2nthrt (problem 3.4.6)

Percentage Accurate: 53.2% → 85.9%

Time: 28.2s

Alternatives: 10

Speedup: 1.6×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;n \leq -22000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 0.0022:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.32 \cdot 10^{+66}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log1p x) (log x)) n)))
   (if (<= n -22000000000.0)
     t_0
     (if (<= n 0.0022)
       (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))
       (if (<= n 1.32e+66) (/ (exp (/ (log x) n)) (* n x)) t_0)))))

C

double code(double x, double n) {
	double t_0 = (log1p(x) - log(x)) / n;
	double tmp;
	if (n <= -22000000000.0) {
		tmp = t_0;
	} else if (n <= 0.0022) {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	} else if (n <= 1.32e+66) {
		tmp = exp((log(x) / n)) / (n * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (Math.log1p(x) - Math.log(x)) / n;
	double tmp;
	if (n <= -22000000000.0) {
		tmp = t_0;
	} else if (n <= 0.0022) {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	} else if (n <= 1.32e+66) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (math.log1p(x) - math.log(x)) / n
	tmp = 0
	if n <= -22000000000.0:
		tmp = t_0
	elif n <= 0.0022:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	elif n <= 1.32e+66:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (n <= -22000000000.0)
		tmp = t_0;
	elseif (n <= 0.0022)
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	elseif (n <= 1.32e+66)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -22000000000.0], t$95$0, If[LessEqual[n, 0.0022], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.32e+66], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.2e10 or 1.32000000000000009e66 < n

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf 81.4%

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

      1. log1p-def 81.4%

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

   5. Simplified 81.4%

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

   if -2.2e10 < n < 0.00220000000000000013

   1. Initial program 86.6%

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

   3. Taylor expanded in n around 0 86.6%

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

      1. log1p-def 99.1%

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

   5. Simplified 99.1%

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

   if 0.00220000000000000013 < n < 1.32000000000000009e66

   1. Initial program 12.9%

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

   3. Taylor expanded in x around inf 61.7%

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

      1. mul-1-neg 61.7%

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

      2. log-rec 61.7%

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

      3. mul-1-neg 61.7%

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

      4. distribute-neg-frac 61.7%

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

      5. mul-1-neg 61.7%

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

      6. remove-double-neg 61.7%

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

      7. *-commutative 61.7%

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

   5. Simplified 61.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -22000000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;n \leq 0.0022:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.32 \cdot 10^{+66}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -33000000 \lor \neg \left(n \leq 65000000\right):\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -33000000.0) (not (<= n 65000000.0)))
   (+
    (/ (log (/ (+ x 1.0) 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)) (pow x (/ 1.0 n)))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -33000000.0) || !(n <= 65000000.0)) {
		tmp = (log(((x + 1.0) / 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)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -33000000.0) || !(n <= 65000000.0)) {
		tmp = (Math.log(((x + 1.0) / 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)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -33000000.0) or not (n <= 65000000.0):
		tmp = (math.log(((x + 1.0) / 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)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -33000000.0) || !(n <= 65000000.0))
		tmp = Float64(Float64(log(Float64(Float64(x + 1.0) / 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)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -33000000.0], N[Not[LessEqual[n, 65000000.0]], $MachinePrecision]], N[(N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / 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] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.3e7 or 6.5e7 < n

