2nthrt (problem 3.4.6)

Percentage Accurate: 54.1% → 86.8%

Time: 21.6s

Alternatives: 17

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -6 \cdot 10^{-23}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -6e-23)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-10)
       (+
        (/ (- (log1p x) (log x)) n)
        (*
         0.5
         (-
          (/ (pow (log1p x) 2.0) (pow n 2.0))
          (/ (pow (log x) 2.0) (pow n 2.0)))))
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -6e-23) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-10) {
		tmp = ((log1p(x) - log(x)) / n) + (0.5 * ((pow(log1p(x), 2.0) / pow(n, 2.0)) - (pow(log(x), 2.0) / pow(n, 2.0))));
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -6e-23) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-10) {
		tmp = ((Math.log1p(x) - Math.log(x)) / n) + (0.5 * ((Math.pow(Math.log1p(x), 2.0) / Math.pow(n, 2.0)) - (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))));
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -6e-23:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-10:
		tmp = ((math.log1p(x) - math.log(x)) / n) + (0.5 * ((math.pow(math.log1p(x), 2.0) / math.pow(n, 2.0)) - (math.pow(math.log(x), 2.0) / math.pow(n, 2.0))))
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -6e-23)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(Float64(Float64(log1p(x) - log(x)) / n) + Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) / (n ^ 2.0)) - Float64((log(x) ^ 2.0) / (n ^ 2.0)))));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -6e-23], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -6.00000000000000006e-23

   1. Initial program 95.9%

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

   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. mul-1-neg 97.8%

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

      2. log-rec 97.8%

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

      3. mul-1-neg 97.8%

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

      4. distribute-neg-frac 97.8%

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

      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 inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. log-rec 97.8%

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

      3. *-rgt-identity 97.8%

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

      4. associate-*r/ 97.8%

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

      5. distribute-lft-neg-in 97.8%

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

      6. remove-double-neg 97.8%

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

      7. exp-to-pow 97.9%

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

   7. Simplified 97.9%

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

   if -6.00000000000000006e-23 < (/.f64 1 n) < 2.00000000000000007e-10

   1. Initial program 26.8%

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

   2. Taylor expanded in n around inf 81.4%

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

      1. associate--l+ 76.2%

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

      2. +-commutative 76.2%

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

      3. associate--r+ 81.4%

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

      4. div-sub 81.5%

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

      5. remove-double-neg 81.5%

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

      6. mul-1-neg 81.5%

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

      7. distribute-lft-out-- 81.5%

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

      8. distribute-neg-frac 81.5%

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

      9. mul-1-neg 81.5%

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

   4. Simplified 81.5%

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

   if 2.00000000000000007e-10 < (/.f64 1 n)

   1. Initial program 45.1%

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

   2. Taylor expanded in n around 0 45.1%

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

      1. log1p-def 97.5%

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

   4. Simplified 97.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -6 \cdot 10^{-23}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (- (log (/ x (+ 1.0 x)))) n)
       (log (exp (- (exp (/ (log1p x) n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -log((x / (1.0 + x))) / n;
	} else {
		tmp = log(exp((exp((log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -Math.log((x / (1.0 + x))) / n;
	} else {
		tmp = Math.log(Math.exp((Math.exp((Math.log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = -math.log((x / (1.0 + x))) / n
	else:
		tmp = math.log(math.exp((math.exp((math.log1p(x) / n)) - t_0)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n);
	else
		tmp = log(exp(Float64(exp(Float64(log1p(x) / n)) - t_0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[Log[N[Exp[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. associate-*r/ 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. remove-double-neg 98.9%

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

      7. exp-to-pow 98.9%

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

   7. Simplified 98.9%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.99999999999999988e-11

   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 80.6%

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

      1. +-rgt-identity 80.6%

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

      2. +-rgt-identity 80.6%

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

      3. log1p-def 80.6%

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

   4. Simplified 80.6%

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

      1. log1p-udef 80.6%

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

      2. diff-log 80.7%

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

      3. +-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

      1. clear-num 80.7%

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

      2. log-rec 80.7%

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

   8. Applied egg-rr 80.7%

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

   if 1.99999999999999988e-11 < (/.f64 1 n)

   1. Initial program 45.6%

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

      1. add-log-exp 45.7%

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

      2. add-exp-log 45.7%

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

      3. log-pow 45.7%

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

      4. +-commutative 45.7%

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

      5. log1p-udef 96.7%

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

      6. *-commutative 96.7%

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

      7. un-div-inv 96.7%

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

   3. Applied egg-rr 96.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array} \]

Alternative 3: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (- (log (/ x (+ 1.0 x)))) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -log((x / (1.0 + x))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -Math.log((x / (1.0 + x))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = -math.log((x / (1.0 + x))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. associate-*r/ 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. remove-double-neg 98.9%

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

      7. exp-to-pow 98.9%

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

   7. Simplified 98.9%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.99999999999999988e-11

   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 80.6%

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

      1. +-rgt-identity 80.6%

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

      2. +-rgt-identity 80.6%

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

      3. log1p-def 80.6%

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

   4. Simplified 80.6%

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

      1. log1p-udef 80.6%

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

      2. diff-log 80.7%

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

      3. +-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

      1. clear-num 80.7%

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

      2. log-rec 80.7%

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

   8. Applied egg-rr 80.7%

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

   if 1.99999999999999988e-11 < (/.f64 1 n)

   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 0 45.6%

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

      1. log1p-def 96.6%

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

   4. Simplified 96.6%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (- (log (/ x (+ 1.0 x)))) n)
       (if (<= (/ 1.0 n) 5e+125)
         (- (+ 1.0 (/ x n)) t_0)
         (sqrt (pow (* n x) -2.0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -log((x / (1.0 + x))) / n;
	} else if ((1.0 / n) <= 5e+125) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-11)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-11) then
        tmp = -log((x / (1.0d0 + x))) / n
    else if ((1.0d0 / n) <= 5d+125) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = sqrt(((n * x) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -Math.log((x / (1.0 + x))) / n;
	} else if ((1.0 / n) <= 5e+125) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = -math.log((x / (1.0 + x))) / n
	elif (1.0 / n) <= 5e+125:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n);
	elseif (Float64(1.0 / n) <= 5e+125)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-11)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-11)
		tmp = -log((x / (1.0 + x))) / n;
	elseif ((1.0 / n) <= 5e+125)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = sqrt(((n * x) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+125], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. associate-*r/ 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. remove-double-neg 98.9%

