2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 86.4%

Time: 59.9s

Alternatives: 23

Speedup: 1.8×

• 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: 53.4% 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: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-41)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2.0)
         (/ (exp (/ (log x) n)) (* n x))
         (- (expm1 (log1p (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) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-41) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = exp((log(x) / n)) / (n * x);
	} else {
		tmp = expm1(log1p(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) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-41) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else {
		tmp = Math.expm1(Math.log1p(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) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-41:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2.0:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	else:
		tmp = math.expm1(math.log1p(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) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-41)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	else
		tmp = Float64(expm1(log1p(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], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-41], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[N[Log[1 + N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. div-inv 99.9%

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

      3. exp-to-pow 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   9. Simplified 100.0%

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

   if -2e-16 < (/.f64 1 n) < 2.00000000000000001e-41

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 85.0%

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

      1. log1p-define 85.0%

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

   5. Simplified 85.0%

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

      1. log1p-undefine 85.0%

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

      2. diff-log 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. +-commutative 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.0%

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

       2. log-div 85.0%

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

       3. metadata-eval 85.0%

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

       4. +-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

       1. neg-sub0 85.0%

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

   13. Simplified 85.0%

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

   if 2.00000000000000001e-41 < (/.f64 1 n) < 2

   1. Initial program 5.7%

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

   3. Taylor expanded in x around inf 98.0%

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

      1. mul-1-neg 98.0%

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

      2. log-rec 98.0%

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

      3. mul-1-neg 98.0%

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

      4. distribute-neg-frac 98.0%

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

      5. mul-1-neg 98.0%

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

      6. remove-double-neg 98.0%

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

      7. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   if 2 < (/.f64 1 n)

   1. Initial program 52.6%

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

      1. expm1-log1p-u 52.6%

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

      2. expm1-undefine 52.6%

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

      3. pow-to-exp 52.6%

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

      4. un-div-inv 52.6%

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

      5. +-commutative 52.6%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. expm1-define 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-41)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2.0)
         (/ (exp (/ (log x) n)) (* n x))
         (- (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) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-41) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = exp((log(x) / n)) / (n * x);
	} 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) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-41) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} 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) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-41:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2.0:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	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) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-41)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	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], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-41], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. div-inv 99.9%

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

      3. exp-to-pow 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   9. Simplified 100.0%

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

   if -2e-16 < (/.f64 1 n) < 2.00000000000000001e-41

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 85.0%

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

      1. log1p-define 85.0%

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

   5. Simplified 85.0%

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

      1. log1p-undefine 85.0%

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

      2. diff-log 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. +-commutative 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.0%

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

       2. log-div 85.0%

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

       3. metadata-eval 85.0%

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

       4. +-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

       1. neg-sub0 85.0%

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

   13. Simplified 85.0%

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

   if 2.00000000000000001e-41 < (/.f64 1 n) < 2

   1. Initial program 5.7%

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

   3. Taylor expanded in x around inf 98.0%

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

      1. mul-1-neg 98.0%

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

      2. log-rec 98.0%

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

      3. mul-1-neg 98.0%

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

      4. distribute-neg-frac 98.0%

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

      5. mul-1-neg 98.0%

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

      6. remove-double-neg 98.0%

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

      7. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   if 2 < (/.f64 1 n)

   1. Initial program 52.6%

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

   3. Taylor expanded in n around 0 52.6%

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

      1. log1p-define 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. associate-*l/ 99.9%

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

      4. associate-/l* 99.9%

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

      5. exp-to-pow 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \left(\frac{1}{n} \cdot \frac{1}{n}\right) + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-16)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-41)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2.0)
         (/ (exp (/ (log x) n)) (* n x))
         (-
          (+
           1.0
           (*
            x
            (+
             (/ 1.0 n)
             (* x (+ (* 0.5 (* (/ 1.0 n) (/ 1.0 n))) (* 0.5 (/ -1.0 n)))))))
          t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-41) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = exp((log(x) / n)) / (n * x);
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * ((1.0 / n) * (1.0 / n))) + (0.5 * (-1.0 / 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) <= (-2d-16)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-41) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 2.0d0) then
        tmp = exp((log(x) / n)) / (n * x)
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (x * ((0.5d0 * ((1.0d0 / n) * (1.0d0 / n))) + (0.5d0 * ((-1.0d0) / 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) <= -2e-16) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-41) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * ((1.0 / n) * (1.0 / n))) + (0.5 * (-1.0 / n))))))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-41:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2.0:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * ((1.0 / n) * (1.0 / n))) + (0.5 * (-1.0 / 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) <= -2e-16)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-41)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 * Float64(Float64(1.0 / n) * Float64(1.0 / n))) + Float64(0.5 * Float64(-1.0 / 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) <= -2e-16)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-41)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 2.0)
		tmp = exp((log(x) / n)) / (n * x);
	else
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * ((1.0 / n) * (1.0 / n))) + (0.5 * (-1.0 / 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], -2e-16], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-41], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(0.5 * N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. div-inv 99.9%

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

      3. exp-to-pow 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   9. Simplified 100.0%

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

   if -2e-16 < (/.f64 1 n) < 2.00000000000000001e-41

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 85.0%

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

      1. log1p-define 85.0%

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

   5. Simplified 85.0%

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

      1. log1p-undefine 85.0%

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

      2. diff-log 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. +-commutative 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.0%

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

       2. log-div 85.0%

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

       3. metadata-eval 85.0%

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

       4. +-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

       1. neg-sub0 85.0%

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

   13. Simplified 85.0%

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

   if 2.00000000000000001e-41 < (/.f64 1 n) < 2

   1. Initial program 5.7%

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

   3. Taylor expanded in x around inf 98.0%

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

      1. mul-1-neg 98.0%

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

      2. log-rec 98.0%

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

      3. mul-1-neg 98.0%

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

      4. distribute-neg-frac 98.0%

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

      5. mul-1-neg 98.0%

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

      6. remove-double-neg 98.0%

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

      7. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   if 2 < (/.f64 1 n)