   1. Initial program 26.5%

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

   3. Taylor expanded in n around inf 76.7%

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

      1. associate--l+ 70.9%

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

      2. +-commutative 70.9%

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

      3. associate--r+ 76.7%

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

      4. div-sub 76.7%

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

      5. remove-double-neg 76.7%

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

      6. mul-1-neg 76.7%

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

      7. distribute-lft-out-- 76.7%

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

      8. distribute-neg-frac 76.7%

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

      9. mul-1-neg 76.7%

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

   5. Simplified 76.7%

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

      1. log1p-udef 76.7%

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

      2. diff-log 76.8%

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

      3. +-commutative 76.8%

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

   7. Applied egg-rr 76.8%

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

   if -3.3e7 < n < 6.5e7

   1. Initial program 85.4%

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

   3. Taylor expanded in n around 0 85.4%

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

      1. log1p-def 97.7%

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

   5. Simplified 97.7%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -33000000 \lor \neg \left(n \leq 65000000\right):\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -50:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|1 - t_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -50.0)
     t_1
     (if (<= (/ 1.0 n) 5e-69)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e-26) t_1 (fabs (- 1.0 t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -50.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-26) {
		tmp = t_1;
	} else {
		tmp = fabs((1.0 - t_0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -50.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-26) {
		tmp = t_1;
	} else {
		tmp = Math.abs((1.0 - t_0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -50.0:
		tmp = t_1
	elif (1.0 / n) <= 5e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-26:
		tmp = t_1
	else:
		tmp = math.fabs((1.0 - t_0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -50.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-26)
		tmp = t_1;
	else
		tmp = abs(Float64(1.0 - t_0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -50.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-26], t$95$1, N[Abs[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -50 or 5.00000000000000033e-69 < (/.f64 1 n) < 2.0000000000000001e-26

   1. Initial program 84.1%

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

   3. Taylor expanded in n around 0 84.1%

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

      1. log1p-def 84.1%

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

   5. Simplified 84.1%

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

      1. rem-cbrt-cube 84.1%

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

      2. pow1/3 45.2%

         \[\leadsto \color{blue}{{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

      3. cube-mult 45.2%

         \[\leadsto {\color{blue}{\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)}}^{0.3333333333333333} \]

      4. unpow-prod-down 45.2%

         \[\leadsto \color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333} \cdot {\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{0.3333333333333333}} \]

      5. pow1/3 84.1%

         \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}} \cdot {\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{0.3333333333333333} \]

      6. pow2 84.1%

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

   7. Applied egg-rr 84.1%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot {\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{2}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 84.1%

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

   9. Simplified 84.1%

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

   10. Taylor expanded in x around inf 94.7%

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

       1. mul-1-neg 94.7%

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

       2. log-rec 94.7%

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

       3. distribute-neg-frac 94.7%

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

       4. remove-double-neg 94.7%

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

       5. *-rgt-identity 94.7%

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

       6. associate-*r/ 94.7%

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

       7. exp-to-pow 94.7%

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

       8. *-commutative 94.7%

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

   12. Simplified 94.7%

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

   if -50 < (/.f64 1 n) < 5.00000000000000033e-69

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf 80.9%

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

      1. log1p-def 80.9%

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

   5. Simplified 80.9%

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

   if 2.0000000000000001e-26 < (/.f64 1 n)

   1. Initial program 61.3%

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

   3. Taylor expanded in x around 0 57.2%

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

      1. add-sqr-sqrt 57.2%

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

      2. sqrt-unprod 59.4%

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

      3. pow2 59.4%

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

   5. Applied egg-rr 59.4%

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

      1. unpow2 59.4%

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

      2. rem-sqrt-square 59.5%

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

   7. Simplified 59.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left|1 - {x}^{\left(\frac{1}{n}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;n \leq -3600000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 0.0034:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{+66}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log1p x) (log x)) n)))
   (if (<= n -3600000.0)
     t_0
     (if (<= n 0.0034)
       (- (exp (/ x n)) (pow x (/ 1.0 n)))
       (if (<= n 1.95e+66) (/ (exp (/ (log x) n)) (* n x)) t_0)))))

C

double code(double x, double n) {
	double t_0 = (log1p(x) - log(x)) / n;
	double tmp;
	if (n <= -3600000.0) {
		tmp = t_0;
	} else if (n <= 0.0034) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else if (n <= 1.95e+66) {
		tmp = exp((log(x) / n)) / (n * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (Math.log1p(x) - Math.log(x)) / n;
	double tmp;
	if (n <= -3600000.0) {
		tmp = t_0;
	} else if (n <= 0.0034) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else if (n <= 1.95e+66) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (math.log1p(x) - math.log(x)) / n
	tmp = 0
	if n <= -3600000.0:
		tmp = t_0
	elif n <= 0.0034:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	elif n <= 1.95e+66:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (n <= -3600000.0)
		tmp = t_0;
	elseif (n <= 0.0034)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (n <= 1.95e+66)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -3600000.0], t$95$0, If[LessEqual[n, 0.0034], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.95e+66], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6e6 or 1.9500000000000002e66 < n

