2nthrt (problem 3.4.6)

Percentage Accurate: 53.2% → 80.8%

Time: 20.0s

Alternatives: 15

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: 53.2% 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: 80.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ t_1 := -0.16666666666666666 \cdot {t_0}^{3}\\ t_2 := {x}^{\left(\frac{1}{n}\right)}\\ t_3 := \sqrt[3]{t_2}\\ \mathbf{if}\;x \leq 8.2 \cdot 10^{-153}:\\ \;\;\;\;\left(-0.5 \cdot {\left(\log \left(t_3 \cdot t_3\right) + \log t_3\right)}^{2} + t_1\right) - t_0\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-125}:\\ \;\;\;\;e^{\frac{x}{n}} - t_2\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(t_1 + -0.5 \cdot {t_0}^{2}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n))
        (t_1 (* -0.16666666666666666 (pow t_0 3.0)))
        (t_2 (pow x (/ 1.0 n)))
        (t_3 (cbrt t_2)))
   (if (<= x 8.2e-153)
     (- (+ (* -0.5 (pow (+ (log (* t_3 t_3)) (log t_3)) 2.0)) t_1) t_0)
     (if (<= x 4.7e-125)
       (- (exp (/ x n)) t_2)
       (if (<= x 1.0)
         (- (+ t_1 (* -0.5 (pow t_0 2.0))) t_0)
         (/ t_2 (* x n)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double t_1 = -0.16666666666666666 * pow(t_0, 3.0);
	double t_2 = pow(x, (1.0 / n));
	double t_3 = cbrt(t_2);
	double tmp;
	if (x <= 8.2e-153) {
		tmp = ((-0.5 * pow((log((t_3 * t_3)) + log(t_3)), 2.0)) + t_1) - t_0;
	} else if (x <= 4.7e-125) {
		tmp = exp((x / n)) - t_2;
	} else if (x <= 1.0) {
		tmp = (t_1 + (-0.5 * pow(t_0, 2.0))) - t_0;
	} else {
		tmp = t_2 / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double t_1 = -0.16666666666666666 * Math.pow(t_0, 3.0);
	double t_2 = Math.pow(x, (1.0 / n));
	double t_3 = Math.cbrt(t_2);
	double tmp;
	if (x <= 8.2e-153) {
		tmp = ((-0.5 * Math.pow((Math.log((t_3 * t_3)) + Math.log(t_3)), 2.0)) + t_1) - t_0;
	} else if (x <= 4.7e-125) {
		tmp = Math.exp((x / n)) - t_2;
	} else if (x <= 1.0) {
		tmp = (t_1 + (-0.5 * Math.pow(t_0, 2.0))) - t_0;
	} else {
		tmp = t_2 / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	t_1 = Float64(-0.16666666666666666 * (t_0 ^ 3.0))
	t_2 = x ^ Float64(1.0 / n)
	t_3 = cbrt(t_2)
	tmp = 0.0
	if (x <= 8.2e-153)
		tmp = Float64(Float64(Float64(-0.5 * (Float64(log(Float64(t_3 * t_3)) + log(t_3)) ^ 2.0)) + t_1) - t_0);
	elseif (x <= 4.7e-125)
		tmp = Float64(exp(Float64(x / n)) - t_2);
	elseif (x <= 1.0)
		tmp = Float64(Float64(t_1 + Float64(-0.5 * (t_0 ^ 2.0))) - t_0);
	else
		tmp = Float64(t_2 / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 1/3], $MachinePrecision]}, If[LessEqual[x, 8.2e-153], N[(N[(N[(-0.5 * N[Power[N[(N[Log[N[(t$95$3 * t$95$3), $MachinePrecision]], $MachinePrecision] + N[Log[t$95$3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 4.7e-125], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(t$95$1 + N[(-0.5 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$2 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.2e-153

   1. Initial program 39.8%

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

   2. Taylor expanded in x around 0 39.8%

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

   3. Taylor expanded in n around inf 76.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n} + \left(-0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + -0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)} \]
   4. Step-by-step derivation

      1. neg-mul-1 76.0%

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

      2. +-commutative 76.0%

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

      3. unsub-neg 76.0%

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

   5. Simplified 76.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {\left(\frac{\log x}{n}\right)}^{2}, -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) - \frac{\log x}{n}} \]
   6. Step-by-step derivation

      1. fma-udef 76.0%

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

   7. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {\left(\frac{\log x}{n}\right)}^{2} + -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right)} - \frac{\log x}{n} \]
   8. Step-by-step derivation

      1. add-log-exp 82.5%

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

      2. div-inv 82.5%

         \[\leadsto \left(-0.5 \cdot {\log \left(e^{\color{blue}{\log x \cdot \frac{1}{n}}}\right)}^{2} + -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) - \frac{\log x}{n} \]

      3. pow-to-exp 82.5%

         \[\leadsto \left(-0.5 \cdot {\log \color{blue}{\left({x}^{\left(\frac{1}{n}\right)}\right)}}^{2} + -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) - \frac{\log x}{n} \]

      4. add-cube-cbrt 82.5%

         \[\leadsto \left(-0.5 \cdot {\log \color{blue}{\left(\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}}^{2} + -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) - \frac{\log x}{n} \]

      5. log-prod 82.5%

         \[\leadsto \left(-0.5 \cdot {\color{blue}{\left(\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) + \log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}}^{2} + -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) - \frac{\log x}{n} \]

   9. Applied egg-rr 82.5%

      \[\leadsto \left(-0.5 \cdot {\color{blue}{\left(\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) + \log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}}^{2} + -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) - \frac{\log x}{n} \]

   if 8.2e-153 < x < 4.7e-125

   1. Initial program 76.5%

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

   2. Taylor expanded in n around 0 76.5%

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

      1. log1p-def 91.0%

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

   4. Simplified 91.0%

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

   5. Taylor expanded in x around 0 91.0%

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

   if 4.7e-125 < x < 1

   1. Initial program 23.5%

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

   2. Taylor expanded in x around 0 23.5%

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

   3. Taylor expanded in n around inf 65.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n} + \left(-0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + -0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)} \]
   4. Step-by-step derivation