   1. Initial program 52.6%

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

   3. Taylor expanded in x around 0 75.1%

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

      1. inv-pow 75.1%

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

      2. unpow2 75.1%

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

      3. unpow-prod-down 75.1%

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

      4. inv-pow 75.1%

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

      5. inv-pow 75.1%

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

   5. Applied egg-rr 75.1%

      \[\leadsto \left(1 + x \cdot \left(x \cdot \left(0.5 \cdot \color{blue}{\left(\frac{1}{n} \cdot \frac{1}{n}\right)} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \left(\frac{1}{n} \cdot \frac{1}{n}\right) + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \left(\frac{1}{n} \cdot \frac{1}{n}\right) + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -2e-16)
     t_1
     (if (<= (/ 1.0 n) 2e-41)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2.0)
         t_1
         (-
          (+
           1.0
           (*
            x
            (+
             (/ 1.0 n)
             (* x (+ (* 0.5 (* (/ 1.0 n) (/ 1.0 n))) (* 0.5 (/ -1.0 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) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * ((1.0 / n) * (1.0 / n))) + (0.5 * (-1.0 / n))))))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-2d-16)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-41) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 2.0d0) then
        tmp = t_1
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (x * ((0.5d0 * ((1.0d0 / n) * (1.0d0 / n))) + (0.5d0 * ((-1.0d0) / n))))))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * ((1.0 / n) * (1.0 / n))) + (0.5 * (-1.0 / 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) <= -2e-16:
		tmp = t_1
	elif (1.0 / n) <= 2e-41:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2.0:
		tmp = t_1
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * ((1.0 / n) * (1.0 / n))) + (0.5 * (-1.0 / 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) <= -2e-16)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-41)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 * Float64(Float64(1.0 / n) * Float64(1.0 / n))) + Float64(0.5 * Float64(-1.0 / n))))))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-16)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-41)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 2.0)
		tmp = t_1;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * ((1.0 / n) * (1.0 / n))) + (0.5 * (-1.0 / n))))))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-41], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], t$95$1, N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(0.5 * N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2e-16 or 2.00000000000000001e-41 < (/.f64 1 n) < 2

   1. Initial program 87.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. log-rec 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. mul-1-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. div-inv 99.7%

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

      3. exp-to-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   9. Simplified 99.8%

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

   if -2e-16 < (/.f64 1 n) < 2.00000000000000001e-41

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 85.0%

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

      1. log1p-define 85.0%

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

   5. Simplified 85.0%

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

      1. log1p-undefine 85.0%

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

      2. diff-log 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. +-commutative 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.0%

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

       2. log-div 85.0%

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

       3. metadata-eval 85.0%

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

       4. +-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

       1. neg-sub0 85.0%

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

   13. Simplified 85.0%

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

   if 2 < (/.f64 1 n)

   1. Initial program 52.6%

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

   3. Taylor expanded in x around 0 75.1%

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

      1. inv-pow 75.1%

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

      2. unpow2 75.1%

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

      3. unpow-prod-down 75.1%

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

      4. inv-pow 75.1%

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

      5. inv-pow 75.1%

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

   5. Applied egg-rr 75.1%

      \[\leadsto \left(1 + x \cdot \left(x \cdot \left(0.5 \cdot \color{blue}{\left(\frac{1}{n} \cdot \frac{1}{n}\right)} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \left(\frac{1}{n} \cdot \frac{1}{n}\right) + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -2e-16)
     t_1
     (if (<= (/ 1.0 n) 2e-41)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2.0)
         t_1
         (-
          (+ 1.0 (* x (+ (/ 1.0 n) (/ (+ (* x -0.5) (* 0.5 (/ x n))) 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) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-2d-16)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-41) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 2.0d0) then
        tmp = t_1
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (((x * (-0.5d0)) + (0.5d0 * (x / n))) / n)))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / 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) <= -2e-16:
		tmp = t_1
	elif (1.0 / n) <= 2e-41:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2.0:
		tmp = t_1
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / 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) <= -2e-16)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-41)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n))) / n)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-16)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-41)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 2.0)
		tmp = t_1;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-41], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], t$95$1, N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2e-16 or 2.00000000000000001e-41 < (/.f64 1 n) < 2

   1. Initial program 87.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. log-rec 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. mul-1-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. div-inv 99.7%

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

      3. exp-to-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   9. Simplified 99.8%

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

   if -2e-16 < (/.f64 1 n) < 2.00000000000000001e-41

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 85.0%

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

      1. log1p-define 85.0%

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

   5. Simplified 85.0%

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

      1. log1p-undefine 85.0%

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

      2. diff-log 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. +-commutative 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.0%

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

       2. log-div 85.0%

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

       3. metadata-eval 85.0%

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

       4. +-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

       1. neg-sub0 85.0%

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

   13. Simplified 85.0%

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

   if 2 < (/.f64 1 n)

   1. Initial program 52.6%

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

   3. Taylor expanded in x around 0 75.1%

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

   4. Taylor expanded in n around inf 73.0%

      \[\leadsto \left(1 + x \cdot \left(\color{blue}{\frac{-0.5 \cdot x + 0.5 \cdot \frac{x}{n}}{n}} + \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+219}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{n}{1 + \frac{0.25}{{x}^{3}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -2e-16)
     t_1
     (if (<= (/ 1.0 n) 2e-41)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2.0)
         t_1
         (if (<= (/ 1.0 n) 1e+219)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 1.0 (* x (/ n (+ 1.0 (/ 0.25 (pow x 3.0))))))))))))

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) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+219) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (x * (n / (1.0 + (0.25 / 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) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-2d-16)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-41) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 2.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+219) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (x * (n / (1.0d0 + (0.25d0 / (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 t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+219) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (x * (n / (1.0 + (0.25 / Math.pow(x, 3.0)))));
	}
	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) <= -2e-16:
		tmp = t_1
	elif (1.0 / n) <= 2e-41:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2.0:
		tmp = t_1
	elif (1.0 / n) <= 1e+219:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (x * (n / (1.0 + (0.25 / math.pow(x, 3.0)))))
	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) <= -2e-16)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-41)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+219)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(x * Float64(n / Float64(1.0 + Float64(0.25 / (x ^ 3.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-16)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-41)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 2.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+219)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (x * (n / (1.0 + (0.25 / (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]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-41], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+219], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(x * N[(n / N[(1.0 + N[(0.25 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16 or 2.00000000000000001e-41 < (/.f64 1 n) < 2

   1. Initial program 87.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. log-rec 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. mul-1-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. div-inv 99.7%

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

      3. exp-to-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   9. Simplified 99.8%

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

   if -2e-16 < (/.f64 1 n) < 2.00000000000000001e-41

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 85.0%

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

      1. log1p-define 85.0%

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

   5. Simplified 85.0%

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

      1. log1p-undefine 85.0%

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

      2. diff-log 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. +-commutative 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.0%

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

       2. log-div 85.0%

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

       3. metadata-eval 85.0%

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

       4. +-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

       1. neg-sub0 85.0%

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

   13. Simplified 85.0%

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

   if 2 < (/.f64 1 n) < 9.99999999999999965e218

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0 64.6%

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

   if 9.99999999999999965e218 < (/.f64 1 n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 8.0%

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

      1. log1p-define 8.0%

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

   5. Simplified 8.0%

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

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   8. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{{\left(\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 88.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}}} \]

      2. associate-/r/ 88.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1} \cdot x}} \]

      3. +-commutative 88.6%

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}} \cdot x} \]

      4. +-commutative 88.6%

         \[\leadsto \frac{1}{\frac{n}{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}} \cdot x} \]