   1. Initial program 29.0%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-def 80.7%

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

   5. Simplified 80.7%

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

   if -3.6e6 < n < 0.00339999999999999981

   1. Initial program 87.3%

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

   3. Taylor expanded in n around 0 87.3%

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

      1. log1p-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

   if 0.00339999999999999981 < n < 1.9500000000000002e66

   1. Initial program 12.9%

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

   3. Taylor expanded in x around inf 61.7%

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

      1. mul-1-neg 61.7%

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

      2. log-rec 61.7%

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

      3. mul-1-neg 61.7%

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

      4. distribute-neg-frac 61.7%

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

      5. mul-1-neg 61.7%

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

      6. remove-double-neg 61.7%

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

      7. *-commutative 61.7%

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

   5. Simplified 61.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3600000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;n \leq 0.0034:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{+66}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -50:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -50.0)
     t_1
     (if (<= (/ 1.0 n) 5e-69)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 50.0) t_1 (- (+ 1.0 (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -50.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 50.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -50.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 50.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -50.0:
		tmp = t_1
	elif (1.0 / n) <= 5e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 50.0:
		tmp = t_1
	else:
		tmp = (1.0 + (x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -50.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 50.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -50.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50.0], t$95$1, N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -50 or 5.00000000000000033e-69 < (/.f64 1 n) < 50

   1. Initial program 81.7%

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

   3. Taylor expanded in n around 0 81.7%

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

      1. log1p-def 81.7%

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

   5. Simplified 81.7%

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

      1. rem-cbrt-cube 81.7%

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

      2. pow1/3 44.8%

         \[\leadsto \color{blue}{{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

      3. cube-mult 44.8%

         \[\leadsto {\color{blue}{\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)}}^{0.3333333333333333} \]

      4. unpow-prod-down 44.8%

         \[\leadsto \color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333} \cdot {\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{0.3333333333333333}} \]

      5. pow1/3 81.7%

         \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}} \cdot {\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{0.3333333333333333} \]

      6. pow2 81.7%

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

   7. Applied egg-rr 81.7%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot {\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{2}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 81.7%

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

   9. Simplified 81.7%

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

   10. Taylor expanded in x around inf 91.9%

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

       1. mul-1-neg 91.9%

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

       2. log-rec 91.9%

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

       3. distribute-neg-frac 91.9%

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

       4. remove-double-neg 91.9%

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

       5. *-rgt-identity 91.9%

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

       6. associate-*r/ 91.9%

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

       7. exp-to-pow 91.9%

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

       8. *-commutative 91.9%

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

   12. Simplified 91.9%

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

   if -50 < (/.f64 1 n) < 5.00000000000000033e-69

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf 80.9%

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

      1. log1p-def 80.9%

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

   5. Simplified 80.9%

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

   if 50 < (/.f64 1 n)