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

      7. exp-to-pow 98.9%

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

   7. Simplified 98.9%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.99999999999999988e-11

   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 80.6%

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

      1. +-rgt-identity 80.6%

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

      2. +-rgt-identity 80.6%

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

      3. log1p-def 80.6%

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

   4. Simplified 80.6%

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

      1. log1p-udef 80.6%

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

      2. diff-log 80.7%

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

      3. +-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

      1. clear-num 80.7%

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

      2. log-rec 80.7%

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

   8. Applied egg-rr 80.7%

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

   if 1.99999999999999988e-11 < (/.f64 1 n) < 4.99999999999999962e125

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0 74.2%

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

   if 4.99999999999999962e125 < (/.f64 1 n)

   1. Initial program 18.8%

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

   2. Taylor expanded in n around inf 7.8%

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

      1. +-rgt-identity 7.8%

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

      2. +-rgt-identity 7.8%

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

      3. log1p-def 7.8%

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

   4. Simplified 7.8%

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

   5. Taylor expanded in x around inf 62.2%

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

      1. add-sqr-sqrt 62.2%

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

      2. sqrt-unprod 85.5%

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

      3. inv-pow 85.5%

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

      4. inv-pow 85.5%

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

      5. pow-prod-up 85.5%

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

      6. *-commutative 85.5%

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

      7. metadata-eval 85.5%

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

   7. Applied egg-rr 85.5%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]

Alternative 5: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (- (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 tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		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) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-11)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-11) 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 tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		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))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		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)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(-log(Float64(x / Float64(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);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-11)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-11)
		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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $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 1 n) < -5.00000000000000018e-11

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. associate-*r/ 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. remove-double-neg 98.9%

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

      7. exp-to-pow 98.9%

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

   7. Simplified 98.9%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.99999999999999988e-11

   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 80.6%

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

      1. +-rgt-identity 80.6%

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

      2. +-rgt-identity 80.6%

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

      3. log1p-def 80.6%

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

   4. Simplified 80.6%

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

      1. log1p-udef 80.6%

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

      2. diff-log 80.7%

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

      3. +-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

      1. clear-num 80.7%

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

      2. log-rec 80.7%

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

   8. Applied egg-rr 80.7%

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

   if 1.99999999999999988e-11 < (/.f64 1 n)

   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 0 45.6%

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

      1. log1p-def 96.6%

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

   4. Simplified 96.6%

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

   5. Taylor expanded in x around 0 96.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6: 56.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 4 \cdot 10^{-272}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.04 \cdot 10^{-232}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-196}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (- (log x)) n)))
   (if (<= x 4e-272)
     t_0
     (if (<= x 1.1e-241)
       t_1
       (if (<= x 1.04e-232)
         (/ 1.0 (* n x))
         (if (<= x 2.5e-196)
           (* (log x) (/ 1.0 (- n)))
           (if (<= x 6e-167)
             t_0
             (if (<= x 3.1e-103)
               t_1
               (if (<= x 3.9e-75)
                 t_0
                 (if (<= x 2.6e-35) t_1 (if (<= x 1.0) t_0 (/ 0.0 n))))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = -log(x) / n;
	double tmp;
	if (x <= 4e-272) {
		tmp = t_0;
	} else if (x <= 1.1e-241) {
		tmp = t_1;
	} else if (x <= 1.04e-232) {
		tmp = 1.0 / (n * x);
	} else if (x <= 2.5e-196) {
		tmp = log(x) * (1.0 / -n);
	} else if (x <= 6e-167) {
		tmp = t_0;
	} else if (x <= 3.1e-103) {
		tmp = t_1;
	} else if (x <= 3.9e-75) {
		tmp = t_0;
	} else if (x <= 2.6e-35) {
		tmp = t_1;
	} else if (x <= 1.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = -log(x) / n
    if (x <= 4d-272) then
        tmp = t_0
    else if (x <= 1.1d-241) then
        tmp = t_1
    else if (x <= 1.04d-232) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 2.5d-196) then
        tmp = log(x) * (1.0d0 / -n)
    else if (x <= 6d-167) then
        tmp = t_0
    else if (x <= 3.1d-103) then
        tmp = t_1
    else if (x <= 3.9d-75) then
        tmp = t_0
    else if (x <= 2.6d-35) then
        tmp = t_1
    else if (x <= 1.0d0) then
        tmp = t_0
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = -Math.log(x) / n;
	double tmp;
	if (x <= 4e-272) {
		tmp = t_0;
	} else if (x <= 1.1e-241) {
		tmp = t_1;
	} else if (x <= 1.04e-232) {
		tmp = 1.0 / (n * x);
	} else if (x <= 2.5e-196) {
		tmp = Math.log(x) * (1.0 / -n);
	} else if (x <= 6e-167) {
		tmp = t_0;
	} else if (x <= 3.1e-103) {
		tmp = t_1;
	} else if (x <= 3.9e-75) {
		tmp = t_0;
	} else if (x <= 2.6e-35) {
		tmp = t_1;
	} else if (x <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = -math.log(x) / n
	tmp = 0
	if x <= 4e-272:
		tmp = t_0
	elif x <= 1.1e-241:
		tmp = t_1
	elif x <= 1.04e-232:
		tmp = 1.0 / (n * x)
	elif x <= 2.5e-196:
		tmp = math.log(x) * (1.0 / -n)
	elif x <= 6e-167:
		tmp = t_0
	elif x <= 3.1e-103:
		tmp = t_1
	elif x <= 3.9e-75:
		tmp = t_0
	elif x <= 2.6e-35:
		tmp = t_1
	elif x <= 1.0:
		tmp = t_0
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 4e-272)
		tmp = t_0;
	elseif (x <= 1.1e-241)
		tmp = t_1;
	elseif (x <= 1.04e-232)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 2.5e-196)
		tmp = Float64(log(x) * Float64(1.0 / Float64(-n)));
	elseif (x <= 6e-167)
		tmp = t_0;
	elseif (x <= 3.1e-103)
		tmp = t_1;
	elseif (x <= 3.9e-75)
		tmp = t_0;
	elseif (x <= 2.6e-35)
		tmp = t_1;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = -log(x) / n;
	tmp = 0.0;
	if (x <= 4e-272)
		tmp = t_0;
	elseif (x <= 1.1e-241)
		tmp = t_1;
	elseif (x <= 1.04e-232)
		tmp = 1.0 / (n * x);
	elseif (x <= 2.5e-196)
		tmp = log(x) * (1.0 / -n);
	elseif (x <= 6e-167)
		tmp = t_0;
	elseif (x <= 3.1e-103)
		tmp = t_1;
	elseif (x <= 3.9e-75)
		tmp = t_0;
	elseif (x <= 2.6e-35)
		tmp = t_1;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 4e-272], t$95$0, If[LessEqual[x, 1.1e-241], t$95$1, If[LessEqual[x, 1.04e-232], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-196], N[(N[Log[x], $MachinePrecision] * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-167], t$95$0, If[LessEqual[x, 3.1e-103], t$95$1, If[LessEqual[x, 3.9e-75], t$95$0, If[LessEqual[x, 2.6e-35], t$95$1, If[LessEqual[x, 1.0], t$95$0, N[(0.0 / n), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 3.99999999999999972e-272 or 2.5000000000000002e-196 < x < 5.9999999999999996e-167 or 3.1000000000000001e-103 < x < 3.9000000000000001e-75 or 2.60000000000000005e-35 < x < 1