   10. Simplified 88.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}} \cdot x}} \]

   11. Taylor expanded in x around 0 88.6%

       \[\leadsto \frac{1}{\frac{n}{1 + \color{blue}{\frac{0.25}{{x}^{3}}}} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+219}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{n}{1 + \frac{0.25}{{x}^{3}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+219}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{n \cdot {x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -2e-16)
     t_1
     (if (<= (/ 1.0 n) 2e-41)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2.0)
         t_1
         (if (<= (/ 1.0 n) 1e+219)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 0.25 (* n (pow x 4.0)))))))))

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) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+219) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.25 / (n * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-2d-16)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-41) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 2.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+219) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 0.25d0 / (n * (x ** 4.0d0))
    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 t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+219) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.25 / (n * Math.pow(x, 4.0));
	}
	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) <= -2e-16:
		tmp = t_1
	elif (1.0 / n) <= 2e-41:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2.0:
		tmp = t_1
	elif (1.0 / n) <= 1e+219:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 0.25 / (n * math.pow(x, 4.0))
	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) <= -2e-16)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-41)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+219)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.25 / Float64(n * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-16)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-41)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 2.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+219)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 0.25 / (n * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-41], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+219], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.25 / N[(n * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16 or 2.00000000000000001e-41 < (/.f64 1 n) < 2

   1. Initial program 87.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. log-rec 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. mul-1-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. div-inv 99.7%

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

      3. exp-to-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   9. Simplified 99.8%

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

   if -2e-16 < (/.f64 1 n) < 2.00000000000000001e-41

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 85.0%

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

      1. log1p-define 85.0%

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

   5. Simplified 85.0%

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

      1. log1p-undefine 85.0%

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

      2. diff-log 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. +-commutative 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.0%

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

       2. log-div 85.0%

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

       3. metadata-eval 85.0%

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

       4. +-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

       1. neg-sub0 85.0%

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

   13. Simplified 85.0%

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

   if 2 < (/.f64 1 n) < 9.99999999999999965e218

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0 64.6%

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

   if 9.99999999999999965e218 < (/.f64 1 n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 8.0%

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

      1. log1p-define 8.0%

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

   5. Simplified 8.0%

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

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   8. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{{\left(\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 88.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}}} \]

      2. associate-/r/ 88.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1} \cdot x}} \]

      3. +-commutative 88.6%

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}} \cdot x} \]

      4. +-commutative 88.6%

         \[\leadsto \frac{1}{\frac{n}{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}} \cdot x} \]

   10. Simplified 88.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}} \cdot x}} \]

   11. Taylor expanded in x around 0 88.6%

       \[\leadsto \color{blue}{\frac{0.25}{n \cdot {x}^{4}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+219}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{n \cdot {x}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+219}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{n \cdot {x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -2e-16)
     t_1
     (if (<= (/ 1.0 n) 2e-41)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2.0)
         t_1
         (if (<= (/ 1.0 n) 1e+219) (- 1.0 t_0) (/ 0.25 (* n (pow x 4.0)))))))))

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) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+219) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.25 / (n * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / n) / x
    if ((1.0d0 / n) <= (-2d-16)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-41) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 2.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+219) then
        tmp = 1.0d0 - t_0
    else
        tmp = 0.25d0 / (n * (x ** 4.0d0))
    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 t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+219) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.25 / (n * Math.pow(x, 4.0));
	}
	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) <= -2e-16:
		tmp = t_1
	elif (1.0 / n) <= 2e-41:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2.0:
		tmp = t_1
	elif (1.0 / n) <= 1e+219:
		tmp = 1.0 - t_0
	else:
		tmp = 0.25 / (n * math.pow(x, 4.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-41)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+219)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.25 / Float64(n * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / n) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -2e-16)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-41)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 2.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+219)
		tmp = 1.0 - t_0;
	else
		tmp = 0.25 / (n * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-41], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+219], N[(1.0 - t$95$0), $MachinePrecision], N[(0.25 / N[(n * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16 or 2.00000000000000001e-41 < (/.f64 1 n) < 2

   1. Initial program 87.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. associate-/r* 99.7%

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

      2. mul-1-neg 99.7%

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

      3. log-rec 99.7%

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

      4. mul-1-neg 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. mul-1-neg 99.7%

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

      7. remove-double-neg 99.7%

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

      8. *-rgt-identity 99.7%

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

      9. associate-/l* 99.7%

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

      10. exp-to-pow 99.8%

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

   5. Simplified 99.8%

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

   if -2e-16 < (/.f64 1 n) < 2.00000000000000001e-41

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 85.0%

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

      1. log1p-define 85.0%

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

   5. Simplified 85.0%

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

      1. log1p-undefine 85.0%

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

      2. diff-log 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. +-commutative 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.0%

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

       2. log-div 85.0%

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

       3. metadata-eval 85.0%

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

       4. +-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

       1. neg-sub0 85.0%

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

   13. Simplified 85.0%

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

   if 2 < (/.f64 1 n) < 9.99999999999999965e218

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0 61.0%

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

      1. *-rgt-identity 61.0%

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

      2. associate-*l/ 61.0%

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

      3. associate-/l* 61.0%

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

      4. exp-to-pow 61.0%

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

   5. Simplified 61.0%

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

   if 9.99999999999999965e218 < (/.f64 1 n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 8.0%

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

      1. log1p-define 8.0%

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

   5. Simplified 8.0%

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

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   8. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{{\left(\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 88.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}}} \]

      2. associate-/r/ 88.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1} \cdot x}} \]

      3. +-commutative 88.6%

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}} \cdot x} \]

      4. +-commutative 88.6%

         \[\leadsto \frac{1}{\frac{n}{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}} \cdot x} \]

   10. Simplified 88.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}} \cdot x}} \]

   11. Taylor expanded in x around 0 88.6%

       \[\leadsto \color{blue}{\frac{0.25}{n \cdot {x}^{4}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+219}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{n \cdot {x}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+219}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{n \cdot {x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -2e-16)
     t_1
     (if (<= (/ 1.0 n) 2e-41)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 2.0)
         t_1
         (if (<= (/ 1.0 n) 1e+219) (- 1.0 t_0) (/ 0.25 (* n (pow x 4.0)))))))))

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) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+219) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.25 / (n * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-2d-16)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-41) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 2.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+219) then
        tmp = 1.0d0 - t_0
    else
        tmp = 0.25d0 / (n * (x ** 4.0d0))
    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 t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-41) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 2.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+219) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.25 / (n * Math.pow(x, 4.0));
	}
	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) <= -2e-16:
		tmp = t_1
	elif (1.0 / n) <= 2e-41:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 2.0:
		tmp = t_1
	elif (1.0 / n) <= 1e+219:
		tmp = 1.0 - t_0
	else:
		tmp = 0.25 / (n * math.pow(x, 4.0))
	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) <= -2e-16)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-41)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 2.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+219)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.25 / Float64(n * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-16)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-41)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 2.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+219)
		tmp = 1.0 - t_0;
	else
		tmp = 0.25 / (n * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-41], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+219], N[(1.0 - t$95$0), $MachinePrecision], N[(0.25 / N[(n * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e-16 or 2.00000000000000001e-41 < (/.f64 1 n) < 2

