2nthrt (problem 3.4.6)

Percentage Accurate: 53.3% → 98.1%

Time: 18.7s

Alternatives: 14

Speedup: 2.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% 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: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-6)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log1p (/ 1.0 x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-6:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-6)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-6], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -5.00000000000000041e-6 < (/.f64 1 n) < 1.99999999999999996e-12

   1. Initial program 31.3%

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

   2. Taylor expanded in n around inf 81.0%

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

      1. log1p-udef 81.0%

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

      2. expm1-log1p-u 80.3%

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

   4. Applied egg-rr 80.3%

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

   5. Taylor expanded in n around 0 81.0%

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

      1. log-div 81.5%

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

      2. *-lft-identity 81.5%

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

      3. associate-*l/ 77.6%

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

      4. +-commutative 77.6%

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

      5. distribute-lft-in 77.6%

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

      6. lft-mult-inverse 81.5%

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

      7. *-rgt-identity 81.5%

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

      8. log1p-def 97.8%

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

   7. Simplified 97.8%

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

   if 1.99999999999999996e-12 < (/.f64 1 n)

   1. Initial program 61.6%

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

   2. Taylor expanded in n around 0 61.6%

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

      1. log1p-def 96.2%

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

   4. Simplified 96.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-6)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 4e+212)
         (- (pow (+ 1.0 x) (/ 1.0 n)) t_0)
         (log1p (expm1 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = log1p(expm1((1.0 / (n * x))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((1.0 / (n * x))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-6:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 4e+212:
		tmp = math.pow((1.0 + x), (1.0 / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((1.0 / (n * x))))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-6)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 4e+212)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(1.0 / Float64(n * x))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-6], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+212], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -5.00000000000000041e-6 < (/.f64 1 n) < 1.99999999999999996e-12

   1. Initial program 31.3%

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

   2. Taylor expanded in n around inf 81.0%

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

      1. log1p-udef 81.0%

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

      2. expm1-log1p-u 80.3%

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

   4. Applied egg-rr 80.3%

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

   5. Taylor expanded in n around 0 81.0%

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

      1. log-div 81.5%

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

      2. *-lft-identity 81.5%

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

      3. associate-*l/ 77.6%

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

      4. +-commutative 77.6%

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

      5. distribute-lft-in 77.6%

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

      6. lft-mult-inverse 81.5%

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

      7. *-rgt-identity 81.5%

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

      8. log1p-def 97.8%

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

   7. Simplified 97.8%

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

   if 1.99999999999999996e-12 < (/.f64 1 n) < 3.9999999999999996e212

   1. Initial program 74.3%

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

   if 3.9999999999999996e212 < (/.f64 1 n)

   1. Initial program 38.8%

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

   2. Taylor expanded in n around inf 7.3%

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

   3. Taylor expanded in x around inf 71.8%

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

      1. *-commutative 71.8%

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

   5. Simplified 71.8%

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

      1. log1p-expm1-u 90.3%

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

   7. Applied egg-rr 90.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \end{array} \]

Alternative 3: 93.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-6)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log1p (/ 1.0 x)) n)
       (-
        (+ (* (- (/ 0.5 (* n n)) (/ 0.5 n)) (* x x)) (+ 1.0 (/ x n)))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-6:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-6)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)) * Float64(x * x)) + Float64(1.0 + Float64(x / n))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-6], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -5.00000000000000041e-6 < (/.f64 1 n) < 1.99999999999999996e-12

   1. Initial program 31.3%

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

   2. Taylor expanded in n around inf 81.0%

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

      1. log1p-udef 81.0%

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

      2. expm1-log1p-u 80.3%

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

   4. Applied egg-rr 80.3%

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

   5. Taylor expanded in n around 0 81.0%

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

      1. log-div 81.5%

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

      2. *-lft-identity 81.5%

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

      3. associate-*l/ 77.6%

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

      4. +-commutative 77.6%

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

      5. distribute-lft-in 77.6%

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

      6. lft-mult-inverse 81.5%

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

      7. *-rgt-identity 81.5%

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

      8. log1p-def 97.8%

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

   7. Simplified 97.8%

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

   if 1.99999999999999996e-12 < (/.f64 1 n)