      1. neg-mul-1 65.9%

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

      2. +-commutative 65.9%

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

      3. unsub-neg 65.9%

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

   5. Simplified 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {\left(\frac{\log x}{n}\right)}^{2}, -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) - \frac{\log x}{n}} \]
   6. Step-by-step derivation

      1. fma-udef 65.9%

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

   7. Applied egg-rr 65.9%

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

   if 1 < x

   1. Initial program 61.1%

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

   2. Taylor expanded in x around inf 97.8%

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

      1. log-rec 97.8%

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

      2. mul-1-neg 97.8%

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

      3. associate-*r/ 97.8%

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

      4. neg-mul-1 97.8%

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

      5. mul-1-neg 97.8%

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

      6. remove-double-neg 97.8%

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

      7. *-commutative 97.8%

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

   4. Simplified 97.8%

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

   5. Taylor expanded in x around 0 97.8%

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

      1. *-rgt-identity 97.8%

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

      2. associate-*r/ 97.8%

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

      3. exp-to-pow 97.8%

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

      4. *-commutative 97.8%

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

   7. Simplified 97.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-153}:\\ \;\;\;\;\left(-0.5 \cdot {\left(\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) + \log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}^{2} + -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-125}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(-0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3} + -0.5 \cdot {\left(\frac{\log x}{n}\right)}^{2}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 2: 79.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ t_1 := {t_0}^{2}\\ t_2 := -0.16666666666666666 \cdot {t_0}^{3}\\ t_3 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 8.2 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, \frac{{\log x}^{4}}{{n}^{4}}, \mathsf{fma}\left(-0.5, t_1, t_2\right)\right) - t_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-125}:\\ \;\;\;\;e^{\frac{x}{n}} - t_3\\ \mathbf{elif}\;x \leq 0.94:\\ \;\;\;\;\left(t_2 + -0.5 \cdot t_1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n))
        (t_1 (pow t_0 2.0))
        (t_2 (* -0.16666666666666666 (pow t_0 3.0)))
        (t_3 (pow x (/ 1.0 n))))
   (if (<= x 8.2e-153)
     (-
      (fma
       -0.041666666666666664
       (/ (pow (log x) 4.0) (pow n 4.0))
       (fma -0.5 t_1 t_2))
      t_0)
     (if (<= x 4.3e-125)
       (- (exp (/ x n)) t_3)
       (if (<= x 0.94) (- (+ t_2 (* -0.5 t_1)) t_0) (/ t_3 (* x n)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double t_1 = pow(t_0, 2.0);
	double t_2 = -0.16666666666666666 * pow(t_0, 3.0);
	double t_3 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 8.2e-153) {
		tmp = fma(-0.041666666666666664, (pow(log(x), 4.0) / pow(n, 4.0)), fma(-0.5, t_1, t_2)) - t_0;
	} else if (x <= 4.3e-125) {
		tmp = exp((x / n)) - t_3;
	} else if (x <= 0.94) {
		tmp = (t_2 + (-0.5 * t_1)) - t_0;
	} else {
		tmp = t_3 / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	t_1 = t_0 ^ 2.0
	t_2 = Float64(-0.16666666666666666 * (t_0 ^ 3.0))
	t_3 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 8.2e-153)
		tmp = Float64(fma(-0.041666666666666664, Float64((log(x) ^ 4.0) / (n ^ 4.0)), fma(-0.5, t_1, t_2)) - t_0);
	elseif (x <= 4.3e-125)
		tmp = Float64(exp(Float64(x / n)) - t_3);
	elseif (x <= 0.94)
		tmp = Float64(Float64(t_2 + Float64(-0.5 * t_1)) - t_0);
	else
		tmp = Float64(t_3 / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(-0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 8.2e-153], N[(N[(-0.041666666666666664 * N[(N[Power[N[Log[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 4.3e-125], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision], If[LessEqual[x, 0.94], N[(N[(t$95$2 + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$3 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.2e-153

   1. Initial program 39.8%

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

   2. Taylor expanded in x around 0 39.8%

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

   3. Taylor expanded in n around inf 78.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n} + \left(-0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + \left(-0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + -0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)\right)} \]
   4. Step-by-step derivation

      1. neg-mul-1 78.5%

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

      2. +-commutative 78.5%

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

      3. unsub-neg 78.5%

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

   5. Simplified 78.5%

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

   if 8.2e-153 < x < 4.3000000000000002e-125

   1. Initial program 76.5%

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

   2. Taylor expanded in n around 0 76.5%

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

      1. log1p-def 91.0%

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

   4. Simplified 91.0%

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

   5. Taylor expanded in x around 0 91.0%

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

   if 4.3000000000000002e-125 < x < 0.93999999999999995

   1. Initial program 23.5%

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

   2. Taylor expanded in x around 0 23.5%

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

   3. Taylor expanded in n around inf 65.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n} + \left(-0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + -0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{3}}\right)} \]
   4. Step-by-step derivation

      1. neg-mul-1 65.9%

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

      2. +-commutative 65.9%

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

      3. unsub-neg 65.9%

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

   5. Simplified 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {\left(\frac{\log x}{n}\right)}^{2}, -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right) - \frac{\log x}{n}} \]
   6. Step-by-step derivation

      1. fma-udef 65.9%

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

   7. Applied egg-rr 65.9%

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

   if 0.93999999999999995 < x

   1. Initial program 61.1%

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

   2. Taylor expanded in x around inf 97.8%

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

      1. log-rec 97.8%

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

      2. mul-1-neg 97.8%

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

      3. associate-*r/ 97.8%

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

      4. neg-mul-1 97.8%

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

      5. mul-1-neg 97.8%

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

      6. remove-double-neg 97.8%

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

      7. *-commutative 97.8%

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

   4. Simplified 97.8%

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

   5. Taylor expanded in x around 0 97.8%

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

      1. *-rgt-identity 97.8%

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

      2. associate-*r/ 97.8%

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

      3. exp-to-pow 97.8%

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

      4. *-commutative 97.8%

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

   7. Simplified 97.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, \frac{{\log x}^{4}}{{n}^{4}}, \mathsf{fma}\left(-0.5, {\left(\frac{\log x}{n}\right)}^{2}, -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\right)\right) - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-125}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.94:\\ \;\;\;\;\left(-0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3} + -0.5 \cdot {\left(\frac{\log x}{n}\right)}^{2}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 3: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\log \left(e^{-{x}^{\left({n}^{-1}\right)}} \cdot e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)\\ \mathbf{elif}\;t_1 \leq 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