   1. Initial program 87.7%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. log-rec 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. mul-1-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. div-inv 99.7%

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

      3. exp-to-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   9. Simplified 99.8%

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

   if -2e-16 < (/.f64 1 n) < 2.00000000000000001e-41

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf 85.0%

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

      1. log1p-define 85.0%

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

   5. Simplified 85.0%

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

      1. log1p-undefine 85.0%

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

      2. diff-log 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. +-commutative 85.0%

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

   9. Simplified 85.0%

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

       1. clear-num 85.0%

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

       2. log-div 85.0%

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

       3. metadata-eval 85.0%

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

       4. +-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

       1. neg-sub0 85.0%

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

   13. Simplified 85.0%

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

   if 2 < (/.f64 1 n) < 9.99999999999999965e218

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0 61.0%

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

      1. *-rgt-identity 61.0%

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

      2. associate-*l/ 61.0%

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

      3. associate-/l* 61.0%

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

      4. exp-to-pow 61.0%

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

   5. Simplified 61.0%

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

   if 9.99999999999999965e218 < (/.f64 1 n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 8.0%

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

      1. log1p-define 8.0%

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

   5. Simplified 8.0%

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

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   8. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{{\left(\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 88.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}}} \]

      2. associate-/r/ 88.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1} \cdot x}} \]

      3. +-commutative 88.6%

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}} \cdot x} \]

      4. +-commutative 88.6%

         \[\leadsto \frac{1}{\frac{n}{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}} \cdot x} \]

   10. Simplified 88.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}} \cdot x}} \]

   11. Taylor expanded in x around 0 88.6%

       \[\leadsto \color{blue}{\frac{0.25}{n \cdot {x}^{4}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+219}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{n \cdot {x}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;n \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-219}:\\ \;\;\;\;\frac{0.25}{n \cdot {x}^{4}}\\ \mathbf{elif}\;n \leq 0.75:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= n -4e+15)
     t_0
     (if (<= n 1.06e-219)
       (/ 0.25 (* n (pow x 4.0)))
       (if (<= n 0.75)
         (- 1.0 (pow x (/ 1.0 n)))
         (if (<= n 4.2e+49) (/ 1.0 (* x (+ n (* 0.5 (/ n x))))) t_0))))))

C

double code(double x, double n) {
	double t_0 = log(((1.0 + x) / x)) / n;
	double tmp;
	if (n <= -4e+15) {
		tmp = t_0;
	} else if (n <= 1.06e-219) {
		tmp = 0.25 / (n * pow(x, 4.0));
	} else if (n <= 0.75) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 4.2e+49) {
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	} 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 <= (-4d+15)) then
        tmp = t_0
    else if (n <= 1.06d-219) then
        tmp = 0.25d0 / (n * (x ** 4.0d0))
    else if (n <= 0.75d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (n <= 4.2d+49) then
        tmp = 1.0d0 / (x * (n + (0.5d0 * (n / x))))
    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 <= -4e+15) {
		tmp = t_0;
	} else if (n <= 1.06e-219) {
		tmp = 0.25 / (n * Math.pow(x, 4.0));
	} else if (n <= 0.75) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (n <= 4.2e+49) {
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if n <= -4e+15:
		tmp = t_0
	elif n <= 1.06e-219:
		tmp = 0.25 / (n * math.pow(x, 4.0))
	elif n <= 0.75:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif n <= 4.2e+49:
		tmp = 1.0 / (x * (n + (0.5 * (n / x))))
	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 <= -4e+15)
		tmp = t_0;
	elseif (n <= 1.06e-219)
		tmp = Float64(0.25 / Float64(n * (x ^ 4.0)));
	elseif (n <= 0.75)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 4.2e+49)
		tmp = Float64(1.0 / Float64(x * Float64(n + Float64(0.5 * Float64(n / x)))));
	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 <= -4e+15)
		tmp = t_0;
	elseif (n <= 1.06e-219)
		tmp = 0.25 / (n * (x ^ 4.0));
	elseif (n <= 0.75)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (n <= 4.2e+49)
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	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, -4e+15], t$95$0, If[LessEqual[n, 1.06e-219], N[(0.25 / N[(n * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.75], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.2e+49], N[(1.0 / N[(x * N[(n + N[(0.5 * N[(n / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4e15 or 4.20000000000000022e49 < n

   1. Initial program 26.2%

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

   3. Taylor expanded in n around inf 85.5%

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

      1. log1p-define 85.5%

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

   5. Simplified 85.5%

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

      1. log1p-undefine 85.5%

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

      2. diff-log 85.5%

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

   7. Applied egg-rr 85.5%

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

      1. +-commutative 85.5%

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

   9. Simplified 85.5%

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

   if -4e15 < n < 1.06e-219

   1. Initial program 87.7%

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

   3. Taylor expanded in n around inf 48.3%

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

      1. log1p-define 48.3%

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

   5. Simplified 48.3%

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

   6. Taylor expanded in x around -inf 4.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   8. Applied egg-rr 44.2%

      \[\leadsto \color{blue}{{\left(\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 44.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}}} \]

      2. associate-/r/ 44.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1} \cdot x}} \]

      3. +-commutative 44.2%

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}} \cdot x} \]

      4. +-commutative 44.2%

         \[\leadsto \frac{1}{\frac{n}{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}} \cdot x} \]

   10. Simplified 44.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}} \cdot x}} \]

   11. Taylor expanded in x around 0 69.0%

       \[\leadsto \color{blue}{\frac{0.25}{n \cdot {x}^{4}}} \]

   if 1.06e-219 < n < 0.75

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0 61.0%

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

      1. *-rgt-identity 61.0%

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

      2. associate-*l/ 61.0%

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

      3. associate-/l* 61.0%

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

      4. exp-to-pow 61.0%

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

   5. Simplified 61.0%

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

   if 0.75 < n < 4.20000000000000022e49

   1. Initial program 5.3%

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

   3. Taylor expanded in n around inf 16.0%

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

      1. log1p-define 16.0%

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

   5. Simplified 16.0%

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

   6. Taylor expanded in x around -inf 72.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   8. Applied egg-rr 72.5%

      \[\leadsto \color{blue}{{\left(\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 72.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}}} \]

      2. associate-/r/ 72.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1} \cdot x}} \]

      3. +-commutative 72.6%

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}} \cdot x} \]

      4. +-commutative 72.6%

         \[\leadsto \frac{1}{\frac{n}{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}} \cdot x} \]