   1. Initial program 61.6%

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

   2. Taylor expanded in x around 0 64.3%

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

      1. associate-+r+ 64.3%

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

      2. +-commutative 64.3%

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

      3. associate-*r/ 64.3%

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

      4. metadata-eval 64.3%

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

      5. unpow2 64.3%

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

      6. associate-*r/ 64.3%

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

      7. metadata-eval 64.3%

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

      8. unpow2 64.3%

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

   4. Simplified 64.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 79.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0}{n} \cdot -0.3333333333333333\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* (/ 0.0 n) -0.3333333333333333))
        (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2.5e+239)
     t_0
     (if (<= (/ 1.0 n) -2e+108)
       t_1
       (if (<= (/ 1.0 n) -2e+19)
         t_0
         (if (<= (/ 1.0 n) 2e-12)
           (/ (log1p (/ 1.0 x)) n)
           (if (<= (/ 1.0 n) 4e+212) t_1 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = (0.0 / n) * -0.3333333333333333;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2.5e+239) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e+108) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e+19) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (0.0 / n) * -0.3333333333333333;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2.5e+239) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e+108) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e+19) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (0.0 / n) * -0.3333333333333333
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2.5e+239:
		tmp = t_0
	elif (1.0 / n) <= -2e+108:
		tmp = t_1
	elif (1.0 / n) <= -2e+19:
		tmp = t_0
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 4e+212:
		tmp = t_1
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(0.0 / n) * -0.3333333333333333)
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.5e+239)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -2e+108)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -2e+19)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 4e+212)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(0.0 / n), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.5e+239], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+108], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+19], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+212], t$95$1, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2.50000000000000003e239 or -2.0000000000000001e108 < (/.f64 1 n) < -2e19

   1. Initial program 100.0%

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

      1. flip3-- 0.0%

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

      2. div-inv 0.0%

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

      3. pow-to-exp 0.0%

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

      4. pow-exp 0.0%

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

      5. un-div-inv 0.0%

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

      6. +-commutative 0.0%

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

      7. log1p-udef 0.0%

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

      8. inv-pow 0.0%

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

   3. Applied egg-rr 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in n around -inf 67.2%

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

      1. *-commutative 67.2%

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

   8. Simplified 28.1%

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

   9. Taylor expanded in x around inf 68.0%

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

       1. log-rec 68.0%

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

       2. log-rec 68.0%

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

       3. distribute-rgt-out 68.0%

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

       4. metadata-eval 68.0%

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

       5. mul0-rgt 68.0%

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

   11. Simplified 68.0%

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

   if -2.50000000000000003e239 < (/.f64 1 n) < -2.0000000000000001e108 or 1.99999999999999996e-12 < (/.f64 1 n) < 3.9999999999999996e212

   1. Initial program 92.1%

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

   2. Taylor expanded in x around 0 67.5%

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

   if -2e19 < (/.f64 1 n) < 1.99999999999999996e-12

   1. Initial program 31.6%

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

   2. Taylor expanded in n around inf 80.0%

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

      1. log1p-udef 80.0%

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

      2. expm1-log1p-u 79.3%

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

   4. Applied egg-rr 79.3%

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

   5. Taylor expanded in n around 0 80.0%

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

      1. log-div 80.5%

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

      2. *-lft-identity 80.5%

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

      3. associate-*l/ 76.6%

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

      4. +-commutative 76.6%

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

      5. distribute-lft-in 76.5%

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

      6. lft-mult-inverse 80.5%

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

      7. *-rgt-identity 80.5%

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

      8. log1p-def 96.7%

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

   7. Simplified 96.7%

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

   if 3.9999999999999996e212 < (/.f64 1 n)

   1. Initial program 38.8%

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

   2. Taylor expanded in n around inf 7.3%

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

   3. Taylor expanded in x around inf 71.8%

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

      1. *-commutative 71.8%

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

   5. Simplified 71.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{+239}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+108}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 5: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -4e+221)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) -2e+108)
       t_0
       (if (<= (/ 1.0 n) -2e+19)
         (* (/ 0.0 n) -0.3333333333333333)
         (if (<= (/ 1.0 n) 2e-12)
           (/ (log1p (/ 1.0 x)) n)
           (if (<= (/ 1.0 n) 4e+212) t_0 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e+221) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= -2e+108) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e+19) {
		tmp = (0.0 / n) * -0.3333333333333333;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e+221) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= -2e+108) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e+19) {
		tmp = (0.0 / n) * -0.3333333333333333;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e+221:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= -2e+108:
		tmp = t_0
	elif (1.0 / n) <= -2e+19:
		tmp = (0.0 / n) * -0.3333333333333333
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 4e+212:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e+221)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= -2e+108)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -2e+19)
		tmp = Float64(Float64(0.0 / n) * -0.3333333333333333);
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 4e+212)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e+221], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+108], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+19], N[(N[(0.0 / n), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+212], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.0000000000000002e221