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-8)
     (log (* (exp (- (pow x (pow n -1.0)))) (exp (exp (/ (log1p x) n)))))
     (if (<= t_1 1e-8) (/ (log (/ (+ x 1.0) x)) n) (- (exp (/ 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-8) {
		tmp = log((exp(-pow(x, pow(n, -1.0))) * exp(exp((log1p(x) / n)))));
	} else if (t_1 <= 1e-8) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((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-8) {
		tmp = Math.log((Math.exp(-Math.pow(x, Math.pow(n, -1.0))) * Math.exp(Math.exp((Math.log1p(x) / n)))));
	} else if (t_1 <= 1e-8) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -5e-8:
		tmp = math.log((math.exp(-math.pow(x, math.pow(n, -1.0))) * math.exp(math.exp((math.log1p(x) / n)))))
	elif t_1 <= 1e-8:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((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-8)
		tmp = log(Float64(exp(Float64(-(x ^ (n ^ -1.0)))) * exp(exp(Float64(log1p(x) / n)))));
	elseif (t_1 <= 1e-8)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(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-8], N[Log[N[(N[Exp[(-N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision])], $MachinePrecision] * N[Exp[N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e-8], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / 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))) < -4.9999999999999998e-8

   1. Initial program 97.5%

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

      1. sub-neg 97.5%

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

      2. +-commutative 97.5%

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

      3. add-log-exp 97.5%

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

      4. add-log-exp 97.5%

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

      5. sum-log 97.7%

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

      6. inv-pow 97.7%

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

      7. pow-to-exp 97.7%

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

      8. un-div-inv 97.7%

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

      9. +-commutative 97.7%

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

      10. log1p-udef 97.7%

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

   3. Applied egg-rr 97.7%

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

   if -4.9999999999999998e-8 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 1e-8

   1. Initial program 39.2%

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

   2. Taylor expanded in n around inf 76.0%

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

      1. +-rgt-identity 76.0%

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

      2. +-rgt-identity 76.0%

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

      3. log1p-def 76.0%

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

   4. Simplified 76.0%

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

      1. log1p-udef 76.0%

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

      2. diff-log 76.1%

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

      3. +-commutative 76.1%

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

   6. Applied egg-rr 76.1%

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

   if 1e-8 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 39.3%

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

   2. Taylor expanded in n around 0 39.3%

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

      1. log1p-def 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\log \left(e^{-{x}^{\left({n}^{-1}\right)}} \cdot e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}}\right)\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 85.7% accurate, 0.3× speedup

Math

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

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-8)
     (- 1.0 t_0)
     (if (<= t_1 1e-8) (/ (log (/ (+ x 1.0) x)) n) (- (exp (/ 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-8) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 1e-8) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -5e-8:
		tmp = 1.0 - t_0
	elif t_1 <= 1e-8:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((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-8)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 1e-8)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -5e-8)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 1e-8)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = exp((x / n)) - t_0;
	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[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-8], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-8], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / 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))) < -4.9999999999999998e-8

   1. Initial program 97.5%

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

   2. Taylor expanded in x around 0 97.5%

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

   if -4.9999999999999998e-8 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 1e-8

   1. Initial program 39.2%

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

   2. Taylor expanded in n around inf 76.0%

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

      1. +-rgt-identity 76.0%

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

      2. +-rgt-identity 76.0%

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

      3. log1p-def 76.0%

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

   4. Simplified 76.0%

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

      1. log1p-udef 76.0%

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

      2. diff-log 76.1%

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

      3. +-commutative 76.1%

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

   6. Applied egg-rr 76.1%

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

   if 1e-8 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 39.3%

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

   2. Taylor expanded in n around 0 39.3%

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

      1. log1p-def 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

4. Final simplification 82.4%

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

Alternative 5: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-18}:\\ \;\;\;\;\frac{t_1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-254}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t_0 \cdot \frac{t_0}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 n) x)) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-18)
     (/ t_1 (* x n))
     (if (<= (/ 1.0 n) 1e-254)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e-190)
         t_0
         (if (<= (/ 1.0 n) 2e-8)
           (/ (log (/ (+ x 1.0) x)) n)
           (if (<= (/ 1.0 n) 2e+141)
             (- (+ 1.0 (/ x n)) t_1)
             (cbrt (* t_0 (/ t_0 (* x n)))))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-18) {
		tmp = t_1 / (x * n);
	} else if ((1.0 / n) <= 1e-254) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+141) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = cbrt((t_0 * (t_0 / (x * n))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-18) {
		tmp = t_1 / (x * n);
	} else if ((1.0 / n) <= 1e-254) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+141) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = Math.cbrt((t_0 * (t_0 / (x * n))));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / n) / x)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-18)
		tmp = Float64(t_1 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-254)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-190)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+141)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = cbrt(Float64(t_0 * Float64(t_0 / Float64(x * n))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-18], N[(t$95$1 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-254], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-190], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+141], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[Power[N[(t$95$0 * N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -4.0000000000000003e-18

   1. Initial program 94.6%

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

   2. Taylor expanded in x around inf 94.4%

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

      1. log-rec 94.4%

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

      2. mul-1-neg 94.4%

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

      3. associate-*r/ 94.4%

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

      4. neg-mul-1 94.4%

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

      5. mul-1-neg 94.4%

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

      6. remove-double-neg 94.4%

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

      7. *-commutative 94.4%

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

   4. Simplified 94.4%

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

   5. Taylor expanded in x around 0 94.4%

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

      1. *-rgt-identity 94.4%

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

      2. associate-*r/ 94.4%

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

      3. exp-to-pow 94.4%

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

      4. *-commutative 94.4%

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

   7. Simplified 94.4%

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

   if -4.0000000000000003e-18 < (/.f64 1 n) < 9.9999999999999991e-255

   1. Initial program 32.8%

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

   2. Taylor expanded in n around inf 81.4%

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

      1. +-rgt-identity 81.4%

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

      2. +-rgt-identity 81.4%

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

      3. log1p-def 81.4%

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

   4. Simplified 81.4%

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

   if 9.9999999999999991e-255 < (/.f64 1 n) < 5.00000000000000034e-190

   1. Initial program 41.3%

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

   2. Taylor expanded in n around inf 52.3%

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

      1. associate--l+ 52.3%

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

      2. fma-def 52.3%

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

      3. log1p-def 52.3%

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

      4. unpow2 52.3%

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

      5. associate--r+ 52.3%

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

      6. +-rgt-identity 52.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 52.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 52.3%