   10. Simplified 72.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}} \cdot x}} \]

   11. Taylor expanded in x around inf 73.9%

       \[\leadsto \frac{1}{\color{blue}{\left(n + 0.5 \cdot \frac{n}{x}\right)} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-219}:\\ \;\;\;\;\frac{0.25}{n \cdot {x}^{4}}\\ \mathbf{elif}\;n \leq 0.75:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3900000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-219}:\\ \;\;\;\;\frac{0.25}{n \cdot {x}^{4}}\\ \mathbf{elif}\;n \leq 0.75:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -3900000000.0)
   (/ (log (/ x (+ 1.0 x))) (- n))
   (if (<= n 1.06e-219)
     (/ 0.25 (* n (pow x 4.0)))
     (if (<= n 0.75)
       (- 1.0 (pow x (/ 1.0 n)))
       (if (<= n 2.6e+53)
         (/ 1.0 (* x (+ n (* 0.5 (/ n x)))))
         (/ (log (/ (+ 1.0 x) x)) n))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -3900000000.0) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if (n <= 1.06e-219) {
		tmp = 0.25 / (n * pow(x, 4.0));
	} else if (n <= 0.75) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 2.6e+53) {
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	} else {
		tmp = log(((1.0 + x) / 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 <= (-3900000000.0d0)) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if (n <= 1.06d-219) then
        tmp = 0.25d0 / (n * (x ** 4.0d0))
    else if (n <= 0.75d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (n <= 2.6d+53) then
        tmp = 1.0d0 / (x * (n + (0.5d0 * (n / x))))
    else
        tmp = log(((1.0d0 + x) / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -3900000000.0) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if (n <= 1.06e-219) {
		tmp = 0.25 / (n * Math.pow(x, 4.0));
	} else if (n <= 0.75) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (n <= 2.6e+53) {
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	} else {
		tmp = Math.log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -3900000000.0:
		tmp = math.log((x / (1.0 + x))) / -n
	elif n <= 1.06e-219:
		tmp = 0.25 / (n * math.pow(x, 4.0))
	elif n <= 0.75:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif n <= 2.6e+53:
		tmp = 1.0 / (x * (n + (0.5 * (n / x))))
	else:
		tmp = math.log(((1.0 + x) / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -3900000000.0)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (n <= 1.06e-219)
		tmp = Float64(0.25 / Float64(n * (x ^ 4.0)));
	elseif (n <= 0.75)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 2.6e+53)
		tmp = Float64(1.0 / Float64(x * Float64(n + Float64(0.5 * Float64(n / x)))));
	else
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -3900000000.0)
		tmp = log((x / (1.0 + x))) / -n;
	elseif (n <= 1.06e-219)
		tmp = 0.25 / (n * (x ^ 4.0));
	elseif (n <= 0.75)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (n <= 2.6e+53)
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	else
		tmp = log(((1.0 + x) / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -3900000000.0], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[n, 1.06e-219], N[(0.25 / N[(n * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.75], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.6e+53], N[(1.0 / N[(x * N[(n + N[(0.5 * N[(n / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -3.9e9

   1. Initial program 23.4%

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

   3. Taylor expanded in n around inf 78.3%

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

      1. log1p-define 78.3%

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

   5. Simplified 78.3%

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

      1. log1p-undefine 78.3%

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

      2. diff-log 78.4%

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

   7. Applied egg-rr 78.4%

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

      1. +-commutative 78.4%

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

   9. Simplified 78.4%

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

       1. clear-num 78.4%

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

       2. log-div 78.4%

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

       3. metadata-eval 78.4%

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

       4. +-commutative 78.4%

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

   11. Applied egg-rr 78.4%

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

       1. neg-sub0 78.4%

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

   13. Simplified 78.4%

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

   if -3.9e9 < n < 1.06e-219

   1. Initial program 89.6%

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

   3. Taylor expanded in n around inf 49.4%

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

      1. log1p-define 49.4%

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

   5. Simplified 49.4%

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

   6. Taylor expanded in x around -inf 2.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   8. Applied egg-rr 43.5%

      \[\leadsto \color{blue}{{\left(\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 43.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}}} \]

      2. associate-/r/ 43.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1} \cdot x}} \]

      3. +-commutative 43.5%

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}} \cdot x} \]

      4. +-commutative 43.5%

         \[\leadsto \frac{1}{\frac{n}{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}} \cdot x} \]

   10. Simplified 43.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}} \cdot x}} \]

   11. Taylor expanded in x around 0 70.5%

       \[\leadsto \color{blue}{\frac{0.25}{n \cdot {x}^{4}}} \]

   if 1.06e-219 < n < 0.75

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0 61.0%

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

      1. *-rgt-identity 61.0%

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

      2. associate-*l/ 61.0%

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

      3. associate-/l* 61.0%

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

      4. exp-to-pow 61.0%

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

   5. Simplified 61.0%

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

   if 0.75 < n < 2.59999999999999998e53

   1. Initial program 5.3%

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

   3. Taylor expanded in n around inf 16.0%

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

      1. log1p-define 16.0%

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

   5. Simplified 16.0%

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

   6. Taylor expanded in x around -inf 72.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   8. Applied egg-rr 72.5%

      \[\leadsto \color{blue}{{\left(\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 72.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}}} \]

      2. associate-/r/ 72.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1} \cdot x}} \]

      3. +-commutative 72.6%

         \[\leadsto \frac{1}{\frac{n}{\color{blue}{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}} \cdot x} \]

      4. +-commutative 72.6%

         \[\leadsto \frac{1}{\frac{n}{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}} \cdot x} \]

   10. Simplified 72.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}} \cdot x}} \]

   11. Taylor expanded in x around inf 73.9%

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

   if 2.59999999999999998e53 < n

   1. Initial program 28.3%

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

   3. Taylor expanded in n around inf 90.2%

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

      1. log1p-define 90.2%

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

   5. Simplified 90.2%

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

      1. log1p-undefine 90.2%

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

      2. diff-log 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. +-commutative 90.2%