   1. Initial program 65.6%

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

   2. Taylor expanded in x around 0 64.2%

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

   if 3.99999999999999972e-272 < x < 1.1e-241 or 5.9999999999999996e-167 < x < 3.1000000000000001e-103 or 3.9000000000000001e-75 < x < 2.60000000000000005e-35

   1. Initial program 23.6%

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

   2. Taylor expanded in x around 0 23.6%

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

   3. Taylor expanded in n around inf 73.0%

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

      1. neg-mul-1 73.0%

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

      2. distribute-neg-frac 73.0%

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

   5. Simplified 73.0%

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

   if 1.1e-241 < x < 1.0399999999999999e-232

   1. Initial program 41.9%

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

   2. Taylor expanded in n around inf 8.9%

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

      1. +-rgt-identity 8.9%

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

      2. +-rgt-identity 8.9%

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

      3. log1p-def 8.9%

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

   4. Simplified 8.9%

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

   5. Taylor expanded in x around inf 100.0%

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

   if 1.0399999999999999e-232 < x < 2.5000000000000002e-196

   1. Initial program 41.8%

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

   2. Taylor expanded in n around inf 65.6%

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

      1. +-rgt-identity 65.6%

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

      2. +-rgt-identity 65.6%

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

      3. log1p-def 65.6%

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

   4. Simplified 65.6%

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

      1. log1p-udef 65.6%

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

      2. diff-log 65.6%

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

      3. +-commutative 65.6%

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

   6. Applied egg-rr 65.6%

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

      1. frac-2neg 65.6%

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

      2. log-div 65.6%

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

      3. +-commutative 65.6%

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

      4. log1p-udef 65.6%

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

      5. div-inv 65.7%

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

      6. log1p-udef 65.7%

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

      7. +-commutative 65.7%

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

      8. log-div 65.7%

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

      9. neg-log 65.7%

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

      10. clear-num 65.7%

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

   8. Applied egg-rr 65.7%

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

   9. Taylor expanded in x around 0 65.7%

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

   if 1 < x

   1. Initial program 70.9%

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

   2. Taylor expanded in n around inf 71.9%

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

      1. +-rgt-identity 71.9%

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

      2. +-rgt-identity 71.9%

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

      3. log1p-def 71.9%

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

   4. Simplified 71.9%

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

      1. log1p-udef 71.9%

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

      2. diff-log 72.0%

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

      3. +-commutative 72.0%

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

   6. Applied egg-rr 72.0%

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

      1. add-cube-cbrt 71.9%

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

      2. log-prod 71.9%

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

      3. pow2 71.9%

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

   8. Applied egg-rr 71.9%

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

      1. log-pow 71.9%

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

      2. distribute-lft1-in 71.9%

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

      3. metadata-eval 71.9%

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

   10. Simplified 71.9%

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

   11. Taylor expanded in x around inf 70.9%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-272}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-241}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.04 \cdot 10^{-232}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-196}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-167}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-103}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-75}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 7: 82.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log t_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(t_0 + -1\right) \cdot \frac{1}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (/ t_1 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (- (log t_0)) n)
       (if (<= (/ 1.0 n) 5e+125)
         (- (+ 1.0 (/ x n)) t_1)
         (* (log1p (+ t_0 -1.0)) (/ 1.0 (- n))))))))

C

double code(double x, double n) {
	double t_0 = x / (1.0 + x);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -log(t_0) / n;
	} else if ((1.0 / n) <= 5e+125) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = log1p((t_0 + -1.0)) * (1.0 / -n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = x / (1.0 + x);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -Math.log(t_0) / n;
	} else if ((1.0 / n) <= 5e+125) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = Math.log1p((t_0 + -1.0)) * (1.0 / -n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / (1.0 + x)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_1 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = -math.log(t_0) / n
	elif (1.0 / n) <= 5e+125:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = math.log1p((t_0 + -1.0)) * (1.0 / -n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(1.0 + x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(-log(t_0)) / n);
	elseif (Float64(1.0 / n) <= 5e+125)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = Float64(log1p(Float64(t_0 + -1.0)) * Float64(1.0 / Float64(-n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[((-N[Log[t$95$0], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+125], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[Log[1 + N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. associate-*r/ 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. remove-double-neg 98.9%

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

      7. exp-to-pow 98.9%

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

   7. Simplified 98.9%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.99999999999999988e-11

   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 80.6%

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

      1. +-rgt-identity 80.6%

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

      2. +-rgt-identity 80.6%

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

      3. log1p-def 80.6%

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

   4. Simplified 80.6%

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

      1. log1p-udef 80.6%

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

      2. diff-log 80.7%

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

      3. +-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

      1. clear-num 80.7%

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

      2. log-rec 80.7%

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

   8. Applied egg-rr 80.7%

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

   if 1.99999999999999988e-11 < (/.f64 1 n) < 4.99999999999999962e125

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0 74.2%

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

   if 4.99999999999999962e125 < (/.f64 1 n)