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

      9. log1p-def 52.3%

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

      10. unpow2 52.3%

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

   4. Simplified 52.3%

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

   5. Taylor expanded in x around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. unsub-neg 88.5%

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

      3. log-rec 88.5%

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

      4. distribute-frac-neg 88.5%

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

      5. unpow2 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in n around inf 88.5%

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

   if 5.00000000000000034e-190 < (/.f64 1 n) < 2e-8

   1. Initial program 16.2%

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

   2. Taylor expanded in n around inf 68.9%

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

      1. +-rgt-identity 68.9%

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

      2. +-rgt-identity 68.9%

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

      3. log1p-def 68.9%

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

   4. Simplified 68.9%

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

      1. log1p-udef 68.9%

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

      2. diff-log 69.2%

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

      3. +-commutative 69.2%

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

   6. Applied egg-rr 69.2%

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

   if 2e-8 < (/.f64 1 n) < 2.00000000000000003e141

   1. Initial program 71.1%

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

   2. Taylor expanded in x around 0 72.7%

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

   if 2.00000000000000003e141 < (/.f64 1 n)

   1. Initial program 17.1%

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

   2. Taylor expanded in n around inf 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

      3. log1p-def 0.0%

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

      4. unpow2 0.0%

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

      5. associate--r+ 0.0%

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

      6. +-rgt-identity 0.0%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 0.0%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 0.0%

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

      9. log1p-def 0.0%

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

      10. unpow2 0.0%

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

   4. Simplified 0.0%

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

   5. Taylor expanded in x around inf 0.1%

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

      1. mul-1-neg 0.1%

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

      2. unsub-neg 0.1%

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

      3. log-rec 0.1%

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

      4. distribute-frac-neg 0.1%

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

      5. unpow2 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in n around inf 63.5%

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

      1. add-cbrt-cube 83.2%

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

      2. associate-/l/ 83.2%

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

      3. associate-/l/ 83.2%

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

      4. associate-/l/ 83.2%

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

   10. Applied egg-rr 83.2%

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

       1. associate-*l* 83.2%

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

       2. associate-/l/ 83.2%

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

       3. associate-*l/ 83.2%

          \[\leadsto \sqrt[3]{\frac{\frac{1}{n}}{x} \cdot \color{blue}{\frac{1 \cdot \frac{1}{x \cdot n}}{x \cdot n}}} \]

       4. *-lft-identity 83.2%

          \[\leadsto \sqrt[3]{\frac{\frac{1}{n}}{x} \cdot \frac{\color{blue}{\frac{1}{x \cdot n}}}{x \cdot n}} \]

       5. associate-/l/ 83.2%

          \[\leadsto \sqrt[3]{\frac{\frac{1}{n}}{x} \cdot \frac{\color{blue}{\frac{\frac{1}{n}}{x}}}{x \cdot n}} \]

   12. Simplified 83.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-18}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-254}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{n}}{x} \cdot \frac{\frac{\frac{1}{n}}{x}}{x \cdot n}}\\ \end{array} \]

Alternative 6: 81.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-18}:\\ \;\;\;\;\frac{t_1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t_0 \cdot \frac{t_0}{x \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 n) x))
        (t_1 (pow x (/ 1.0 n)))
        (t_2 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -4e-18)
     (/ t_1 (* x n))
     (if (<= (/ 1.0 n) 1e-254)
       t_2
       (if (<= (/ 1.0 n) 5e-190)
         t_0
         (if (<= (/ 1.0 n) 2e-8)
           t_2
           (if (<= (/ 1.0 n) 2e+141)
             (- (+ 1.0 (/ x n)) t_1)
             (cbrt (* t_0 (/ t_0 (* x n)))))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double t_1 = pow(x, (1.0 / n));
	double t_2 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-18) {
		tmp = t_1 / (x * n);
	} else if ((1.0 / n) <= 1e-254) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = t_2;
	} else if ((1.0 / n) <= 2e+141) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = cbrt((t_0 * (t_0 / (x * n))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-18) {
		tmp = t_1 / (x * n);
	} else if ((1.0 / n) <= 1e-254) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = t_2;
	} else if ((1.0 / n) <= 2e+141) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = Math.cbrt((t_0 * (t_0 / (x * n))));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / n) / x)
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-18)
		tmp = Float64(t_1 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-254)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 5e-190)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 2e+141)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = cbrt(Float64(t_0 * Float64(t_0 / Float64(x * n))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-18], N[(t$95$1 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-254], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-190], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+141], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[Power[N[(t$95$0 * N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.0000000000000003e-18

   1. Initial program 94.6%

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

   2. Taylor expanded in x around inf 94.4%

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

      1. log-rec 94.4%

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

      2. mul-1-neg 94.4%

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

      3. associate-*r/ 94.4%

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

      4. neg-mul-1 94.4%

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

      5. mul-1-neg 94.4%

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

      6. remove-double-neg 94.4%

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

      7. *-commutative 94.4%

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

   4. Simplified 94.4%

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

   5. Taylor expanded in x around 0 94.4%

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

      1. *-rgt-identity 94.4%

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

      2. associate-*r/ 94.4%

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

      3. exp-to-pow 94.4%

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

      4. *-commutative 94.4%

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

   7. Simplified 94.4%

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

   if -4.0000000000000003e-18 < (/.f64 1 n) < 9.9999999999999991e-255 or 5.00000000000000034e-190 < (/.f64 1 n) < 2e-8