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf 63.4%

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

      1. diff-log 63.4%

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

   4. Applied egg-rr 63.4%

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

   if -4.0000000000000002e221 < (/.f64 1 n) < -2.0000000000000001e108 or 1.99999999999999996e-12 < (/.f64 1 n) < 3.9999999999999996e212

   1. Initial program 91.3%

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

   2. Taylor expanded in x around 0 69.3%

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

   if -2.0000000000000001e108 < (/.f64 1 n) < -2e19

   1. Initial program 100.0%

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

      1. flip3-- 0.0%

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

      2. div-inv 0.0%

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

      3. pow-to-exp 0.0%

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

      4. pow-exp 0.0%

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

      5. un-div-inv 0.0%

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

      6. +-commutative 0.0%

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

      7. log1p-udef 0.0%

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

      8. inv-pow 0.0%

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

   3. Applied egg-rr 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in n around -inf 67.8%

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

      1. *-commutative 67.8%

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

   8. Simplified 30.8%

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

   9. Taylor expanded in x around inf 71.9%

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

       1. log-rec 71.9%

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

       2. log-rec 71.9%

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

       3. distribute-rgt-out 71.9%

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

       4. metadata-eval 71.9%

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

       5. mul0-rgt 71.9%

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

   11. Simplified 71.9%

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

   if -2e19 < (/.f64 1 n) < 1.99999999999999996e-12

   1. Initial program 31.6%

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

   2. Taylor expanded in n around inf 80.0%

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

      1. log1p-udef 80.0%

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

      2. expm1-log1p-u 79.3%

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

   4. Applied egg-rr 79.3%

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

   5. Taylor expanded in n around 0 80.0%

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

      1. log-div 80.5%

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

      2. *-lft-identity 80.5%

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

      3. associate-*l/ 76.6%

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

      4. +-commutative 76.6%

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

      5. distribute-lft-in 76.5%

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

      6. lft-mult-inverse 80.5%

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

      7. *-rgt-identity 80.5%

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

      8. log1p-def 96.7%

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

   7. Simplified 96.7%

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

   if 3.9999999999999996e212 < (/.f64 1 n)

   1. Initial program 38.8%

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

   2. Taylor expanded in n around inf 7.3%

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

   3. Taylor expanded in x around inf 71.8%

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

      1. *-commutative 71.8%

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

   5. Simplified 71.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{+221}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+108}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 6: 93.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-6)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 4e+212) (- (+ 1.0 (/ x n)) t_0) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-6:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 4e+212:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-6)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 4e+212)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-6], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+212], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -5.00000000000000041e-6 < (/.f64 1 n) < 1.99999999999999996e-12

   1. Initial program 31.3%

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

   2. Taylor expanded in n around inf 81.0%

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

      1. log1p-udef 81.0%

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

      2. expm1-log1p-u 80.3%

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

   4. Applied egg-rr 80.3%

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

   5. Taylor expanded in n around 0 81.0%

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

      1. log-div 81.5%

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

      2. *-lft-identity 81.5%

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

      3. associate-*l/ 77.6%

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

      4. +-commutative 77.6%

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

      5. distribute-lft-in 77.6%

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

      6. lft-mult-inverse 81.5%

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

      7. *-rgt-identity 81.5%

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

      8. log1p-def 97.8%

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

   7. Simplified 97.8%

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

   if 1.99999999999999996e-12 < (/.f64 1 n) < 3.9999999999999996e212

   1. Initial program 74.3%

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

   2. Taylor expanded in x around 0 70.1%

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

   if 3.9999999999999996e212 < (/.f64 1 n)

   1. Initial program 38.8%

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

   2. Taylor expanded in n around inf 7.3%

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

   3. Taylor expanded in x around inf 71.8%

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

      1. *-commutative 71.8%

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

   5. Simplified 71.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 7: 72.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0}{n} \cdot -0.3333333333333333\\ t_1 := \frac{1}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* (/ 0.0 n) -0.3333333333333333)) (t_1 (/ 1.0 (* n x))))
   (if (<= (/ 1.0 n) -2.5e+239)
     t_0
     (if (<= (/ 1.0 n) -1e+202)
       t_1
       (if (<= (/ 1.0 n) -2e+19)
         t_0
         (if (<= (/ 1.0 n) 4e+212) (/ (log1p (/ 1.0 x)) n) t_1))))))