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

   9. Simplified 90.2%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x + 1}{x}\right)}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3900000000:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-219}:\\ \;\;\;\;\frac{0.25}{n \cdot {x}^{4}}\\ \mathbf{elif}\;n \leq 0.75:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-164}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{-133}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{n \cdot {x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6e-164)
   (/ (log x) (- n))
   (if (<= x 6.9e-133)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.88)
       (/ (- x (log x)) n)
       (if (<= x 4e+172)
         (/
          (/
           (+
            1.0
            (/ (- (/ (- 0.3333333333333333 (* 0.25 (/ 1.0 x))) x) 0.5) x))
           x)
          n)
         (/ -0.25 (* n (pow x 4.0))))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6e-164) {
		tmp = log(x) / -n;
	} else if (x <= 6.9e-133) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4e+172) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = -0.25 / (n * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 6d-164) then
        tmp = log(x) / -n
    else if (x <= 6.9d-133) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4d+172) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - (0.25d0 * (1.0d0 / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = (-0.25d0) / (n * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6e-164) {
		tmp = Math.log(x) / -n;
	} else if (x <= 6.9e-133) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4e+172) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = -0.25 / (n * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6e-164:
		tmp = math.log(x) / -n
	elif x <= 6.9e-133:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 4e+172:
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = -0.25 / (n * math.pow(x, 4.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6e-164)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 6.9e-133)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4e+172)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(0.25 * Float64(1.0 / x))) / x) - 0.5) / x)) / x) / n);
	else
		tmp = Float64(-0.25 / Float64(n * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6e-164)
		tmp = log(x) / -n;
	elseif (x <= 6.9e-133)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 4e+172)
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = -0.25 / (n * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6e-164], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 6.9e-133], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4e+172], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(0.25 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(-0.25 / N[(n * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 6.0000000000000002e-164

   1. Initial program 38.3%

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

   3. Taylor expanded in n around inf 62.9%

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

      1. log1p-define 62.9%

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

   5. Simplified 62.9%

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

   6. Taylor expanded in x around 0 62.9%

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

      1. neg-mul-1 62.9%

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

   8. Simplified 62.9%

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

   if 6.0000000000000002e-164 < x < 6.9000000000000001e-133

   1. Initial program 61.6%

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

   3. Taylor expanded in x around 0 61.6%

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

      1. *-rgt-identity 61.6%

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

      2. associate-*l/ 61.6%

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

      3. associate-/l* 61.6%

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

      4. exp-to-pow 61.6%

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

   5. Simplified 61.6%

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

   if 6.9000000000000001e-133 < x < 0.880000000000000004

   1. Initial program 30.2%

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

   3. Taylor expanded in n around inf 60.2%

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

      1. log1p-define 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in x around 0 59.2%

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

   if 0.880000000000000004 < x < 4.0000000000000003e172

   1. Initial program 56.3%

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

   3. Taylor expanded in n around inf 59.0%

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

      1. log1p-define 59.0%

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

   5. Simplified 59.0%

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

   6. Taylor expanded in x around -inf 59.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   if 4.0000000000000003e172 < x

   1. Initial program 86.9%

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

   3. Taylor expanded in n around inf 86.9%

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

      1. log1p-define 86.9%

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

   5. Simplified 86.9%

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

   6. Taylor expanded in x around -inf 58.8%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in x around 0 86.9%

      \[\leadsto \color{blue}{\frac{-0.25}{n \cdot {x}^{4}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-164}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{-133}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{n \cdot {x}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-142}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1e-142)
   (/ (log x) (- n))
   (if (<= x 4.2e-132)
     (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
     (if (<= x 0.9)
       (/ (- x (log x)) n)
       (/
        (/
         (+ 1.0 (/ (- (/ (- 0.3333333333333333 (* 0.25 (/ 1.0 x))) x) 0.5) x))
         x)
        n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1e-142) {
		tmp = log(x) / -n;
	} else if (x <= 4.2e-132) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / 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 (x <= 1d-142) then
        tmp = log(x) / -n
    else if (x <= 4.2d-132) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - (0.25d0 * (1.0d0 / x))) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1e-142) {
		tmp = Math.log(x) / -n;
	} else if (x <= 4.2e-132) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1e-142:
		tmp = math.log(x) / -n
	elif x <= 4.2e-132:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	elif x <= 0.9:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1e-142)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 4.2e-132)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	elseif (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(0.25 * Float64(1.0 / x))) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1e-142)
		tmp = log(x) / -n;
	elseif (x <= 4.2e-132)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	elseif (x <= 0.9)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1e-142], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 4.2e-132], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(0.25 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1e-142

   1. Initial program 39.8%

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

   3. Taylor expanded in n around inf 60.0%

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

      1. log1p-define 60.0%

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

   5. Simplified 60.0%

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

   6. Taylor expanded in x around 0 60.0%

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

      1. neg-mul-1 60.0%

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

   8. Simplified 60.0%

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

   if 1e-142 < x < 4.2000000000000002e-132

   1. Initial program 67.7%

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

   3. Taylor expanded in n around inf 7.1%

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

      1. log1p-define 7.1%

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

   5. Simplified 7.1%

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

   6. Taylor expanded in x around inf 83.9%

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

      1. associate--l+ 83.9%

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

      2. unpow2 83.9%

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

      3. associate-/r* 83.9%

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

      4. metadata-eval 83.9%

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

      5. associate-*r/ 83.9%

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

      6. associate-*r/ 83.9%

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

      7. metadata-eval 83.9%

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

      8. div-sub 83.9%

         \[\leadsto \frac{\frac{1 + \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x}}{n} \]

      9. sub-neg 83.9%

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

      10. metadata-eval 83.9%

          \[\leadsto \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} + \color{blue}{-0.5}}{x}}{x}}{n} \]

      11. +-commutative 83.9%

          \[\leadsto \frac{\frac{1 + \frac{\color{blue}{-0.5 + 0.3333333333333333 \cdot \frac{1}{x}}}{x}}{x}}{n} \]

      12. associate-*r/ 83.9%

          \[\leadsto \frac{\frac{1 + \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{x}}}{x}}{x}}{n} \]

      13. metadata-eval 83.9%

          \[\leadsto \frac{\frac{1 + \frac{-0.5 + \frac{\color{blue}{0.3333333333333333}}{x}}{x}}{x}}{n} \]

   8. Simplified 83.9%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   if 4.2000000000000002e-132 < x < 0.900000000000000022

   1. Initial program 30.6%

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

   3. Taylor expanded in n around inf 60.9%

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

      1. log1p-define 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in x around 0 59.9%

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

   if 0.900000000000000022 < x

   1. Initial program 68.1%

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

   3. Taylor expanded in n around inf 69.8%

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

      1. log1p-define 69.8%

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

   5. Simplified 69.8%

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

   6. Taylor expanded in x around -inf 59.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-142}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-133}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.6e-164)
   (/ (log x) (- n))
   (if (<= x 9.5e-133)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.88)
       (/ (- x (log x)) n)
       (/
        (/
         (+ 1.0 (/ (- (/ (- 0.3333333333333333 (* 0.25 (/ 1.0 x))) x) 0.5) x))
         x)
        n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.6e-164) {
		tmp = log(x) / -n;
	} else if (x <= 9.5e-133) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / 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 (x <= 6.6d-164) then
        tmp = log(x) / -n
    else if (x <= 9.5d-133) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - (0.25d0 * (1.0d0 / x))) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6.6e-164) {
		tmp = Math.log(x) / -n;
	} else if (x <= 9.5e-133) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6.6e-164:
		tmp = math.log(x) / -n
	elif x <= 9.5e-133:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.6e-164)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 9.5e-133)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(0.25 * Float64(1.0 / x))) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6.6e-164)
		tmp = log(x) / -n;
	elseif (x <= 9.5e-133)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.6e-164], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 9.5e-133], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(0.25 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.6e-164