   1. Initial program 26.3%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. +-rgt-identity 76.5%

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

      2. +-rgt-identity 76.5%

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

      3. log1p-def 76.5%

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

   4. Simplified 76.5%

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

      1. log1p-udef 76.5%

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

      2. diff-log 76.6%

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

      3. +-commutative 76.6%

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

   6. Applied egg-rr 76.6%

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

   if 9.9999999999999991e-255 < (/.f64 1 n) < 5.00000000000000034e-190

   1. Initial program 41.3%

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

   2. Taylor expanded in n around inf 52.3%

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

      1. associate--l+ 52.3%

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

      2. fma-def 52.3%

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

      3. log1p-def 52.3%

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

      4. unpow2 52.3%

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

      5. associate--r+ 52.3%

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

      6. +-rgt-identity 52.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 52.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 52.3%

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

      9. log1p-def 52.3%

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

      10. unpow2 52.3%

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

   4. Simplified 52.3%

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

   5. Taylor expanded in x around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. unsub-neg 88.5%

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

      3. log-rec 88.5%

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

      4. distribute-frac-neg 88.5%

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

      5. unpow2 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in n around inf 88.5%

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

   if 2e-8 < (/.f64 1 n) < 2.00000000000000003e141

   1. Initial program 71.1%

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

   2. Taylor expanded in x around 0 72.7%

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

   if 2.00000000000000003e141 < (/.f64 1 n)

   1. Initial program 17.1%

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

   2. Taylor expanded in n around inf 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

      3. log1p-def 0.0%

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

      4. unpow2 0.0%

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

      5. associate--r+ 0.0%

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

      6. +-rgt-identity 0.0%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 0.0%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 0.0%

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

      9. log1p-def 0.0%

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

      10. unpow2 0.0%

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

   4. Simplified 0.0%

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

   5. Taylor expanded in x around inf 0.1%

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

      1. mul-1-neg 0.1%

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

      2. unsub-neg 0.1%

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

      3. log-rec 0.1%

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

      4. distribute-frac-neg 0.1%

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

      5. unpow2 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in n around inf 63.5%

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

      1. add-cbrt-cube 83.2%

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

      2. associate-/l/ 83.2%

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

      3. associate-/l/ 83.2%

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

      4. associate-/l/ 83.2%

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

   10. Applied egg-rr 83.2%

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

       1. associate-*l* 83.2%

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

       2. associate-/l/ 83.2%

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

       3. associate-*l/ 83.2%

          \[\leadsto \sqrt[3]{\frac{\frac{1}{n}}{x} \cdot \color{blue}{\frac{1 \cdot \frac{1}{x \cdot n}}{x \cdot n}}} \]

       4. *-lft-identity 83.2%

          \[\leadsto \sqrt[3]{\frac{\frac{1}{n}}{x} \cdot \frac{\color{blue}{\frac{1}{x \cdot n}}}{x \cdot n}} \]

       5. associate-/l/ 83.2%

          \[\leadsto \sqrt[3]{\frac{\frac{1}{n}}{x} \cdot \frac{\color{blue}{\frac{\frac{1}{n}}{x}}}{x \cdot n}} \]

   12. Simplified 83.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-18}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-254}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{n}}{x} \cdot \frac{\frac{\frac{1}{n}}{x}}{x \cdot n}}\\ \end{array} \]

Alternative 7: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-18}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -4e-18)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-254)
       t_1
       (if (<= (/ 1.0 n) 5e-190)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 2e-8)
           t_1
           (if (<= (/ 1.0 n) 2e+141)
             (- (+ 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 + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-18) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-254) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+141) {
		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 + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-4d-18)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 1d-254) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-190) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 2d-8) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d+141) 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 + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-18) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-254) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+141) {
		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 + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -4e-18:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 1e-254:
		tmp = t_1
	elif (1.0 / n) <= 5e-190:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 2e-8:
		tmp = t_1
	elif (1.0 / n) <= 2e+141:
		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(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-18)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-254)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-190)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+141)
		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 + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -4e-18)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 1e-254)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-190)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 2e-8)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e+141)
		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[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-18], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-254], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-190], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+141], 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 5 regimes

2. if (/.f64 1 n) < -4.0000000000000003e-18

   1. Initial program 94.6%

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

   2. Taylor expanded in x around inf 94.4%

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

      1. log-rec 94.4%

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

      2. mul-1-neg 94.4%

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

      3. associate-*r/ 94.4%

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

      4. neg-mul-1 94.4%

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

      5. mul-1-neg 94.4%

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

      6. remove-double-neg 94.4%

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

      7. *-commutative 94.4%

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

   4. Simplified 94.4%

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

   5. Taylor expanded in x around 0 94.4%

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

      1. *-rgt-identity 94.4%

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

      2. associate-*r/ 94.4%

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

      3. exp-to-pow 94.4%

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

      4. *-commutative 94.4%

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

   7. Simplified 94.4%

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

   if -4.0000000000000003e-18 < (/.f64 1 n) < 9.9999999999999991e-255 or 5.00000000000000034e-190 < (/.f64 1 n) < 2e-8

   1. Initial program 26.3%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. +-rgt-identity 76.5%

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

      2. +-rgt-identity 76.5%

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

      3. log1p-def 76.5%

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

   4. Simplified 76.5%

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

      1. log1p-udef 76.5%

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

      2. diff-log 76.6%

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

      3. +-commutative 76.6%

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

   6. Applied egg-rr 76.6%

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

   if 9.9999999999999991e-255 < (/.f64 1 n) < 5.00000000000000034e-190

   1. Initial program 41.3%

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

   2. Taylor expanded in n around inf 52.3%

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

      1. associate--l+ 52.3%

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

      2. fma-def 52.3%

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

      3. log1p-def 52.3%

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

      4. unpow2 52.3%

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

      5. associate--r+ 52.3%

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

      6. +-rgt-identity 52.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 52.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 52.3%

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

      9. log1p-def 52.3%

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

      10. unpow2 52.3%

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

   4. Simplified 52.3%

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

   5. Taylor expanded in x around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. unsub-neg 88.5%

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

      3. log-rec 88.5%

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

      4. distribute-frac-neg 88.5%

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

      5. unpow2 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in n around inf 88.5%

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

   if 2e-8 < (/.f64 1 n) < 2.00000000000000003e141

   1. Initial program 71.1%

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

   2. Taylor expanded in x around 0 72.7%

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

   if 2.00000000000000003e141 < (/.f64 1 n)