   1. Initial program 64.2%

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

   3. Taylor expanded in x around 0 60.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := 1 - t_0\\ \mathbf{if}\;x \leq 1.42 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-169}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- 1.0 t_0)))
   (if (<= x 1.42e-204)
     t_1
     (if (<= x 1e-169)
       (- (/ (log x) n))
       (if (<= x 1.2e-141)
         t_1
         (if (<= x 1.7e-13) (* (log x) (/ 1.0 (- n))) (/ t_0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = 1.0 - t_0;
	double tmp;
	if (x <= 1.42e-204) {
		tmp = t_1;
	} else if (x <= 1e-169) {
		tmp = -(log(x) / n);
	} else if (x <= 1.2e-141) {
		tmp = t_1;
	} else if (x <= 1.7e-13) {
		tmp = log(x) * (1.0 / -n);
	} else {
		tmp = t_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 = x ** (1.0d0 / n)
    t_1 = 1.0d0 - t_0
    if (x <= 1.42d-204) then
        tmp = t_1
    else if (x <= 1d-169) then
        tmp = -(log(x) / n)
    else if (x <= 1.2d-141) then
        tmp = t_1
    else if (x <= 1.7d-13) then
        tmp = log(x) * (1.0d0 / -n)
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = 1.0 - t_0;
	double tmp;
	if (x <= 1.42e-204) {
		tmp = t_1;
	} else if (x <= 1e-169) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 1.2e-141) {
		tmp = t_1;
	} else if (x <= 1.7e-13) {
		tmp = Math.log(x) * (1.0 / -n);
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = 1.0 - t_0
	tmp = 0
	if x <= 1.42e-204:
		tmp = t_1
	elif x <= 1e-169:
		tmp = -(math.log(x) / n)
	elif x <= 1.2e-141:
		tmp = t_1
	elif x <= 1.7e-13:
		tmp = math.log(x) * (1.0 / -n)
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if (x <= 1.42e-204)
		tmp = t_1;
	elseif (x <= 1e-169)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 1.2e-141)
		tmp = t_1;
	elseif (x <= 1.7e-13)
		tmp = Float64(log(x) * Float64(1.0 / Float64(-n)));
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = 1.0 - t_0;
	tmp = 0.0;
	if (x <= 1.42e-204)
		tmp = t_1;
	elseif (x <= 1e-169)
		tmp = -(log(x) / n);
	elseif (x <= 1.2e-141)
		tmp = t_1;
	elseif (x <= 1.7e-13)
		tmp = log(x) * (1.0 / -n);
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[x, 1.42e-204], t$95$1, If[LessEqual[x, 1e-169], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 1.2e-141], t$95$1, If[LessEqual[x, 1.7e-13], N[(N[Log[x], $MachinePrecision] * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.4200000000000001e-204 or 1.00000000000000002e-169 < x < 1.2e-141

   1. Initial program 56.8%

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

   3. Taylor expanded in x around 0 56.8%

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

   if 1.4200000000000001e-204 < x < 1.00000000000000002e-169

   1. Initial program 22.0%

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

   3. Taylor expanded in x around 0 22.0%

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

   4. Taylor expanded in n around inf 77.9%

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

      1. neg-mul-1 77.9%

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

      2. distribute-neg-frac 77.9%

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

   6. Simplified 77.9%

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

   if 1.2e-141 < x < 1.70000000000000008e-13

   1. Initial program 31.9%

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

   3. Taylor expanded in x around 0 30.1%

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

   4. Taylor expanded in n around inf 61.6%

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

      1. neg-mul-1 61.6%

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

      2. distribute-neg-frac 61.6%

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

   6. Simplified 61.6%

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

      1. frac-2neg 61.6%

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

      2. div-inv 61.6%

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

      3. remove-double-neg 61.6%

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

   8. Applied egg-rr 61.6%

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

   if 1.70000000000000008e-13 < x

   1. Initial program 68.5%

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

   3. Taylor expanded in n around 0 68.5%

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

      1. log1p-def 68.5%

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

   5. Simplified 68.5%

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

      1. rem-cbrt-cube 68.5%

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

      2. pow1/3 65.7%

         \[\leadsto \color{blue}{{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

      3. cube-mult 65.7%

         \[\leadsto {\color{blue}{\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)}}^{0.3333333333333333} \]

      4. unpow-prod-down 65.7%

         \[\leadsto \color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333} \cdot {\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{0.3333333333333333}} \]

      5. pow1/3 68.5%

         \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}} \cdot {\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{0.3333333333333333} \]

      6. pow2 68.5%

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

   7. Applied egg-rr 68.5%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot {\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{2}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 68.5%