C

double code(double x, double n) {
	double t_0 = (0.0 / n) * -0.3333333333333333;
	double t_1 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2.5e+239) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1e+202) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e+19) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (0.0 / n) * -0.3333333333333333;
	double t_1 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -2.5e+239) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1e+202) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e+19) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (0.0 / n) * -0.3333333333333333
	t_1 = 1.0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -2.5e+239:
		tmp = t_0
	elif (1.0 / n) <= -1e+202:
		tmp = t_1
	elif (1.0 / n) <= -2e+19:
		tmp = t_0
	elif (1.0 / n) <= 4e+212:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(0.0 / n) * -0.3333333333333333)
	t_1 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.5e+239)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -1e+202)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -2e+19)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e+212)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(0.0 / n), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.5e+239], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+202], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+19], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+212], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2.50000000000000003e239 or -9.999999999999999e201 < (/.f64 1 n) < -2e19

   1. Initial program 100.0%

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

      1. flip3-- 0.0%

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

      2. div-inv 0.0%

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

      3. pow-to-exp 0.0%

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

      4. pow-exp 0.0%

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

      5. un-div-inv 0.0%

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

      6. +-commutative 0.0%

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

      7. log1p-udef 0.0%

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

      8. inv-pow 0.0%

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

   3. Applied egg-rr 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in n around -inf 55.4%

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

      1. *-commutative 55.4%

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

   8. Simplified 23.3%

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

   9. Taylor expanded in x around inf 55.0%

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

       1. log-rec 55.0%

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

       2. log-rec 55.0%

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

       3. distribute-rgt-out 55.0%

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

       4. metadata-eval 55.0%

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

       5. mul0-rgt 55.0%

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

   11. Simplified 55.0%

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

   if -2.50000000000000003e239 < (/.f64 1 n) < -9.999999999999999e201 or 3.9999999999999996e212 < (/.f64 1 n)

   1. Initial program 70.9%

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

   2. Taylor expanded in n around inf 20.1%

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

   3. Taylor expanded in x around inf 72.6%

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

      1. *-commutative 72.6%

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

   5. Simplified 72.6%

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

   if -2e19 < (/.f64 1 n) < 3.9999999999999996e212

   1. Initial program 36.3%

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

   2. Taylor expanded in n around inf 72.0%

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

      1. log1p-udef 72.0%

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

      2. expm1-log1p-u 71.4%

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

   4. Applied egg-rr 71.4%

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

   5. Taylor expanded in n around 0 72.0%

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

      1. log-div 72.5%

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

      2. *-lft-identity 72.5%

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

      3. associate-*l/ 69.0%

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

      4. +-commutative 69.0%

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

      5. distribute-lft-in 69.0%

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

      6. lft-mult-inverse 72.5%

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

      7. *-rgt-identity 72.5%

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

      8. log1p-def 86.9%

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

   7. Simplified 86.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{+239}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+202}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 8: 93.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-6)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 4e+212) (- 1.0 t_0) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-6:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 4e+212:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-6)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 4e+212)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-6], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+212], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. un-div-inv 100.0%

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

      2. associate-/r* 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -5.00000000000000041e-6 < (/.f64 1 n) < 1.99999999999999996e-12

   1. Initial program 31.3%

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

   2. Taylor expanded in n around inf 81.0%

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

      1. log1p-udef 81.0%

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

      2. expm1-log1p-u 80.3%

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

   4. Applied egg-rr 80.3%

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

   5. Taylor expanded in n around 0 81.0%

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

      1. log-div 81.5%

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

      2. *-lft-identity 81.5%

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

      3. associate-*l/ 77.6%

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

      4. +-commutative 77.6%

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

      5. distribute-lft-in 77.6%

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

      6. lft-mult-inverse 81.5%

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

      7. *-rgt-identity 81.5%

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

      8. log1p-def 97.8%

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

   7. Simplified 97.8%

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

   if 1.99999999999999996e-12 < (/.f64 1 n) < 3.9999999999999996e212

   1. Initial program 74.3%

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

   2. Taylor expanded in x around 0 68.9%

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

   if 3.9999999999999996e212 < (/.f64 1 n)