   1. Initial program 17.1%

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

   2. Taylor expanded in n around inf 7.1%

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

      1. +-rgt-identity 7.1%

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

      2. +-rgt-identity 7.1%

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

      3. log1p-def 7.1%

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

   4. Simplified 7.1%

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

   5. Taylor expanded in x around inf 63.5%

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

      1. *-commutative 63.5%

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

   7. Simplified 63.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-18}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-254}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \]

Alternative 8: 79.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-18}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+141}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -4e-18)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-254)
       t_1
       (if (<= (/ 1.0 n) 5e-190)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 2e-8)
           t_1
           (if (<= (/ 1.0 n) 2e+141) (- 1.0 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 + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-18) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-254) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+141) {
		tmp = 1.0 - 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 + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-4d-18)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 1d-254) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-190) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 2d-8) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d+141) then
        tmp = 1.0d0 - 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 + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-18) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-254) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-190) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-8) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+141) {
		tmp = 1.0 - 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 + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -4e-18:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 1e-254:
		tmp = t_1
	elif (1.0 / n) <= 5e-190:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 2e-8:
		tmp = t_1
	elif (1.0 / n) <= 2e+141:
		tmp = 1.0 - 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(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-18)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-254)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-190)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-8)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+141)
		tmp = Float64(1.0 - 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 + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -4e-18)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 1e-254)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-190)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 2e-8)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e+141)
		tmp = 1.0 - 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[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-18], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-254], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-190], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-8], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+141], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.0000000000000003e-18

   1. Initial program 94.6%

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

   2. Taylor expanded in x around inf 94.4%

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

      1. log-rec 94.4%

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

      2. mul-1-neg 94.4%

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

      3. associate-*r/ 94.4%

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

      4. neg-mul-1 94.4%

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

      5. mul-1-neg 94.4%

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

      6. remove-double-neg 94.4%

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

      7. *-commutative 94.4%

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

   4. Simplified 94.4%

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

   5. Taylor expanded in x around 0 94.4%

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

      1. *-rgt-identity 94.4%

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

      2. associate-*r/ 94.4%

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

      3. exp-to-pow 94.4%

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

      4. *-commutative 94.4%

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

   7. Simplified 94.4%

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

   if -4.0000000000000003e-18 < (/.f64 1 n) < 9.9999999999999991e-255 or 5.00000000000000034e-190 < (/.f64 1 n) < 2e-8

   1. Initial program 26.3%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. +-rgt-identity 76.5%

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

      2. +-rgt-identity 76.5%

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

      3. log1p-def 76.5%

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

   4. Simplified 76.5%

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

      1. log1p-udef 76.5%

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

      2. diff-log 76.6%

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

      3. +-commutative 76.6%

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

   6. Applied egg-rr 76.6%

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

   if 9.9999999999999991e-255 < (/.f64 1 n) < 5.00000000000000034e-190

   1. Initial program 41.3%

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

   2. Taylor expanded in n around inf 52.3%

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

      1. associate--l+ 52.3%

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

      2. fma-def 52.3%

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

      3. log1p-def 52.3%

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

      4. unpow2 52.3%

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

      5. associate--r+ 52.3%

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

      6. +-rgt-identity 52.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 52.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 52.3%

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

      9. log1p-def 52.3%

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

      10. unpow2 52.3%

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

   4. Simplified 52.3%

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

   5. Taylor expanded in x around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. unsub-neg 88.5%

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

      3. log-rec 88.5%

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

      4. distribute-frac-neg 88.5%

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

      5. unpow2 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in n around inf 88.5%

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

   if 2e-8 < (/.f64 1 n) < 2.00000000000000003e141

   1. Initial program 71.1%

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

   2. Taylor expanded in x around 0 70.9%

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

   if 2.00000000000000003e141 < (/.f64 1 n)

   1. Initial program 17.1%

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

   2. Taylor expanded in n around inf 7.1%

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

      1. +-rgt-identity 7.1%

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

      2. +-rgt-identity 7.1%

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

      3. log1p-def 7.1%

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

   4. Simplified 7.1%

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

   5. Taylor expanded in x around inf 63.5%

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

      1. *-commutative 63.5%

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

   7. Simplified 63.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-18}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-254}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+141}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \]

Alternative 9: 60.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-131}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 780:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.8e-164)
   (/ (- (log x)) n)
   (if (<= x 4.1e-131)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 780.0)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= x 3e+176) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.8e-164) {
		tmp = -log(x) / n;
	} else if (x <= 4.1e-131) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 780.0) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if (x <= 3e+176) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.8d-164) then
        tmp = -log(x) / n
    else if (x <= 4.1d-131) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 780.0d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if (x <= 3d+176) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.8e-164) {
		tmp = -Math.log(x) / n;
	} else if (x <= 4.1e-131) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 780.0) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if (x <= 3e+176) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.8e-164:
		tmp = -math.log(x) / n
	elif x <= 4.1e-131:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 780.0:
		tmp = math.log(((x + 1.0) / x)) / n
	elif x <= 3e+176:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.8e-164)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 4.1e-131)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 780.0)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (x <= 3e+176)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.8e-164)
		tmp = -log(x) / n;
	elseif (x <= 4.1e-131)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 780.0)
		tmp = log(((x + 1.0) / x)) / n;
	elseif (x <= 3e+176)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.8e-164], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 4.1e-131], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 780.0], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3e+176], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 3.79999999999999989e-164