   1. Initial program 18.8%

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

   2. Taylor expanded in n around inf 7.8%

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

      1. +-rgt-identity 7.8%

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

      2. +-rgt-identity 7.8%

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

      3. log1p-def 7.8%

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

   4. Simplified 7.8%

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

      1. log1p-udef 7.8%

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

      2. diff-log 7.8%

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

      3. +-commutative 7.8%

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

   6. Applied egg-rr 7.8%

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

      1. frac-2neg 7.8%

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

      2. log-div 7.8%

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

      3. +-commutative 7.8%

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

      4. log1p-udef 7.8%

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

      5. div-inv 7.8%

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

      6. log1p-udef 7.8%

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

      7. +-commutative 7.8%

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

      8. log-div 7.8%

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

      9. neg-log 7.8%

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

      10. clear-num 7.8%

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

   8. Applied egg-rr 7.8%

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

      1. log1p-expm1-u 81.2%

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

      2. expm1-udef 81.2%

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

      3. add-exp-log 81.2%

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

   10. Applied egg-rr 81.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\frac{x}{1 + x} + -1\right) \cdot \frac{1}{-n}\\ \end{array} \]

Alternative 8: 82.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (- (log (/ x (+ 1.0 x)))) n)
       (if (<= (/ 1.0 n) 5e+125)
         (- (+ 1.0 (/ x n)) t_0)
         (/ 0.3333333333333333 (* n (pow x 3.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -log((x / (1.0 + x))) / n;
	} else if ((1.0 / n) <= 5e+125) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-11)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-11) then
        tmp = -log((x / (1.0d0 + x))) / n
    else if ((1.0d0 / n) <= 5d+125) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -Math.log((x / (1.0 + x))) / n;
	} else if ((1.0 / n) <= 5e+125) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = -math.log((x / (1.0 + x))) / n
	elif (1.0 / n) <= 5e+125:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n);
	elseif (Float64(1.0 / n) <= 5e+125)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-11)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-11)
		tmp = -log((x / (1.0 + x))) / n;
	elseif ((1.0 / n) <= 5e+125)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+125], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. associate-*r/ 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. remove-double-neg 98.9%

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

      7. exp-to-pow 98.9%

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

   7. Simplified 98.9%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.99999999999999988e-11

   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 80.6%

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

      1. +-rgt-identity 80.6%

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

      2. +-rgt-identity 80.6%

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

      3. log1p-def 80.6%

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

   4. Simplified 80.6%

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

      1. log1p-udef 80.6%

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

      2. diff-log 80.7%

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

      3. +-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

      1. clear-num 80.7%

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

      2. log-rec 80.7%

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

   8. Applied egg-rr 80.7%

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

   if 1.99999999999999988e-11 < (/.f64 1 n) < 4.99999999999999962e125

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0 74.2%

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

   if 4.99999999999999962e125 < (/.f64 1 n)

   1. Initial program 18.8%

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

   2. Taylor expanded in n around inf 7.8%

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

      1. +-rgt-identity 7.8%

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

      2. +-rgt-identity 7.8%

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

      3. log1p-def 7.8%

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

   4. Simplified 7.8%

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

   5. Taylor expanded in x around inf 30.3%

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

      1. sub-neg 30.3%

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

      2. +-commutative 30.3%

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

      3. associate-+l+ 30.3%

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

      4. associate-*r/ 30.3%

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

      5. metadata-eval 30.3%

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

      6. associate-*r/ 30.3%

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

      7. metadata-eval 30.3%

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

      8. distribute-neg-frac 30.3%

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

      9. metadata-eval 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in x around 0 80.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   9. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{3} \cdot n}} \]

   10. Simplified 80.8%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{3} \cdot n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]

Alternative 9: 60.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - \log x}{n}\\ \mathbf{if}\;n \leq -1.02 \cdot 10^{+189}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \mathbf{elif}\;n \leq -2.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -2.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-128}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 58000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 7.6 \cdot 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- x (log x)) n)))
   (if (<= n -1.02e+189)
     (* (log x) (/ 1.0 (- n)))
     (if (<= n -2.5e+143)
       (/ (/ 1.0 n) x)
       (if (<= n -2.45)
         t_0
         (if (<= n 3.5e-128)
           (/ 0.3333333333333333 (* n (pow x 3.0)))
           (if (<= n 58000000000.0)
             (- 1.0 (pow x (/ 1.0 n)))
             (if (<= n 7.6e+103)
               t_0
               (if (<= n 3.1e+187) (/ (/ 1.0 x) n) (/ (- (log x)) n))))))))))

C

double code(double x, double n) {
	double t_0 = (x - log(x)) / n;
	double tmp;
	if (n <= -1.02e+189) {
		tmp = log(x) * (1.0 / -n);
	} else if (n <= -2.5e+143) {
		tmp = (1.0 / n) / x;
	} else if (n <= -2.45) {
		tmp = t_0;
	} else if (n <= 3.5e-128) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 58000000000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 7.6e+103) {
		tmp = t_0;
	} else if (n <= 3.1e+187) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = -log(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) :: tmp
    t_0 = (x - log(x)) / n
    if (n <= (-1.02d+189)) then
        tmp = log(x) * (1.0d0 / -n)
    else if (n <= (-2.5d+143)) then
        tmp = (1.0d0 / n) / x
    else if (n <= (-2.45d0)) then
        tmp = t_0
    else if (n <= 3.5d-128) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 58000000000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (n <= 7.6d+103) then
        tmp = t_0
    else if (n <= 3.1d+187) then
        tmp = (1.0d0 / x) / n
    else
        tmp = -log(x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (x - Math.log(x)) / n;
	double tmp;
	if (n <= -1.02e+189) {
		tmp = Math.log(x) * (1.0 / -n);
	} else if (n <= -2.5e+143) {
		tmp = (1.0 / n) / x;
	} else if (n <= -2.45) {
		tmp = t_0;
	} else if (n <= 3.5e-128) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 58000000000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (n <= 7.6e+103) {
		tmp = t_0;
	} else if (n <= 3.1e+187) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = -Math.log(x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (x - math.log(x)) / n
	tmp = 0
	if n <= -1.02e+189:
		tmp = math.log(x) * (1.0 / -n)
	elif n <= -2.5e+143:
		tmp = (1.0 / n) / x
	elif n <= -2.45:
		tmp = t_0
	elif n <= 3.5e-128:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 58000000000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif n <= 7.6e+103:
		tmp = t_0
	elif n <= 3.1e+187:
		tmp = (1.0 / x) / n
	else:
		tmp = -math.log(x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(x - log(x)) / n)
	tmp = 0.0
	if (n <= -1.02e+189)
		tmp = Float64(log(x) * Float64(1.0 / Float64(-n)));
	elseif (n <= -2.5e+143)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (n <= -2.45)
		tmp = t_0;
	elseif (n <= 3.5e-128)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 58000000000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 7.6e+103)
		tmp = t_0;
	elseif (n <= 3.1e+187)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(-log(x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x - log(x)) / n;
	tmp = 0.0;
	if (n <= -1.02e+189)
		tmp = log(x) * (1.0 / -n);
	elseif (n <= -2.5e+143)
		tmp = (1.0 / n) / x;
	elseif (n <= -2.45)
		tmp = t_0;
	elseif (n <= 3.5e-128)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 58000000000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (n <= 7.6e+103)
		tmp = t_0;
	elseif (n <= 3.1e+187)
		tmp = (1.0 / x) / n;
	else
		tmp = -log(x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -1.02e+189], N[(N[Log[x], $MachinePrecision] * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.5e+143], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -2.45], t$95$0, If[LessEqual[n, 3.5e-128], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 58000000000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.6e+103], t$95$0, If[LessEqual[n, 3.1e+187], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if n < -1.02e189