   1. Initial program 38.8%

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

   2. Taylor expanded in n around inf 7.3%

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

   3. Taylor expanded in x around inf 71.8%

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

      1. *-commutative 71.8%

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

   5. Simplified 71.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 9: 93.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-6)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 4e+212) (- 1.0 t_0) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-6) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 4e+212) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-6:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 4e+212:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-6)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 4e+212)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-6], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+212], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -5.00000000000000041e-6 < (/.f64 1 n) < 1.99999999999999996e-12

   1. Initial program 31.3%

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

   2. Taylor expanded in n around inf 81.0%

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

      1. log1p-udef 81.0%

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

      2. expm1-log1p-u 80.3%

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

   4. Applied egg-rr 80.3%

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

   5. Taylor expanded in n around 0 81.0%

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

      1. log-div 81.5%

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

      2. *-lft-identity 81.5%

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

      3. associate-*l/ 77.6%

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

      4. +-commutative 77.6%

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

      5. distribute-lft-in 77.6%

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

      6. lft-mult-inverse 81.5%

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

      7. *-rgt-identity 81.5%

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

      8. log1p-def 97.8%

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

   7. Simplified 97.8%

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

   if 1.99999999999999996e-12 < (/.f64 1 n) < 3.9999999999999996e212

   1. Initial program 74.3%

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

   2. Taylor expanded in x around 0 68.9%

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

   if 3.9999999999999996e212 < (/.f64 1 n)

   1. Initial program 38.8%

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

   2. Taylor expanded in n around inf 7.3%

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

   3. Taylor expanded in x around inf 71.8%

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

      1. *-commutative 71.8%

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

   5. Simplified 71.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+212}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 10: 59.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.68)
   (- (/ (log x) n))
   (if (<= x 1.02e+49)
     (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)
     (* (/ 0.0 n) -0.3333333333333333))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = -(log(x) / n);
	} else if (x <= 1.02e+49) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = (0.0 / n) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = -(log(x) / n)
    else if (x <= 1.02d+49) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = (0.0d0 / n) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 1.02e+49) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = (0.0 / n) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.68:
		tmp = -(math.log(x) / n)
	elif x <= 1.02e+49:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = (0.0 / n) * -0.3333333333333333
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 1.02e+49)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(Float64(0.0 / n) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = -(log(x) / n);
	elseif (x <= 1.02e+49)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = (0.0 / n) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.68], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 1.02e+49], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(0.0 / n), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.680000000000000049