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

   9. Simplified 68.5%

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

   10. Taylor expanded in x around inf 96.3%

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

       1. mul-1-neg 96.3%

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

       2. log-rec 96.3%

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

       3. distribute-neg-frac 96.3%

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

       4. remove-double-neg 96.3%

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

       5. *-rgt-identity 96.3%

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

       6. associate-*r/ 96.2%

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

       7. exp-to-pow 96.2%

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

       8. *-commutative 96.2%

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

   12. Simplified 96.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42 \cdot 10^{-204}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 10^{-169}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-141}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{if}\;x \leq 7.6 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-170}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (+ 1.0 (/ x n)) t_0)))
   (if (<= x 7.6e-205)
     t_1
     (if (<= x 1.65e-170)
       (- (/ (log x) n))
       (if (<= x 2.05e-141)
         t_1
         (if (<= x 1.58e-12) (* (log x) (/ 1.0 (- n))) (/ t_0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (1.0 + (x / n)) - t_0;
	double tmp;
	if (x <= 7.6e-205) {
		tmp = t_1;
	} else if (x <= 1.65e-170) {
		tmp = -(log(x) / n);
	} else if (x <= 2.05e-141) {
		tmp = t_1;
	} else if (x <= 1.58e-12) {
		tmp = log(x) * (1.0 / -n);
	} else {
		tmp = t_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 = x ** (1.0d0 / n)
    t_1 = (1.0d0 + (x / n)) - t_0
    if (x <= 7.6d-205) then
        tmp = t_1
    else if (x <= 1.65d-170) then
        tmp = -(log(x) / n)
    else if (x <= 2.05d-141) then
        tmp = t_1
    else if (x <= 1.58d-12) then
        tmp = log(x) * (1.0d0 / -n)
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (1.0 + (x / n)) - t_0;
	double tmp;
	if (x <= 7.6e-205) {
		tmp = t_1;
	} else if (x <= 1.65e-170) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 2.05e-141) {
		tmp = t_1;
	} else if (x <= 1.58e-12) {
		tmp = Math.log(x) * (1.0 / -n);
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (1.0 + (x / n)) - t_0
	tmp = 0
	if x <= 7.6e-205:
		tmp = t_1
	elif x <= 1.65e-170:
		tmp = -(math.log(x) / n)
	elif x <= 2.05e-141:
		tmp = t_1
	elif x <= 1.58e-12:
		tmp = math.log(x) * (1.0 / -n)
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(1.0 + Float64(x / n)) - t_0)
	tmp = 0.0
	if (x <= 7.6e-205)
		tmp = t_1;
	elseif (x <= 1.65e-170)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 2.05e-141)
		tmp = t_1;
	elseif (x <= 1.58e-12)
		tmp = Float64(log(x) * Float64(1.0 / Float64(-n)));
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (1.0 + (x / n)) - t_0;
	tmp = 0.0;
	if (x <= 7.6e-205)
		tmp = t_1;
	elseif (x <= 1.65e-170)
		tmp = -(log(x) / n);
	elseif (x <= 2.05e-141)
		tmp = t_1;
	elseif (x <= 1.58e-12)
		tmp = log(x) * (1.0 / -n);
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[x, 7.6e-205], t$95$1, If[LessEqual[x, 1.65e-170], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 2.05e-141], t$95$1, If[LessEqual[x, 1.58e-12], N[(N[Log[x], $MachinePrecision] * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.59999999999999983e-205 or 1.65000000000000002e-170 < x < 2.05000000000000001e-141

   1. Initial program 56.8%

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

   3. Taylor expanded in x around 0 57.0%

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

   if 7.59999999999999983e-205 < x < 1.65000000000000002e-170

   1. Initial program 22.0%

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

   3. Taylor expanded in x around 0 22.0%

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

   4. Taylor expanded in n around inf 77.9%

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

      1. neg-mul-1 77.9%

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

      2. distribute-neg-frac 77.9%

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

   6. Simplified 77.9%

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

   if 2.05000000000000001e-141 < x < 1.57999999999999993e-12

   1. Initial program 31.9%

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

   3. Taylor expanded in x around 0 30.1%

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

   4. Taylor expanded in n around inf 61.6%

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

      1. neg-mul-1 61.6%

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

      2. distribute-neg-frac 61.6%

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

   6. Simplified 61.6%

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

      1. frac-2neg 61.6%

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

      2. div-inv 61.6%

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

      3. remove-double-neg 61.6%

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

   8. Applied egg-rr 61.6%

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

   if 1.57999999999999993e-12 < x

   1. Initial program 68.5%

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

   3. Taylor expanded in n around 0 68.5%

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

      1. log1p-def 68.5%

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

   5. Simplified 68.5%

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

      1. rem-cbrt-cube 68.5%

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

      2. pow1/3 65.7%

         \[\leadsto \color{blue}{{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

      3. cube-mult 65.7%

         \[\leadsto {\color{blue}{\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right)}}^{0.3333333333333333} \]

      4. unpow-prod-down 65.7%

         \[\leadsto \color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333} \cdot {\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{0.3333333333333333}} \]