   1. Initial program 34.5%

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

   2. Taylor expanded in n around inf 91.0%

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

      1. +-rgt-identity 91.0%

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

      2. +-rgt-identity 91.0%

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

      3. log1p-def 91.0%

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

   4. Simplified 91.0%

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

      1. log1p-udef 91.0%

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

      2. diff-log 91.0%

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

      3. +-commutative 91.0%

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

   6. Applied egg-rr 91.0%

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

      1. frac-2neg 91.0%

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

      2. log-div 91.0%

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

      3. +-commutative 91.0%

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

      4. log1p-udef 91.0%

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

      5. div-inv 91.0%

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

      6. log1p-udef 91.0%

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

      7. +-commutative 91.0%

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

      8. log-div 91.0%

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

      9. neg-log 91.0%

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

      10. clear-num 91.0%

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

   8. Applied egg-rr 91.0%

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

   9. Taylor expanded in x around 0 63.3%

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

   if -1.02e189 < n < -2.50000000000000006e143

   1. Initial program 26.1%

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

   2. Taylor expanded in n around inf 46.2%

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

      1. +-rgt-identity 46.2%

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

      2. +-rgt-identity 46.2%

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

      3. log1p-def 46.2%

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

   4. Simplified 46.2%

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

   5. Taylor expanded in x around inf 75.9%

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

      1. associate-/r* 81.3%

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

   7. Simplified 81.3%

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

   if -2.50000000000000006e143 < n < -2.4500000000000002 or 5.8e10 < n < 7.5999999999999994e103

   1. Initial program 11.1%

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

   2. Taylor expanded in n around inf 73.7%

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

      1. +-rgt-identity 73.7%

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

      2. +-rgt-identity 73.7%

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

      3. log1p-def 73.7%

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

   4. Simplified 73.7%

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

   5. Taylor expanded in x around 0 68.4%

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

      1. neg-mul-1 68.4%

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

      2. unsub-neg 68.4%

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

   7. Simplified 68.4%

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

   if -2.4500000000000002 < n < 3.5e-128

   1. Initial program 84.7%

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

   2. Taylor expanded in n around inf 41.7%

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

      1. +-rgt-identity 41.7%

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

      2. +-rgt-identity 41.7%

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

      3. log1p-def 41.7%

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

   4. Simplified 41.7%

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

   5. Taylor expanded in x around inf 19.8%

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

      1. sub-neg 19.8%

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

      2. +-commutative 19.8%

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

      3. associate-+l+ 19.8%

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

      4. associate-*r/ 19.8%

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

      5. metadata-eval 19.8%

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

      6. associate-*r/ 19.8%

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

      7. metadata-eval 19.8%

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

      8. distribute-neg-frac 19.8%

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

      9. metadata-eval 19.8%

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

   7. Simplified 19.8%

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

   8. Taylor expanded in x around 0 68.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   9. Step-by-step derivation

      1. *-commutative 68.5%

         \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{3} \cdot n}} \]

   10. Simplified 68.5%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{3} \cdot n}} \]

   if 3.5e-128 < n < 5.8e10

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0 73.7%

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

   if 7.5999999999999994e103 < n < 3.10000000000000012e187

   1. Initial program 43.9%

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

   2. Taylor expanded in n around inf 72.2%

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

      1. +-rgt-identity 72.2%

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

      2. +-rgt-identity 72.2%

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

      3. log1p-def 72.2%

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

   4. Simplified 72.2%

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

   5. Taylor expanded in x around inf 71.4%

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

   if 3.10000000000000012e187 < n

   1. Initial program 32.5%

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

   2. Taylor expanded in x around 0 32.5%

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

   3. Taylor expanded in n around inf 70.1%

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

      1. neg-mul-1 70.1%

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

      2. distribute-neg-frac 70.1%

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

   5. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.02 \cdot 10^{+189}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \mathbf{elif}\;n \leq -2.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -2.45:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-128}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 58000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 7.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log x}{n}\\ \end{array} \]

Alternative 10: 82.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+125}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-11)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (- (log (/ x (+ 1.0 x)))) n)
       (if (<= (/ 1.0 n) 5e+125)
         (- 1.0 t_0)
         (/ 0.3333333333333333 (* n (pow x 3.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -log((x / (1.0 + x))) / n;
	} else if ((1.0 / n) <= 5e+125) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-11)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-11) then
        tmp = -log((x / (1.0d0 + x))) / n
    else if ((1.0d0 / n) <= 5d+125) then
        tmp = 1.0d0 - t_0
    else
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-11) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -Math.log((x / (1.0 + x))) / n;
	} else if ((1.0 / n) <= 5e+125) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-11:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = -math.log((x / (1.0 + x))) / n
	elif (1.0 / n) <= 5e+125:
		tmp = 1.0 - t_0
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-11)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n);
	elseif (Float64(1.0 / n) <= 5e+125)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-11)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-11)
		tmp = -log((x / (1.0 + x))) / n;
	elseif ((1.0 / n) <= 5e+125)
		tmp = 1.0 - t_0;
	else
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-11], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+125], N[(1.0 - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000018e-11

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. associate-*r/ 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. remove-double-neg 98.9%

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

      7. exp-to-pow 98.9%

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

   7. Simplified 98.9%

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

   if -5.00000000000000018e-11 < (/.f64 1 n) < 1.99999999999999988e-11

   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 80.6%

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

      1. +-rgt-identity 80.6%

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

      2. +-rgt-identity 80.6%

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

      3. log1p-def 80.6%

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

   4. Simplified 80.6%

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

      1. log1p-udef 80.6%

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

      2. diff-log 80.7%

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

      3. +-commutative 80.7%

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

   6. Applied egg-rr 80.7%

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

      1. clear-num 80.7%

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

      2. log-rec 80.7%

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

   8. Applied egg-rr 80.7%

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

   if 1.99999999999999988e-11 < (/.f64 1 n) < 4.99999999999999962e125

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0 73.7%

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

   if 4.99999999999999962e125 < (/.f64 1 n)