   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 61.7%

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

      1. +-rgt-identity 61.7%

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

      2. +-rgt-identity 61.7%

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

      3. log1p-def 61.7%

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

   4. Simplified 61.7%

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

   5. Taylor expanded in x around 0 61.7%

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

      1. neg-mul-1 61.7%

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

   7. Simplified 61.7%

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

   if 3.79999999999999989e-164 < x < 4.1000000000000002e-131

   1. Initial program 79.8%

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

   2. Taylor expanded in x around 0 79.8%

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

   if 4.1000000000000002e-131 < x < 780

   1. Initial program 23.4%

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

   2. Taylor expanded in n around inf 56.9%

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

      1. +-rgt-identity 56.9%

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

      2. +-rgt-identity 56.9%

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

      3. log1p-def 56.9%

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

   4. Simplified 56.9%

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

      1. log1p-udef 56.9%

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

      2. diff-log 56.9%

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

      3. +-commutative 56.9%

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

   6. Applied egg-rr 56.9%

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

   if 780 < x < 3e176

   1. Initial program 45.6%

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

   2. Taylor expanded in n around inf 44.8%

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

      1. +-rgt-identity 44.8%

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

      2. +-rgt-identity 44.8%

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

      3. log1p-def 44.8%

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

   4. Simplified 44.8%

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

   5. Taylor expanded in x around inf 72.4%

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

      1. associate-*r/ 72.4%

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

      2. metadata-eval 72.4%

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

      3. unpow2 72.4%

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

   7. Simplified 72.4%

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

   if 3e176 < x

   1. Initial program 83.6%

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

   2. Taylor expanded in n around inf 83.6%

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

      1. +-rgt-identity 83.6%

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

      2. +-rgt-identity 83.6%

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

      3. log1p-def 83.6%

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

   4. Simplified 83.6%

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

      1. log1p-udef 83.6%

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

      2. diff-log 83.6%

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

      3. +-commutative 83.6%

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

   6. Applied egg-rr 83.6%

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

   7. Taylor expanded in x around inf 83.6%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-131}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 780:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 10: 60.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-131}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.6e-164)
   (/ (- (log x)) n)
   (if (<= x 9e-131)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 1.0)
       (/ (- x (log x)) n)
       (if (<= x 3e+176) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.6e-164) {
		tmp = -log(x) / n;
	} else if (x <= 9e-131) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 3e+176) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 6.6d-164) then
        tmp = -log(x) / n
    else if (x <= 9d-131) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 3d+176) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6.6e-164) {
		tmp = -Math.log(x) / n;
	} else if (x <= 9e-131) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 3e+176) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6.6e-164:
		tmp = -math.log(x) / n
	elif x <= 9e-131:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 3e+176:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.6e-164)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 9e-131)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 3e+176)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6.6e-164)
		tmp = -log(x) / n;
	elseif (x <= 9e-131)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 3e+176)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.6e-164], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 9e-131], 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, 3e+176], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 6.6e-164

   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 61.7%

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

      1. +-rgt-identity 61.7%

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

      2. +-rgt-identity 61.7%

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

      3. log1p-def 61.7%

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

   4. Simplified 61.7%

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

   5. Taylor expanded in x around 0 61.7%

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

      1. neg-mul-1 61.7%

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

   7. Simplified 61.7%

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

   if 6.6e-164 < x < 9.0000000000000004e-131

   1. Initial program 79.8%

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

   2. Taylor expanded in x around 0 79.8%

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

   if 9.0000000000000004e-131 < x < 1

   1. Initial program 23.4%

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

   2. Taylor expanded in n around inf 56.9%

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

      1. +-rgt-identity 56.9%

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

      2. +-rgt-identity 56.9%

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

      3. log1p-def 56.9%

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

   4. Simplified 56.9%

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

   5. Taylor expanded in x around 0 54.1%

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

      1. neg-mul-1 54.1%

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

      2. unsub-neg 54.1%

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

   7. Simplified 54.1%

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

   if 1 < x < 3e176

   1. Initial program 45.6%

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

   2. Taylor expanded in n around inf 44.8%

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

      1. +-rgt-identity 44.8%

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

      2. +-rgt-identity 44.8%

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

      3. log1p-def 44.8%

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

   4. Simplified 44.8%

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

   5. Taylor expanded in x around inf 72.4%

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

      1. associate-*r/ 72.4%

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

      2. metadata-eval 72.4%

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

      3. unpow2 72.4%

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

   7. Simplified 72.4%

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

   if 3e176 < x

   1. Initial program 83.6%

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

   2. Taylor expanded in n around inf 83.6%

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

      1. +-rgt-identity 83.6%

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

      2. +-rgt-identity 83.6%

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

      3. log1p-def 83.6%

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

   4. Simplified 83.6%

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

      1. log1p-udef 83.6%

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

      2. diff-log 83.6%

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

      3. +-commutative 83.6%

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

   6. Applied egg-rr 83.6%

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

   7. Taylor expanded in x around inf 83.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-131}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 11: 61.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.94:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.94)
   (/ (- x (log x)) n)
   (if (<= x 1.55e+176) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.94) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.55e+176) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.94d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.55d+176) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.94) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.55e+176) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.94:
		tmp = (x - math.log(x)) / n
	elif x <= 1.55e+176:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.94)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.55e+176)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.94)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.55e+176)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.94], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.55e+176], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.93999999999999995

   1. Initial program 36.8%

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

   2. Taylor expanded in n around inf 54.5%

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

      1. +-rgt-identity 54.5%

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

      2. +-rgt-identity 54.5%

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

      3. log1p-def 54.5%

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

   4. Simplified 54.5%

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

   5. Taylor expanded in x around 0 53.2%

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

      1. neg-mul-1 53.2%

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

      2. unsub-neg 53.2%

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

   7. Simplified 53.2%

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

   if 0.93999999999999995 < x < 1.55e176

   1. Initial program 45.6%

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

   2. Taylor expanded in n around inf 44.8%

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

      1. +-rgt-identity 44.8%

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

      2. +-rgt-identity 44.8%

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

      3. log1p-def 44.8%

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

   4. Simplified 44.8%

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

   5. Taylor expanded in x around inf 72.4%

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

      1. associate-*r/ 72.4%

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

      2. metadata-eval 72.4%

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

      3. unpow2 72.4%

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

   7. Simplified 72.4%

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

   if 1.55e176 < x

   1. Initial program 83.6%

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

   2. Taylor expanded in n around inf 83.6%

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

      1. +-rgt-identity 83.6%

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

      2. +-rgt-identity 83.6%

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

      3. log1p-def 83.6%

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

   4. Simplified 83.6%

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

      1. log1p-udef 83.6%

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

      2. diff-log 83.6%

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

      3. +-commutative 83.6%

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

   6. Applied egg-rr 83.6%

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

   7. Taylor expanded in x around inf 83.6%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.94:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 12: 61.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.7)
   (/ (- (log x)) n)
   (if (<= x 2.4e+176) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -log(x) / n;
	} else if (x <= 2.4e+176) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = -log(x) / n
    else if (x <= 2.4d+176) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.4e+176) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.7:
		tmp = -math.log(x) / n
	elif x <= 2.4e+176:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.4e+176)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = -log(x) / n;
	elseif (x <= 2.4e+176)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.7], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.4e+176], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.69999999999999996