      5. pow1/3 68.5%

         \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}} \cdot {\left(\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{0.3333333333333333} \]

      6. pow2 68.5%

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

   7. Applied egg-rr 68.5%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot {\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{2}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 68.5%

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

   9. Simplified 68.5%

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

   10. Taylor expanded in x around inf 96.3%

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

       1. mul-1-neg 96.3%

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

       2. log-rec 96.3%

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

       3. distribute-neg-frac 96.3%

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

       4. remove-double-neg 96.3%

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

       5. *-rgt-identity 96.3%

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

       6. associate-*r/ 96.2%

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

       7. exp-to-pow 96.2%

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

       8. *-commutative 96.2%

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

   12. Simplified 96.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{-205}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-170}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-141}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -54000000000:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;n \leq 47000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -54000000000.0)
   (- (/ (log x) n))
   (if (<= n 47000000000.0)
     (- 1.0 (pow x (/ 1.0 n)))
     (* (log x) (/ 1.0 (- n))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -54000000000.0) {
		tmp = -(log(x) / n);
	} else if (n <= 47000000000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = log(x) * (1.0 / -n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-54000000000.0d0)) then
        tmp = -(log(x) / n)
    else if (n <= 47000000000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = log(x) * (1.0d0 / -n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -54000000000.0) {
		tmp = -(Math.log(x) / n);
	} else if (n <= 47000000000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log(x) * (1.0 / -n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -54000000000.0:
		tmp = -(math.log(x) / n)
	elif n <= 47000000000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.log(x) * (1.0 / -n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -54000000000.0)
		tmp = Float64(-Float64(log(x) / n));
	elseif (n <= 47000000000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log(x) * Float64(1.0 / Float64(-n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -54000000000.0)
		tmp = -(log(x) / n);
	elseif (n <= 47000000000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = log(x) * (1.0 / -n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -54000000000.0], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[n, 47000000000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[x], $MachinePrecision] * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.4e10

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0 21.5%

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

   4. Taylor expanded in n around inf 60.4%

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

      1. neg-mul-1 60.4%

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

      2. distribute-neg-frac 60.4%

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

   6. Simplified 60.4%

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

   if -5.4e10 < n < 4.7e10

   1. Initial program 85.3%

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

   3. Taylor expanded in x around 0 51.2%

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

   if 4.7e10 < n

   1. Initial program 30.4%

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

   3. Taylor expanded in x around 0 30.4%

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

   4. Taylor expanded in n around inf 50.1%

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

      1. neg-mul-1 50.1%

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

      2. distribute-neg-frac 50.1%

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

   6. Simplified 50.1%

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

      1. frac-2neg 50.1%

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

      2. div-inv 50.1%

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

      3. remove-double-neg 50.1%

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

   8. Applied egg-rr 50.1%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54000000000:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;n \leq 47000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 (- (/ (log x) n)))

C

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

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = -(log(x) / n)
end function

Java

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

Python

def code(x, n):
	return -(math.log(x) / n)

Julia

function code(x, n)
	return Float64(-Float64(log(x) / n))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0 38.2%

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

4. Taylor expanded in n around inf 31.1%

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

   1. neg-mul-1 31.1%

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

   2. distribute-neg-frac 31.1%

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

6. Simplified 31.1%

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

7. Final simplification 31.1%

   \[\leadsto -\frac{\log x}{n} \]
8. Add Preprocessing

Alternative 10: 3.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double n) {
	return Math.log(x) / n;
}

Python

def code(x, n):
	return math.log(x) / n

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0 38.2%

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

4. Taylor expanded in n around inf 31.1%

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

   1. neg-mul-1 31.1%

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

   2. distribute-neg-frac 31.1%

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

6. Simplified 31.1%

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

   1. add-sqr-sqrt 29.4%

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

   2. sqrt-unprod 31.5%

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

   3. sqr-neg 31.5%

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

   4. sqrt-unprod 1.9%

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

   5. add-sqr-sqrt 3.1%

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

   6. clear-num 3.1%

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

   7. associate-/r/ 3.1%

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

8. Applied egg-rr 3.1%

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

9. Taylor expanded in n around 0 3.1%

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

10. Final simplification 3.1%

    \[\leadsto \frac{\log x}{n} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024023 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))