   1. Initial program 38.3%

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

   3. Taylor expanded in n around inf 62.9%

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

      1. log1p-define 62.9%

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

   5. Simplified 62.9%

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

   6. Taylor expanded in x around 0 62.9%

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

      1. neg-mul-1 62.9%

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

   8. Simplified 62.9%

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

   if 6.6e-164 < x < 9.4999999999999992e-133

   1. Initial program 61.6%

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

   3. Taylor expanded in x around 0 61.6%

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

      1. *-rgt-identity 61.6%

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

      2. associate-*l/ 61.6%

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

      3. associate-/l* 61.6%

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

      4. exp-to-pow 61.6%

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

   5. Simplified 61.6%

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

   if 9.4999999999999992e-133 < x < 0.880000000000000004

   1. Initial program 30.2%

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

   3. Taylor expanded in n around inf 60.2%

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

      1. log1p-define 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in x around 0 59.2%

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

   if 0.880000000000000004 < x

   1. Initial program 68.1%

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

   3. Taylor expanded in n around inf 69.8%

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

      1. log1p-define 69.8%

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

   5. Simplified 69.8%

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

   6. Taylor expanded in x around -inf 59.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-133}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 9.6 \cdot 10^{-143}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 9.6e-143)
     t_0
     (if (<= x 2.3e-132)
       (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
       (if (<= x 0.7)
         t_0
         (/
          (/
           (+
            1.0
            (/ (- (/ (- 0.3333333333333333 (* 0.25 (/ 1.0 x))) x) 0.5) x))
           x)
          n))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 9.6e-143) {
		tmp = t_0;
	} else if (x <= 2.3e-132) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if (x <= 0.7) {
		tmp = t_0;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / 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 = log(x) / -n
    if (x <= 9.6d-143) then
        tmp = t_0
    else if (x <= 2.3d-132) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else if (x <= 0.7d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - (0.25d0 * (1.0d0 / x))) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 9.6e-143) {
		tmp = t_0;
	} else if (x <= 2.3e-132) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if (x <= 0.7) {
		tmp = t_0;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 9.6e-143:
		tmp = t_0
	elif x <= 2.3e-132:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	elif x <= 0.7:
		tmp = t_0
	else:
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 9.6e-143)
		tmp = t_0;
	elseif (x <= 2.3e-132)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	elseif (x <= 0.7)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(0.25 * Float64(1.0 / x))) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 9.6e-143)
		tmp = t_0;
	elseif (x <= 2.3e-132)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	elseif (x <= 0.7)
		tmp = t_0;
	else
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 9.6e-143], t$95$0, If[LessEqual[x, 2.3e-132], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 0.7], t$95$0, N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(0.25 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.5999999999999995e-143 or 2.30000000000000003e-132 < x < 0.69999999999999996

   1. Initial program 35.5%

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

   3. Taylor expanded in n around inf 60.4%

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

      1. log1p-define 60.4%

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

   5. Simplified 60.4%

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

   6. Taylor expanded in x around 0 59.4%

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

      1. neg-mul-1 59.4%

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

   8. Simplified 59.4%

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

   if 9.5999999999999995e-143 < x < 2.30000000000000003e-132

   1. Initial program 67.7%

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

   3. Taylor expanded in n around inf 7.1%

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

      1. log1p-define 7.1%

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

   5. Simplified 7.1%

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

   6. Taylor expanded in x around inf 83.9%

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

      1. associate--l+ 83.9%

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

      2. unpow2 83.9%

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

      3. associate-/r* 83.9%

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

      4. metadata-eval 83.9%

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

      5. associate-*r/ 83.9%

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

      6. associate-*r/ 83.9%

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

      7. metadata-eval 83.9%

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

      8. div-sub 83.9%

         \[\leadsto \frac{\frac{1 + \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x}}{n} \]

      9. sub-neg 83.9%

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

      10. metadata-eval 83.9%

          \[\leadsto \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} + \color{blue}{-0.5}}{x}}{x}}{n} \]

      11. +-commutative 83.9%

          \[\leadsto \frac{\frac{1 + \frac{\color{blue}{-0.5 + 0.3333333333333333 \cdot \frac{1}{x}}}{x}}{x}}{n} \]

      12. associate-*r/ 83.9%

          \[\leadsto \frac{\frac{1 + \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{x}}}{x}}{x}}{n} \]

      13. metadata-eval 83.9%

          \[\leadsto \frac{\frac{1 + \frac{-0.5 + \frac{\color{blue}{0.3333333333333333}}{x}}{x}}{x}}{n} \]

   8. Simplified 83.9%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   if 0.69999999999999996 < x

   1. Initial program 68.1%

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

   3. Taylor expanded in n around inf 69.8%

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

      1. log1p-define 69.8%

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

   5. Simplified 69.8%

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

   6. Taylor expanded in x around -inf 59.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.0% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n} \cdot \frac{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (*
  (/ 1.0 n)
  (/ (+ 1.0 (/ (+ -0.5 (/ (+ (/ 0.25 x) -0.3333333333333333) x)) x)) x)))

C

double code(double x, double n) {
	return (1.0 / n) * ((1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) * ((1.0d0 + (((-0.5d0) + (((0.25d0 / x) + (-0.3333333333333333d0)) / x)) / x)) / x)
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) * ((1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x);
}

Python

def code(x, n):
	return (1.0 / n) * ((1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x)

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) * Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(Float64(Float64(0.25 / x) + -0.3333333333333333) / x)) / x)) / x))
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) * ((1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x);
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] * N[(N[(1.0 + N[(N[(-0.5 + N[(N[(N[(0.25 / x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in n around inf 63.1%

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

   1. log1p-define 63.1%

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

5. Simplified 63.1%

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

6. Taylor expanded in x around -inf 25.0%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

8. Applied egg-rr 40.2%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x} \cdot \frac{1}{n}} \]

9. Final simplification 40.2%

   \[\leadsto \frac{1}{n} \cdot \frac{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{x} \]
10. Add Preprocessing

Alternative 17: 46.0% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{n} \cdot \left(1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}\right) \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (*
  (/ (/ 1.0 x) n)
  (+ 1.0 (/ (+ -0.5 (/ (+ (/ 0.25 x) -0.3333333333333333) x)) x))))

C

double code(double x, double n) {
	return ((1.0 / x) / n) * (1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / x) / n) * (1.0d0 + (((-0.5d0) + (((0.25d0 / x) + (-0.3333333333333333d0)) / x)) / x))
end function

Java

public static double code(double x, double n) {
	return ((1.0 / x) / n) * (1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x));
}

