

      9. pow-pow 0.0%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      2. metadata-eval 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. +-commutative 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} \]

      4. pow-sqr 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      5. sqr-neg 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)}\right)} \]

      6. +-commutative 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left(\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      7. associate-+r+ 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + {x}^{\left(\frac{2}{n}\right)}\right) + {\left(x \cdot \left(1 + x\right)\right)}^{\left(\frac{1}{n}\right)}}} \]

   6. Taylor expanded in n around inf 56.7%

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

      1. cancel-sign-sub-inv 56.7%

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

      2. log1p-def 56.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\color{blue}{\mathsf{log1p}\left(x\right)} + 2 \cdot \log \left(1 + x\right)\right) + \left(-3\right) \cdot \log x}{n} \]

      3. log1p-def 56.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\mathsf{log1p}\left(x\right) + 2 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right) + \left(-3\right) \cdot \log x}{n} \]

      4. distribute-rgt1-in 56.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\left(2 + 1\right) \cdot \mathsf{log1p}\left(x\right)} + \left(-3\right) \cdot \log x}{n} \]

      5. metadata-eval 56.7%

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

      6. metadata-eval 56.7%

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

      7. *-commutative 56.7%

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

   8. Simplified 56.7%

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

   9. Taylor expanded in x around inf 56.9%

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

       1. log-rec 56.9%

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

       2. log-rec 56.9%

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

       3. distribute-rgt-out 56.9%

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

       4. metadata-eval 56.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\left(-\log x\right) \cdot \color{blue}{0}}{n} \]

       5. mul0-rgt 56.9%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]

   11. Simplified 56.9%

       \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.4 \lor \neg \left(n \leq -3.2 \cdot 10^{-265}\right):\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \end{array} \]

Alternative 15: 47.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -40.0) (* 0.3333333333333333 (/ 0.0 n)) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = 0.3333333333333333 * (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) <= (-40.0d0)) then
        tmp = 0.3333333333333333d0 * (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) <= -40.0) {
		tmp = 0.3333333333333333 * (0.0 / n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -40.0:
		tmp = 0.3333333333333333 * (0.0 / n)
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40.0)
		tmp = Float64(0.3333333333333333 * 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) <= -40.0)
		tmp = 0.3333333333333333 * (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], -40.0], N[(0.3333333333333333 * N[(0.0 / n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -40

   1. Initial program 100.0%

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

      1. flip3-- 0.0%

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

      2. div-inv 0.0%

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

      3. add-exp-log 0.0%

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

      4. log-pow 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-udef 0.0%

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

      7. *-commutative 0.0%

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

      8. un-div-inv 0.0%

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

      9. pow-pow 0.0%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      2. metadata-eval 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. +-commutative 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} \]

      4. pow-sqr 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      5. sqr-neg 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)}\right)} \]

      6. +-commutative 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left(\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      7. associate-+r+ 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + {x}^{\left(\frac{2}{n}\right)}\right) + {\left(x \cdot \left(1 + x\right)\right)}^{\left(\frac{1}{n}\right)}}} \]

   6. Taylor expanded in n around inf 55.9%

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

      1. cancel-sign-sub-inv 55.9%

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

      2. log1p-def 55.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\color{blue}{\mathsf{log1p}\left(x\right)} + 2 \cdot \log \left(1 + x\right)\right) + \left(-3\right) \cdot \log x}{n} \]

      3. log1p-def 55.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\mathsf{log1p}\left(x\right) + 2 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right) + \left(-3\right) \cdot \log x}{n} \]

      4. distribute-rgt1-in 55.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\left(2 + 1\right) \cdot \mathsf{log1p}\left(x\right)} + \left(-3\right) \cdot \log x}{n} \]

      5. metadata-eval 55.9%

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

      6. metadata-eval 55.9%

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

      7. *-commutative 55.9%

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

   8. Simplified 55.9%

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

   9. Taylor expanded in x around inf 54.2%

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

       1. log-rec 54.2%

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

       2. log-rec 54.2%

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

       3. distribute-rgt-out 54.2%

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

       4. metadata-eval 54.2%

          \[\leadsto 0.3333333333333333 \cdot \frac{\left(-\log x\right) \cdot \color{blue}{0}}{n} \]

       5. mul0-rgt 54.2%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]

   11. Simplified 54.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]

   if -40 < (/.f64 1 n)

   1. Initial program 34.5%

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

   2. Taylor expanded in n around inf 64.1%

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

      1. +-rgt-identity 64.1%

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

      2. +-rgt-identity 64.1%

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

      3. log1p-def 64.1%

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

   4. Simplified 64.1%

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

      1. log1p-udef 64.1%

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

      2. +-commutative 64.1%

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

      3. diff-log 64.1%

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

   6. Applied egg-rr 64.1%

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

   7. Taylor expanded in x around inf 44.4%

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

      1. associate-/r* 45.4%

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

   9. Simplified 45.4%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 16: 47.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -40.0) (* 0.3333333333333333 (/ 0.0 n)) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = 0.3333333333333333 * (0.0 / n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-40.0d0)) then
        tmp = 0.3333333333333333d0 * (0.0d0 / n)
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = 0.3333333333333333 * (0.0 / n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -40.0:
		tmp = 0.3333333333333333 * (0.0 / n)
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40.0)
		tmp = Float64(0.3333333333333333 * Float64(0.0 / n));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -40.0)
		tmp = 0.3333333333333333 * (0.0 / n);
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -40.0], N[(0.3333333333333333 * N[(0.0 / n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -40