   1. Initial program 18.8%

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

   2. Taylor expanded in n around inf 7.8%

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

      1. +-rgt-identity 7.8%

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

      2. +-rgt-identity 7.8%

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

      3. log1p-def 7.8%

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

   4. Simplified 7.8%

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

   5. Taylor expanded in x around inf 30.3%

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

      1. sub-neg 30.3%

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

      2. +-commutative 30.3%

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

      3. associate-+l+ 30.3%

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

      4. associate-*r/ 30.3%

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

      5. metadata-eval 30.3%

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

      6. associate-*r/ 30.3%

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

      7. metadata-eval 30.3%

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

      8. distribute-neg-frac 30.3%

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

      9. metadata-eval 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in x around 0 80.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   9. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{3} \cdot n}} \]

   10. Simplified 80.8%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{3} \cdot n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+125}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]

Alternative 11: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 16200000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -4.8e-135)
   (/ (log (/ (+ 1.0 x) x)) n)
   (if (<= n 3.3e-130)
     (/ 0.3333333333333333 (* n (pow x 3.0)))
     (if (<= n 16200000000.0)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (- (log (/ x (+ 1.0 x)))) n)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -4.8e-135) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (n <= 3.3e-130) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 16200000000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = -log((x / (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 <= (-4.8d-135)) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if (n <= 3.3d-130) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 16200000000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = -log((x / (1.0d0 + x))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -4.8e-135) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (n <= 3.3e-130) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 16200000000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = -Math.log((x / (1.0 + x))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -4.8e-135:
		tmp = math.log(((1.0 + x) / x)) / n
	elif n <= 3.3e-130:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 16200000000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = -math.log((x / (1.0 + x))) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -4.8e-135)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (n <= 3.3e-130)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 16200000000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -4.8e-135)
		tmp = log(((1.0 + x) / x)) / n;
	elseif (n <= 3.3e-130)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 16200000000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = -log((x / (1.0 + x))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -4.8e-135], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 3.3e-130], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 16200000000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.7999999999999997e-135

   1. Initial program 54.3%

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

   2. Taylor expanded in n around inf 69.0%

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

      1. +-rgt-identity 69.0%

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

      2. +-rgt-identity 69.0%

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

      3. log1p-def 69.0%

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

   4. Simplified 69.0%

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

      1. log1p-udef 69.0%

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

      2. diff-log 69.9%

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

      3. +-commutative 69.9%

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

   6. Applied egg-rr 69.9%

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

   if -4.7999999999999997e-135 < n < 3.2999999999999998e-130

   1. Initial program 73.8%

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

   2. Taylor expanded in n around inf 31.3%

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

      1. +-rgt-identity 31.3%

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

      2. +-rgt-identity 31.3%

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

      3. log1p-def 31.3%

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

   4. Simplified 31.3%

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

   5. Taylor expanded in x around inf 25.3%

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

      1. sub-neg 25.3%

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

      2. +-commutative 25.3%

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

      3. associate-+l+ 25.3%

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

      4. associate-*r/ 25.3%

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

      5. metadata-eval 25.3%

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

      6. associate-*r/ 25.3%

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

      7. metadata-eval 25.3%

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

      8. distribute-neg-frac 25.3%

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

      9. metadata-eval 25.3%

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

   7. Simplified 25.3%

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

   8. Taylor expanded in x around 0 81.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   9. Step-by-step derivation

      1. *-commutative 81.5%

         \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{3} \cdot n}} \]

   10. Simplified 81.5%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{3} \cdot n}} \]

   if 3.2999999999999998e-130 < n < 1.62e10

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0 73.7%

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

   if 1.62e10 < n

   1. Initial program 28.0%

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

   2. Taylor expanded in n around inf 80.9%

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

      1. +-rgt-identity 80.9%

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

      2. +-rgt-identity 80.9%

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

      3. log1p-def 80.9%

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

   4. Simplified 80.9%

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

      1. log1p-udef 80.9%

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

      2. diff-log 81.0%

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

      3. +-commutative 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. clear-num 81.0%

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

      2. log-rec 81.1%

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

   8. Applied egg-rr 81.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 16200000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \end{array} \]

Alternative 12: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;x \leq 1.85 \cdot 10^{-274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-240}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6000000000000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))))
   (if (<= x 1.85e-274)
     t_0
     (if (<= x 1.7e-240)
       (/ (- (log x)) n)
       (if (<= x 1e-232)
         t_0
         (if (<= x 6000000000000.0) (/ (- x (log x)) n) (/ 0.0 n)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (x <= 1.85e-274) {
		tmp = t_0;
	} else if (x <= 1.7e-240) {
		tmp = -log(x) / n;
	} else if (x <= 1e-232) {
		tmp = t_0;
	} else if (x <= 6000000000000.0) {
		tmp = (x - log(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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    if (x <= 1.85d-274) then
        tmp = t_0
    else if (x <= 1.7d-240) then
        tmp = -log(x) / n
    else if (x <= 1d-232) then
        tmp = t_0
    else if (x <= 6000000000000.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (x <= 1.85e-274) {
		tmp = t_0;
	} else if (x <= 1.7e-240) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1e-232) {
		tmp = t_0;
	} else if (x <= 6000000000000.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	tmp = 0
	if x <= 1.85e-274:
		tmp = t_0
	elif x <= 1.7e-240:
		tmp = -math.log(x) / n
	elif x <= 1e-232:
		tmp = t_0
	elif x <= 6000000000000.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (x <= 1.85e-274)
		tmp = t_0;
	elseif (x <= 1.7e-240)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1e-232)
		tmp = t_0;
	elseif (x <= 6000000000000.0)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	tmp = 0.0;
	if (x <= 1.85e-274)
		tmp = t_0;
	elseif (x <= 1.7e-240)
		tmp = -log(x) / n;
	elseif (x <= 1e-232)
		tmp = t_0;
	elseif (x <= 6000000000000.0)
		tmp = (x - log(x)) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.85e-274], t$95$0, If[LessEqual[x, 1.7e-240], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1e-232], t$95$0, If[LessEqual[x, 6000000000000.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.84999999999999992e-274 or 1.69999999999999995e-240 < x < 1.00000000000000002e-232