Python

def code(x, n):
	return ((1.0 / x) / n) * (1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x))

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / x) / n) * Float64(1.0 + Float64(Float64(-0.5 + Float64(Float64(Float64(0.25 / x) + -0.3333333333333333) / x)) / x)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / x) / n) * (1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x));
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 + N[(N[(N[(0.25 / x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in n around inf 63.1%

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

   1. log1p-define 63.1%

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

5. Simplified 63.1%

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

6. Taylor expanded in x around -inf 25.0%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

8. Applied egg-rr 39.8%

   \[\leadsto \color{blue}{{\left(\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}\right)}^{-1}} \]
9. Step-by-step derivation

   1. unpow-1 39.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}}} \]

   2. associate-/r/ 39.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{n}{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1} \cdot x}} \]

   3. +-commutative 39.8%

      \[\leadsto \frac{1}{\frac{n}{\color{blue}{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}} \cdot x} \]

   4. +-commutative 39.8%

      \[\leadsto \frac{1}{\frac{n}{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}} \cdot x} \]

10. Simplified 39.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{n}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}} \cdot x}} \]
11. Step-by-step derivation

    1. associate-*l/ 39.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{n \cdot x}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}}} \]

    2. *-commutative 39.8%

       \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot n}}{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}} \]

    3. associate-/r/ 39.8%

       \[\leadsto \color{blue}{\frac{1}{x \cdot n} \cdot \left(1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}\right)} \]

    4. associate-/r* 40.2%

       \[\leadsto \color{blue}{\frac{\frac{1}{x}}{n}} \cdot \left(1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}\right) \]

12. Applied egg-rr 40.2%

    \[\leadsto \color{blue}{\frac{\frac{1}{x}}{n} \cdot \left(1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}\right)} \]

13. Final simplification 40.2%

    \[\leadsto \frac{\frac{1}{x}}{n} \cdot \left(1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}\right) \]
14. Add Preprocessing

Alternative 18: 45.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (+ 1.0 (/ (+ -0.5 (/ (+ (/ 0.25 x) -0.3333333333333333) x)) x)) (* n x)))

C

double code(double x, double n) {
	return (1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x)) / (n * x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 + (((-0.5d0) + (((0.25d0 / x) + (-0.3333333333333333d0)) / x)) / x)) / (n * x)
end function

Java

public static double code(double x, double n) {
	return (1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x)) / (n * x);
}

Python

def code(x, n):
	return (1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x)) / (n * x)

Julia

function code(x, n)
	return Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(Float64(Float64(0.25 / x) + -0.3333333333333333) / x)) / x)) / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 + ((-0.5 + (((0.25 / x) + -0.3333333333333333) / x)) / x)) / (n * x);
end

Wolfram

code[x_, n_] := N[(N[(1.0 + N[(N[(-0.5 + N[(N[(N[(0.25 / x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in n around inf 63.1%

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

   1. log1p-define 63.1%

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

5. Simplified 63.1%

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

6. Taylor expanded in x around -inf 25.0%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

8. Applied egg-rr 40.2%

   \[\leadsto \color{blue}{1 \cdot \frac{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}} \]
9. Step-by-step derivation

   1. *-lft-identity 40.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}} \]

   2. associate-/l/ 39.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1}{n \cdot x}} \]

   3. +-commutative 39.8%

      \[\leadsto \frac{\color{blue}{1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}}{n \cdot x} \]

   4. +-commutative 39.8%

      \[\leadsto \frac{1 + \frac{\color{blue}{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}}{x}}{n \cdot x} \]

   5. *-commutative 39.8%

      \[\leadsto \frac{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{\color{blue}{x \cdot n}} \]

10. Simplified 39.8%

    \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{x \cdot n}} \]

11. Final simplification 39.8%

    \[\leadsto \frac{1 + \frac{-0.5 + \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{n \cdot x} \]
12. Add Preprocessing

Alternative 19: 45.2% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n))

C

double code(double x, double n) {
	return ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
end function

Java

public static double code(double x, double n) {
	return ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
}

Python

def code(x, n):
	return ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in n around inf 63.1%

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

   1. log1p-define 63.1%

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

5. Simplified 63.1%

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

6. Taylor expanded in x around inf 38.9%

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

   1. associate--l+ 38.9%

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

   2. unpow2 38.9%

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

   3. associate-/r* 38.9%

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

   4. metadata-eval 38.9%

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

   5. associate-*r/ 38.9%

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

   6. associate-*r/ 38.9%

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

   7. metadata-eval 38.9%

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

   8. div-sub 38.9%

      \[\leadsto \frac{\frac{1 + \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x}}{n} \]

   9. sub-neg 38.9%

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

   10. metadata-eval 38.9%

       \[\leadsto \frac{\frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{x} + \color{blue}{-0.5}}{x}}{x}}{n} \]

   11. +-commutative 38.9%

       \[\leadsto \frac{\frac{1 + \frac{\color{blue}{-0.5 + 0.3333333333333333 \cdot \frac{1}{x}}}{x}}{x}}{n} \]

   12. associate-*r/ 38.9%

       \[\leadsto \frac{\frac{1 + \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{x}}}{x}}{x}}{n} \]

   13. metadata-eval 38.9%

       \[\leadsto \frac{\frac{1 + \frac{-0.5 + \frac{\color{blue}{0.3333333333333333}}{x}}{x}}{x}}{n} \]

8. Simplified 38.9%

   \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

9. Final simplification 38.9%

   \[\leadsto \frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n} \]
10. Add Preprocessing

Alternative 20: 39.0% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in n around inf 63.1%

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

   1. log1p-define 63.1%

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

5. Simplified 63.1%

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

6. Taylor expanded in x around inf 34.2%

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

   1. *-commutative 34.2%

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

8. Simplified 34.2%

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

9. Final simplification 34.2%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Alternative 21: 39.5% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in n around inf 63.1%

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

   1. log1p-define 63.1%

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

5. Simplified 63.1%

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

   1. div-inv 63.0%

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

7. Applied egg-rr 63.0%

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

8. Taylor expanded in x around inf 34.2%

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

   1. associate-/r* 34.6%

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

10. Simplified 34.6%

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

11. Final simplification 34.6%

    \[\leadsto \frac{\frac{1}{n}}{x} \]
12. Add Preprocessing

Alternative 22: 39.5% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in n around inf 63.1%

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

   1. log1p-define 63.1%

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

5. Simplified 63.1%

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

6. Taylor expanded in x around inf 34.6%

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

7. Final simplification 34.6%

   \[\leadsto \frac{\frac{1}{x}}{n} \]
8. Add Preprocessing

Alternative 23: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in x around 0 27.8%

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

4. Taylor expanded in x around inf 4.6%

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

5. Final simplification 4.6%

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

Reproduce

herbie shell --seed 2024076 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))