   1. Initial program 44.3%

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

   2. Taylor expanded in n around inf 55.2%

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

   3. Taylor expanded in x around 0 54.4%

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

      1. neg-mul-1 54.4%

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

   5. Simplified 54.4%

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

   if 0.680000000000000049 < x < 1.02e49

   1. Initial program 29.8%

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

   2. Taylor expanded in n around inf 35.4%

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

   3. Taylor expanded in x around inf 63.6%

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

      1. associate-*r/ 63.6%

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

      2. metadata-eval 63.6%

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

      3. unpow2 63.6%

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

   5. Simplified 63.6%

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

   if 1.02e49 < x

   1. Initial program 82.2%

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

      1. flip3-- 42.2%

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

      2. div-inv 42.2%

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

      3. pow-to-exp 42.2%

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

      4. pow-exp 42.1%

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

      5. un-div-inv 42.1%

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

      6. +-commutative 42.1%

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

      7. log1p-udef 42.1%

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

      8. inv-pow 42.1%

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

   3. Applied egg-rr 42.1%

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

   5. Simplified 42.1%

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

   6. Taylor expanded in n around -inf 82.2%

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

      1. *-commutative 82.2%

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

   8. Simplified 35.5%

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

   9. Taylor expanded in x around inf 82.2%

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

       1. log-rec 82.2%

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

       2. log-rec 82.2%

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

       3. distribute-rgt-out 82.2%

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

       4. metadata-eval 82.2%

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

       5. mul0-rgt 82.2%

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

   11. Simplified 82.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 11: 42.7% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{-3}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.02e+49)
   (* -0.3333333333333333 (/ -3.0 (* n x)))
   (* (/ 0.0 n) -0.3333333333333333)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.02e+49) {
		tmp = -0.3333333333333333 * (-3.0 / (n * x));
	} else {
		tmp = (0.0 / n) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.02d+49) then
        tmp = (-0.3333333333333333d0) * ((-3.0d0) / (n * x))
    else
        tmp = (0.0d0 / n) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.02e+49) {
		tmp = -0.3333333333333333 * (-3.0 / (n * x));
	} else {
		tmp = (0.0 / n) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.02e+49:
		tmp = -0.3333333333333333 * (-3.0 / (n * x))
	else:
		tmp = (0.0 / n) * -0.3333333333333333
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.02e+49)
		tmp = Float64(-0.3333333333333333 * Float64(-3.0 / Float64(n * x)));
	else
		tmp = Float64(Float64(0.0 / n) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.02e+49)
		tmp = -0.3333333333333333 * (-3.0 / (n * x));
	else
		tmp = (0.0 / n) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.02e+49], N[(-0.3333333333333333 * N[(-3.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 / n), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.02e49

   1. Initial program 42.6%

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

      1. flip3-- 12.5%

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

      2. div-inv 12.5%

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

      3. pow-to-exp 12.5%

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

      4. pow-exp 12.5%

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

      5. un-div-inv 12.5%

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

      6. +-commutative 12.5%

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

      7. log1p-udef 17.7%

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

      8. inv-pow 17.7%

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

   3. Applied egg-rr 12.9%

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

   5. Simplified 12.3%

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

   6. Taylor expanded in n around -inf 52.7%

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

      1. *-commutative 52.7%

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

   8. Simplified 51.6%

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

   9. Taylor expanded in x around inf 27.9%

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

       1. distribute-rgt-out 27.9%

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

       2. log-rec 27.9%

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

       3. metadata-eval 27.9%

          \[\leadsto \left(\frac{-\log x}{n} \cdot \color{blue}{0} - 3 \cdot \frac{1}{n \cdot x}\right) \cdot -0.3333333333333333 \]

       4. mul0-rgt 27.9%

          \[\leadsto \left(\color{blue}{0} - 3 \cdot \frac{1}{n \cdot x}\right) \cdot -0.3333333333333333 \]

       5. neg-sub0 27.9%

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

       6. associate-*r/ 28.0%

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

       7. metadata-eval 28.0%

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

       8. *-commutative 28.0%

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

       9. distribute-neg-frac 28.0%

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

       10. metadata-eval 28.0%

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

   11. Simplified 28.0%

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

   if 1.02e49 < x

   1. Initial program 82.2%

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

      1. flip3-- 42.2%

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

      2. div-inv 42.2%

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

      3. pow-to-exp 42.2%

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

      4. pow-exp 42.1%

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

      5. un-div-inv 42.1%

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

      6. +-commutative 42.1%

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

      7. log1p-udef 42.1%

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

      8. inv-pow 42.1%

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

   3. Applied egg-rr 42.1%

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

   5. Simplified 42.1%

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

   6. Taylor expanded in n around -inf 82.2%

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

      1. *-commutative 82.2%

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

   8. Simplified 35.5%

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

   9. Taylor expanded in x around inf 82.2%

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

       1. log-rec 82.2%

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

       2. log-rec 82.2%

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

       3. distribute-rgt-out 82.2%

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

       4. metadata-eval 82.2%

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

       5. mul0-rgt 82.2%

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

   11. Simplified 82.2%

       \[\leadsto \frac{\color{blue}{0}}{n} \cdot -0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{-3}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 12: 42.7% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.02e+49) (/ 1.0 (* n x)) (* (/ 0.0 n) -0.3333333333333333)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.02e+49) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = (0.0 / n) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.02d+49) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = (0.0d0 / n) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.02e+49) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = (0.0 / n) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.02e+49:
		tmp = 1.0 / (n * x)
	else:
		tmp = (0.0 / n) * -0.3333333333333333
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.02e+49)
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = Float64(Float64(0.0 / n) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.02e+49)
		tmp = 1.0 / (n * x);
	else
		tmp = (0.0 / n) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.02e+49], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 / n), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.02e49