   1. Initial program 64.9%

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

   2. Taylor expanded in n around inf 23.4%

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

      1. +-rgt-identity 23.4%

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

      2. +-rgt-identity 23.4%

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

      3. log1p-def 23.4%

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

   4. Simplified 23.4%

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

   5. Taylor expanded in x around inf 78.0%

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

   if 1.84999999999999992e-274 < x < 1.69999999999999995e-240

   1. Initial program 23.6%

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

   2. Taylor expanded in x around 0 23.6%

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

   3. Taylor expanded in n around inf 69.6%

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

      1. neg-mul-1 69.6%

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

      2. distribute-neg-frac 69.6%

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

   5. Simplified 69.6%

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

   if 1.00000000000000002e-232 < x < 6e12

   1. Initial program 42.2%

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

   2. Taylor expanded in n around inf 52.6%

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

      1. +-rgt-identity 52.6%

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

      2. +-rgt-identity 52.6%

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

      3. log1p-def 52.6%

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

   4. Simplified 52.6%

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

   5. Taylor expanded in x around 0 51.8%

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

      1. neg-mul-1 51.8%

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

      2. unsub-neg 51.8%

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

   7. Simplified 51.8%

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

   if 6e12 < x

   1. Initial program 72.3%

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

   2. Taylor expanded in n around inf 72.3%

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

      1. +-rgt-identity 72.3%

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

      2. +-rgt-identity 72.3%

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

      3. log1p-def 72.3%

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

   4. Simplified 72.3%

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

      1. log1p-udef 72.3%

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

      2. diff-log 72.3%

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

      3. +-commutative 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. add-cube-cbrt 72.3%

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

      2. log-prod 72.3%

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

      3. pow2 72.3%

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

   8. Applied egg-rr 72.3%

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

      1. log-pow 72.3%

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

      2. distribute-lft1-in 72.3%

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

      3. metadata-eval 72.3%

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

   10. Simplified 72.3%

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

   11. Taylor expanded in x around inf 72.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-274}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-240}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 10^{-232}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 6000000000000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 13: 72.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;n \leq -4.5 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 15000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= n -4.5e-135)
     t_0
     (if (<= n 3.3e-130)
       (/ 0.3333333333333333 (* n (pow x 3.0)))
       (if (<= n 15000000000.0) (- 1.0 (pow x (/ 1.0 n))) t_0)))))

C

double code(double x, double n) {
	double t_0 = log(((1.0 + x) / x)) / n;
	double tmp;
	if (n <= -4.5e-135) {
		tmp = t_0;
	} else if (n <= 3.3e-130) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 15000000000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(((1.0d0 + x) / x)) / n
    if (n <= (-4.5d-135)) then
        tmp = t_0
    else if (n <= 3.3d-130) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 15000000000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if (n <= -4.5e-135) {
		tmp = t_0;
	} else if (n <= 3.3e-130) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 15000000000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if n <= -4.5e-135:
		tmp = t_0
	elif n <= 3.3e-130:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 15000000000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (n <= -4.5e-135)
		tmp = t_0;
	elseif (n <= 3.3e-130)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 15000000000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if (n <= -4.5e-135)
		tmp = t_0;
	elseif (n <= 3.3e-130)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 15000000000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -4.5e-135], t$95$0, If[LessEqual[n, 3.3e-130], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 15000000000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.49999999999999987e-135 or 1.5e10 < n

   1. Initial program 45.0%

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

   2. Taylor expanded in n around inf 73.2%

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

      1. +-rgt-identity 73.2%

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

      2. +-rgt-identity 73.2%

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

      3. log1p-def 73.2%

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

   4. Simplified 73.2%

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

      1. log1p-udef 73.2%

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

      2. diff-log 73.8%

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

      3. +-commutative 73.8%

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

   6. Applied egg-rr 73.8%

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

   if -4.49999999999999987e-135 < n < 3.2999999999999998e-130

   1. Initial program 73.8%

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

   2. Taylor expanded in n around inf 31.3%

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

      1. +-rgt-identity 31.3%

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

      2. +-rgt-identity 31.3%

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

      3. log1p-def 31.3%

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

   4. Simplified 31.3%

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

   5. Taylor expanded in x around inf 25.3%

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

      1. sub-neg 25.3%

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

      2. +-commutative 25.3%

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

      3. associate-+l+ 25.3%

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

      4. associate-*r/ 25.3%

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

      5. metadata-eval 25.3%

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

      6. associate-*r/ 25.3%

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

      7. metadata-eval 25.3%

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

      8. distribute-neg-frac 25.3%

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

      9. metadata-eval 25.3%

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

   7. Simplified 25.3%

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

   8. Taylor expanded in x around 0 81.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   9. Step-by-step derivation

      1. *-commutative 81.5%

         \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{3} \cdot n}} \]

   10. Simplified 81.5%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{3} \cdot n}} \]

   if 3.2999999999999998e-130 < n < 1.5e10

   1. Initial program 73.8%

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

   2. Taylor expanded in x around 0 73.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.5 \cdot 10^{-135}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 15000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \]

Alternative 14: 53.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 6.5 \cdot 10^{-274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))) (t_1 (/ (- (log x)) n)))
   (if (<= x 6.5e-274)
     t_0
     (if (<= x 3.7e-240)
       t_1
       (if (<= x 1e-232) t_0 (if (<= x 1.85e-25) t_1 (/ (/ 1.0 n) x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = -log(x) / n;
	double tmp;
	if (x <= 6.5e-274) {
		tmp = t_0;
	} else if (x <= 3.7e-240) {
		tmp = t_1;
	} else if (x <= 1e-232) {
		tmp = t_0;
	} else if (x <= 1.85e-25) {
		tmp = t_1;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    t_1 = -log(x) / n
    if (x <= 6.5d-274) then
        tmp = t_0
    else if (x <= 3.7d-240) then
        tmp = t_1
    else if (x <= 1d-232) then
        tmp = t_0
    else if (x <= 1.85d-25) then
        tmp = t_1
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = -Math.log(x) / n;
	double tmp;
	if (x <= 6.5e-274) {
		tmp = t_0;
	} else if (x <= 3.7e-240) {
		tmp = t_1;
	} else if (x <= 1e-232) {
		tmp = t_0;
	} else if (x <= 1.85e-25) {
		tmp = t_1;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	t_1 = -math.log(x) / n
	tmp = 0
	if x <= 6.5e-274:
		tmp = t_0
	elif x <= 3.7e-240:
		tmp = t_1
	elif x <= 1e-232:
		tmp = t_0
	elif x <= 1.85e-25:
		tmp = t_1
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 6.5e-274)
		tmp = t_0;
	elseif (x <= 3.7e-240)
		tmp = t_1;
	elseif (x <= 1e-232)
		tmp = t_0;
	elseif (x <= 1.85e-25)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	t_1 = -log(x) / n;
	tmp = 0.0;
	if (x <= 6.5e-274)
		tmp = t_0;
	elseif (x <= 3.7e-240)
		tmp = t_1;
	elseif (x <= 1e-232)
		tmp = t_0;
	elseif (x <= 1.85e-25)
		tmp = t_1;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 6.5e-274], t$95$0, If[LessEqual[x, 3.7e-240], t$95$1, If[LessEqual[x, 1e-232], t$95$0, If[LessEqual[x, 1.85e-25], t$95$1, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.49999999999999959e-274 or 3.7000000000000002e-240 < x < 1.00000000000000002e-232