   1. Initial program 36.8%

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

   2. Taylor expanded in n around inf 54.5%

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

      1. +-rgt-identity 54.5%

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

      2. +-rgt-identity 54.5%

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

      3. log1p-def 54.5%

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

   4. Simplified 54.5%

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

   5. Taylor expanded in x around 0 51.9%

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

      1. neg-mul-1 51.9%

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

   7. Simplified 51.9%

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

   if 0.69999999999999996 < x < 2.4000000000000001e176

   1. Initial program 45.6%

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

   2. Taylor expanded in n around inf 44.8%

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

      1. +-rgt-identity 44.8%

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

      2. +-rgt-identity 44.8%

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

      3. log1p-def 44.8%

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

   4. Simplified 44.8%

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

   5. Taylor expanded in x around inf 72.4%

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

      1. associate-*r/ 72.4%

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

      2. metadata-eval 72.4%

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

      3. unpow2 72.4%

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

   7. Simplified 72.4%

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

   if 2.4000000000000001e176 < x

   1. Initial program 83.6%

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

   2. Taylor expanded in n around inf 83.6%

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

      1. +-rgt-identity 83.6%

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

      2. +-rgt-identity 83.6%

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

      3. log1p-def 83.6%

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

   4. Simplified 83.6%

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

      1. log1p-udef 83.6%

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

      2. diff-log 83.6%

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

      3. +-commutative 83.6%

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

   6. Applied egg-rr 83.6%

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

   7. Taylor expanded in x around inf 83.6%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 13: 47.0% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.55:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -9 \cdot 10^{-216}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -3.55)
   (/ (/ 1.0 n) x)
   (if (<= n -9e-216) (/ 0.0 n) (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -3.55) {
		tmp = (1.0 / n) / x;
	} else if (n <= -9e-216) {
		tmp = 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 (n <= (-3.55d0)) then
        tmp = (1.0d0 / n) / x
    else if (n <= (-9d-216)) then
        tmp = 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 (n <= -3.55) {
		tmp = (1.0 / n) / x;
	} else if (n <= -9e-216) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -3.55:
		tmp = (1.0 / n) / x
	elif n <= -9e-216:
		tmp = 0.0 / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -3.55)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (n <= -9e-216)
		tmp = 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 (n <= -3.55)
		tmp = (1.0 / n) / x;
	elseif (n <= -9e-216)
		tmp = 0.0 / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -3.55], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -9e-216], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.5499999999999998

   1. Initial program 34.0%

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

   2. Taylor expanded in n around inf 75.2%

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

      1. associate--l+ 68.2%

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

      2. fma-def 68.2%

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

      3. log1p-def 68.2%

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

      4. unpow2 68.2%

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

      5. associate--r+ 75.2%

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

      6. +-rgt-identity 75.2%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 75.2%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 75.2%

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

      9. log1p-def 75.2%

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

      10. unpow2 75.2%

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

   4. Simplified 75.2%

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

   5. Taylor expanded in x around inf 53.1%

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

      1. mul-1-neg 53.1%

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

      2. unsub-neg 53.1%

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

      3. log-rec 53.1%

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

      4. distribute-frac-neg 53.1%

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

      5. unpow2 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in n around inf 52.7%

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

   if -3.5499999999999998 < n < -8.9999999999999997e-216

   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 54.5%

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

      1. +-rgt-identity 54.5%

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

      2. +-rgt-identity 54.5%

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

      3. log1p-def 54.5%

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

   4. Simplified 54.5%

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

      1. log1p-udef 54.5%

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

      2. diff-log 54.5%

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

      3. +-commutative 54.5%

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

   6. Applied egg-rr 54.5%

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

   7. Taylor expanded in x around inf 56.0%

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

   if -8.9999999999999997e-216 < n

   1. Initial program 40.0%

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

   2. Taylor expanded in n around inf 49.5%

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

      1. +-rgt-identity 49.5%

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

      2. +-rgt-identity 49.5%

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

      3. log1p-def 49.5%

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

   4. Simplified 49.5%

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

   5. Taylor expanded in x around inf 49.4%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.55:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -9 \cdot 10^{-216}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 14: 41.0% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* x n)))

C

double code(double x, double n) {
	return 1.0 / (x * n);
}

Fortran

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

Java

public static double code(double x, double n) {
	return 1.0 / (x * n);
}

Python

def code(x, n):
	return 1.0 / (x * n)

Julia

function code(x, n)
	return Float64(1.0 / Float64(x * n))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (x * n);
end

Wolfram

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

Derivation

1. Initial program 47.2%

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

2. Taylor expanded in n around inf 57.1%

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

   1. +-rgt-identity 57.1%

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

   2. +-rgt-identity 57.1%

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

   3. log1p-def 57.1%

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

4. Simplified 57.1%

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

5. Taylor expanded in x around inf 44.1%

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

   1. *-commutative 44.1%

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

7. Simplified 44.1%

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

8. Final simplification 44.1%

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

Alternative 15: 41.5% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

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

Fortran

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

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.2%

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

2. Taylor expanded in n around inf 51.9%

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

   1. associate--l+ 49.2%

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

   2. fma-def 49.2%

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

   3. log1p-def 49.2%

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

   4. unpow2 49.2%

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

   5. associate--r+ 51.9%

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

   6. +-rgt-identity 51.9%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

   7. div-sub 51.9%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

   8. +-rgt-identity 51.9%

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

   9. log1p-def 51.9%

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

   10. unpow2 51.9%

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

4. Simplified 51.9%

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

5. Taylor expanded in x around inf 42.6%

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

   1. mul-1-neg 42.6%

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

   2. unsub-neg 42.6%

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

   3. log-rec 42.6%

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

   4. distribute-frac-neg 42.6%

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

   5. unpow2 42.6%

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

7. Simplified 42.6%

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

8. Taylor expanded in n around inf 44.9%

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

9. Final simplification 44.9%

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

Reproduce

herbie shell --seed 2023201 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))