   1. Initial program 42.6%

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

   2. Taylor expanded in n around inf 52.9%

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

   3. Taylor expanded in x around inf 27.9%

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

      1. *-commutative 27.9%

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

   5. Simplified 27.9%

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

   if 1.02e49 < x

   1. Initial program 82.2%

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

      1. flip3-- 42.2%

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

      2. div-inv 42.2%

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

      3. pow-to-exp 42.2%

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

      4. pow-exp 42.1%

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

      5. un-div-inv 42.1%

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

      6. +-commutative 42.1%

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

      7. log1p-udef 42.1%

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

      8. inv-pow 42.1%

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

   3. Applied egg-rr 42.1%

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

   5. Simplified 42.1%

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

   6. Taylor expanded in n around -inf 82.2%

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

      1. *-commutative 82.2%

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

   8. Simplified 35.5%

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

   9. Taylor expanded in x around inf 82.2%

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

       1. log-rec 82.2%

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

       2. log-rec 82.2%

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

       3. distribute-rgt-out 82.2%

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

       4. metadata-eval 82.2%

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

       5. mul0-rgt 82.2%

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

   11. Simplified 82.2%

       \[\leadsto \frac{\color{blue}{0}}{n} \cdot -0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 13: 42.7% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.02e+49) (/ (/ 1.0 x) n) (* (/ 0.0 n) -0.3333333333333333)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.02e+49) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (0.0 / n) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.02d+49) then
        tmp = (1.0d0 / x) / n
    else
        tmp = (0.0d0 / n) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.02e+49) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (0.0 / n) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.02e+49:
		tmp = (1.0 / x) / n
	else:
		tmp = (0.0 / n) * -0.3333333333333333
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.02e+49)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(0.0 / n) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.02e+49)
		tmp = (1.0 / x) / n;
	else
		tmp = (0.0 / n) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.02e+49], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(0.0 / n), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.02e49

   1. Initial program 42.6%

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

   2. Taylor expanded in n around inf 52.9%

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

   3. Taylor expanded in x around inf 27.9%

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

   if 1.02e49 < x

   1. Initial program 82.2%

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

      1. flip3-- 42.2%

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

      2. div-inv 42.2%

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

      3. pow-to-exp 42.2%

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

      4. pow-exp 42.1%

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

      5. un-div-inv 42.1%

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

      6. +-commutative 42.1%

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

      7. log1p-udef 42.1%

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

      8. inv-pow 42.1%

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

   3. Applied egg-rr 42.1%

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

   5. Simplified 42.1%

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

   6. Taylor expanded in n around -inf 82.2%

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

      1. *-commutative 82.2%

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

   8. Simplified 35.5%

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

   9. Taylor expanded in x around inf 82.2%

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

       1. log-rec 82.2%

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

       2. log-rec 82.2%

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

       3. distribute-rgt-out 82.2%

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

       4. metadata-eval 82.2%

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

       5. mul0-rgt 82.2%

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

   11. Simplified 82.2%

       \[\leadsto \frac{\color{blue}{0}}{n} \cdot -0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 14: 30.8% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{0}{n} \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (* (/ 0.0 n) -0.3333333333333333))

C

double code(double x, double n) {
	return (0.0 / n) * -0.3333333333333333;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (0.0d0 / n) * (-0.3333333333333333d0)
end function

Java

public static double code(double x, double n) {
	return (0.0 / n) * -0.3333333333333333;
}

Python

def code(x, n):
	return (0.0 / n) * -0.3333333333333333

Julia

function code(x, n)
	return Float64(Float64(0.0 / n) * -0.3333333333333333)
end

MATLAB

function tmp = code(x, n)
	tmp = (0.0 / n) * -0.3333333333333333;
end

Wolfram

code[x_, n_] := N[(N[(0.0 / n), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 56.5%

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

   1. flip3-- 22.9%

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

   2. div-inv 22.9%

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

   3. pow-to-exp 22.9%

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

   4. pow-exp 22.9%

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

   5. un-div-inv 22.9%

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

   6. +-commutative 22.9%

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

   7. log1p-udef 26.3%

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

   8. inv-pow 26.3%

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

3. Applied egg-rr 23.1%

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

5. Simplified 22.8%

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

6. Taylor expanded in n around -inf 63.1%

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

   1. *-commutative 63.1%

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

8. Simplified 45.9%

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

9. Taylor expanded in x around inf 33.3%

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

    1. log-rec 33.3%

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

    2. log-rec 33.3%

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

    3. distribute-rgt-out 33.3%

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

    4. metadata-eval 33.3%

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

    5. mul0-rgt 33.3%

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

11. Simplified 33.3%

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

12. Final simplification 33.3%

    \[\leadsto \frac{0}{n} \cdot -0.3333333333333333 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))