   1. Initial program 100.0%

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

      1. flip3-- 0.0%

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

      2. div-inv 0.0%

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

      3. add-exp-log 0.0%

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

      4. log-pow 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-udef 0.0%

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

      7. *-commutative 0.0%

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

      8. un-div-inv 0.0%

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

      9. pow-pow 0.0%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      2. metadata-eval 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. +-commutative 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} \]

      4. pow-sqr 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}\right)} \]

      5. sqr-neg 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)}\right)} \]

      6. +-commutative 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left(\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      7. associate-+r+ 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}}} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + {x}^{\left(\frac{2}{n}\right)}\right) + {\left(x \cdot \left(1 + x\right)\right)}^{\left(\frac{1}{n}\right)}}} \]

   6. Taylor expanded in n around inf 55.9%

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

      1. cancel-sign-sub-inv 55.9%

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

      2. log1p-def 55.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\color{blue}{\mathsf{log1p}\left(x\right)} + 2 \cdot \log \left(1 + x\right)\right) + \left(-3\right) \cdot \log x}{n} \]

      3. log1p-def 55.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\left(\mathsf{log1p}\left(x\right) + 2 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right) + \left(-3\right) \cdot \log x}{n} \]

      4. distribute-rgt1-in 55.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\left(2 + 1\right) \cdot \mathsf{log1p}\left(x\right)} + \left(-3\right) \cdot \log x}{n} \]

      5. metadata-eval 55.9%

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

      6. metadata-eval 55.9%

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

      7. *-commutative 55.9%

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

   8. Simplified 55.9%

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

   9. Taylor expanded in x around inf 54.2%

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

       1. log-rec 54.2%

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

       2. log-rec 54.2%

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

       3. distribute-rgt-out 54.2%

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

       4. metadata-eval 54.2%

          \[\leadsto 0.3333333333333333 \cdot \frac{\left(-\log x\right) \cdot \color{blue}{0}}{n} \]

       5. mul0-rgt 54.2%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]

   11. Simplified 54.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]

   if -40 < (/.f64 1 n)

   1. Initial program 34.5%

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

   2. Taylor expanded in n around inf 64.1%

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

      1. +-rgt-identity 64.1%

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

      2. +-rgt-identity 64.1%

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

      3. log1p-def 64.1%

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

   4. Simplified 64.1%

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

   5. Taylor expanded in x around inf 45.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;0.3333333333333333 \cdot \frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 17: 31.1% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{0}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (* 0.3333333333333333 (/ 0.0 n)))

C

double code(double x, double n) {
	return 0.3333333333333333 * (0.0 / n);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.3333333333333333d0 * (0.0d0 / n)
end function

Java

public static double code(double x, double n) {
	return 0.3333333333333333 * (0.0 / n);
}

Python

def code(x, n):
	return 0.3333333333333333 * (0.0 / n)

Julia

function code(x, n)
	return Float64(0.3333333333333333 * Float64(0.0 / n))
end

MATLAB

function tmp = code(x, n)
	tmp = 0.3333333333333333 * (0.0 / n);
end

Wolfram

code[x_, n_] := N[(0.3333333333333333 * N[(0.0 / n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

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

   1. flip3-- 24.9%

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

   2. div-inv 24.9%

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

   3. add-exp-log 24.9%

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

   4. log-pow 24.9%

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

   5. +-commutative 24.9%

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

   6. log1p-udef 29.4%

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

   7. *-commutative 29.4%

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

   8. un-div-inv 29.4%

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

   9. pow-pow 29.5%

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

3. Applied egg-rr 25.6%

   \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
4. Step-by-step derivation

   1. associate-*l/ 25.6%

      \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

   2. metadata-eval 25.6%

      \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({x}^{\left(2 \cdot \frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

   3. +-commutative 25.6%

      \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}} \]

   4. pow-sqr 25.6%

      \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}}\right)} \]

   5. sqr-neg 25.6%

      \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \color{blue}{\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)}\right)} \]

   6. +-commutative 25.6%

      \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \color{blue}{\left(\left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

   7. associate-+r+ 25.6%

      \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left(-{x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(-{x}^{\left(\frac{1}{n}\right)}\right)\right) + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}}} \]

5. Simplified 25.6%

   \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}\right) \cdot \frac{1}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + {x}^{\left(\frac{2}{n}\right)}\right) + {\left(x \cdot \left(1 + x\right)\right)}^{\left(\frac{1}{n}\right)}}} \]

6. Taylor expanded in n around inf 61.7%

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

   1. cancel-sign-sub-inv 61.7%

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

   2. log1p-def 61.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{\left(\color{blue}{\mathsf{log1p}\left(x\right)} + 2 \cdot \log \left(1 + x\right)\right) + \left(-3\right) \cdot \log x}{n} \]

   3. log1p-def 61.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{\left(\mathsf{log1p}\left(x\right) + 2 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right) + \left(-3\right) \cdot \log x}{n} \]

   4. distribute-rgt1-in 61.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\left(2 + 1\right) \cdot \mathsf{log1p}\left(x\right)} + \left(-3\right) \cdot \log x}{n} \]

   5. metadata-eval 61.7%

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

   6. metadata-eval 61.7%

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

   7. *-commutative 61.7%

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

8. Simplified 61.7%

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

9. Taylor expanded in x around inf 32.3%

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

    1. log-rec 32.3%

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

    2. log-rec 32.3%

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

    3. distribute-rgt-out 32.3%

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

    4. metadata-eval 32.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{\left(-\log x\right) \cdot \color{blue}{0}}{n} \]

    5. mul0-rgt 32.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]

11. Simplified 32.3%

    \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{0}}{n} \]

12. Final simplification 32.3%

    \[\leadsto 0.3333333333333333 \cdot \frac{0}{n} \]

Reproduce

herbie shell --seed 2023298 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))