   1. Initial program 64.9%

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

   2. Taylor expanded in n around inf 23.4%

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

      1. +-rgt-identity 23.4%

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

      2. +-rgt-identity 23.4%

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

      3. log1p-def 23.4%

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

   4. Simplified 23.4%

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

   5. Taylor expanded in x around inf 78.0%

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

   if 6.49999999999999959e-274 < x < 3.7000000000000002e-240 or 1.00000000000000002e-232 < x < 1.85000000000000004e-25

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 38.0%

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

   3. Taylor expanded in n around inf 57.0%

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

      1. neg-mul-1 57.0%

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

      2. distribute-neg-frac 57.0%

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

   5. Simplified 57.0%

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

   if 1.85000000000000004e-25 < x

   1. Initial program 71.7%

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

   2. Taylor expanded in n around inf 67.2%

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

      1. +-rgt-identity 67.2%

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

      2. +-rgt-identity 67.2%

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

      3. log1p-def 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in x around inf 54.0%

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

      1. associate-/r* 54.9%

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

   7. Simplified 54.9%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-274}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-240}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 10^{-232}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-25}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 15: 57.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 2.4 \cdot 10^{-274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))) (t_1 (/ (- (log x)) n)))
   (if (<= x 2.4e-274)
     t_0
     (if (<= x 2.8e-240)
       t_1
       (if (<= x 1e-232) t_0 (if (<= x 1.0) t_1 (/ 0.0 n)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = -log(x) / n;
	double tmp;
	if (x <= 2.4e-274) {
		tmp = t_0;
	} else if (x <= 2.8e-240) {
		tmp = t_1;
	} else if (x <= 1e-232) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    t_1 = -log(x) / n
    if (x <= 2.4d-274) then
        tmp = t_0
    else if (x <= 2.8d-240) then
        tmp = t_1
    else if (x <= 1d-232) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = t_1
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = -Math.log(x) / n;
	double tmp;
	if (x <= 2.4e-274) {
		tmp = t_0;
	} else if (x <= 2.8e-240) {
		tmp = t_1;
	} else if (x <= 1e-232) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = t_1;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	t_1 = -math.log(x) / n
	tmp = 0
	if x <= 2.4e-274:
		tmp = t_0
	elif x <= 2.8e-240:
		tmp = t_1
	elif x <= 1e-232:
		tmp = t_0
	elif x <= 1.0:
		tmp = t_1
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 2.4e-274)
		tmp = t_0;
	elseif (x <= 2.8e-240)
		tmp = t_1;
	elseif (x <= 1e-232)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = t_1;
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	t_1 = -log(x) / n;
	tmp = 0.0;
	if (x <= 2.4e-274)
		tmp = t_0;
	elseif (x <= 2.8e-240)
		tmp = t_1;
	elseif (x <= 1e-232)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = t_1;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 2.4e-274], t$95$0, If[LessEqual[x, 2.8e-240], t$95$1, If[LessEqual[x, 1e-232], t$95$0, If[LessEqual[x, 1.0], t$95$1, N[(0.0 / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.4e-274 or 2.7999999999999999e-240 < x < 1.00000000000000002e-232

   1. Initial program 64.9%

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

   2. Taylor expanded in n around inf 23.4%

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

      1. +-rgt-identity 23.4%

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

      2. +-rgt-identity 23.4%

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

      3. log1p-def 23.4%

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

   4. Simplified 23.4%

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

   5. Taylor expanded in x around inf 78.0%

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

   if 2.4e-274 < x < 2.7999999999999999e-240 or 1.00000000000000002e-232 < x < 1

   1. Initial program 40.7%

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

   2. Taylor expanded in x around 0 40.0%

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

   3. Taylor expanded in n around inf 54.3%

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

      1. neg-mul-1 54.3%

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

      2. distribute-neg-frac 54.3%

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

   5. Simplified 54.3%

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

   if 1 < x

   1. Initial program 70.9%

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

   2. Taylor expanded in n around inf 71.9%

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

      1. +-rgt-identity 71.9%

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

      2. +-rgt-identity 71.9%

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

      3. log1p-def 71.9%

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

   4. Simplified 71.9%

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

      1. log1p-udef 71.9%

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

      2. diff-log 72.0%

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

      3. +-commutative 72.0%

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

   6. Applied egg-rr 72.0%

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

      1. add-cube-cbrt 71.9%

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

      2. log-prod 71.9%

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

      3. pow2 71.9%

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

   8. Applied egg-rr 71.9%

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

      1. log-pow 71.9%

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

      2. distribute-lft1-in 71.9%

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

      3. metadata-eval 71.9%

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

   10. Simplified 71.9%

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

   11. Taylor expanded in x around inf 70.9%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-240}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 10^{-232}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 16: 39.7% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in n around inf 58.3%

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

   1. +-rgt-identity 58.3%

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

   2. +-rgt-identity 58.3%

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

   3. log1p-def 58.3%

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

4. Simplified 58.3%

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

5. Taylor expanded in x around inf 38.3%

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

6. Final simplification 38.3%

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

Alternative 17: 40.3% 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 54.1%

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

2. Taylor expanded in n around inf 58.3%

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

   1. +-rgt-identity 58.3%

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

   2. +-rgt-identity 58.3%

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

   3. log1p-def 58.3%

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

4. Simplified 58.3%

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

5. Taylor expanded in x around inf 38.3%

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

   1. associate-/r* 38.7%

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

7. Simplified 38.7%

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

8. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2023322 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))