2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 85.5%

Time: 27.8s

Alternatives: 10

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: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{t_0}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-85}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-44)
     (* (/ 1.0 n) (/ t_0 x))
     (if (<= (/ 1.0 n) 1e-85)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e-13)
         (/ (/ t_0 n) x)
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-44) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-85) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = (t_0 / n) / x;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-44) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-85) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = (t_0 / n) / x;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-44:
		tmp = (1.0 / n) * (t_0 / x)
	elif (1.0 / n) <= 1e-85:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-13:
		tmp = (t_0 / n) / x
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-44)
		tmp = Float64(Float64(1.0 / n) * Float64(t_0 / x));
	elseif (Float64(1.0 / n) <= 1e-85)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = Float64(Float64(t_0 / n) / x);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-44], N[(N[(1.0 / n), $MachinePrecision] * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-85], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -3.99999999999999981e-44

   1. Initial program 89.1%

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

   2. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. log-rec 96.6%

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

      3. mul-1-neg 96.6%

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

      4. distribute-neg-frac 96.6%

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

      5. mul-1-neg 96.6%

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

      6. remove-double-neg 96.6%

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

      7. *-commutative 96.6%

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

   4. Simplified 96.6%

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

      1. associate-/r* 96.6%

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

      2. div-inv 96.6%

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

      3. div-inv 96.6%

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

      4. pow-to-exp 96.6%

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

   6. Applied egg-rr 96.6%

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

   if -3.99999999999999981e-44 < (/.f64 1 n) < 9.9999999999999998e-86

   1. Initial program 31.8%

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

   2. Taylor expanded in n around inf 80.2%

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

      1. log1p-def 80.2%

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

   4. Simplified 80.2%

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

   if 9.9999999999999998e-86 < (/.f64 1 n) < 2.0000000000000001e-13

   1. Initial program 13.9%

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

   2. Taylor expanded in x around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. log-rec 75.3%

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

      3. mul-1-neg 75.3%

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

      4. distribute-neg-frac 75.3%

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

      5. mul-1-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-commutative 75.3%

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

   4. Simplified 75.3%

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

      1. associate-/r* 75.6%

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

      2. div-inv 75.6%

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

      3. div-inv 75.6%

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

      4. pow-to-exp 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. un-div-inv 75.6%

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

      2. associate-/l/ 75.3%

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

      3. associate-/r* 75.7%

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

   8. Applied egg-rr 75.7%

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

   if 2.0000000000000001e-13 < (/.f64 1 n)

   1. Initial program 65.4%

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

   2. Taylor expanded in n around 0 65.4%

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

      1. log1p-def 96.8%

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

   4. Simplified 96.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-85}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{t_0}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-85}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-44)
     (* (/ 1.0 n) (/ t_0 x))
     (if (<= (/ 1.0 n) 1e-85)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e-13) (/ (/ t_0 n) x) (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-44) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-85) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = (t_0 / n) / x;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-44) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-85) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = (t_0 / n) / x;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-44:
		tmp = (1.0 / n) * (t_0 / x)
	elif (1.0 / n) <= 1e-85:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-13:
		tmp = (t_0 / n) / x
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-44)
		tmp = Float64(Float64(1.0 / n) * Float64(t_0 / x));
	elseif (Float64(1.0 / n) <= 1e-85)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = Float64(Float64(t_0 / n) / x);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-44], N[(N[(1.0 / n), $MachinePrecision] * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-85], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -3.99999999999999981e-44

   1. Initial program 89.1%

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

   2. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. log-rec 96.6%

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

      3. mul-1-neg 96.6%

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

      4. distribute-neg-frac 96.6%

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

      5. mul-1-neg 96.6%

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

      6. remove-double-neg 96.6%

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

      7. *-commutative 96.6%

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

   4. Simplified 96.6%

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

      1. associate-/r* 96.6%

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

      2. div-inv 96.6%

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

      3. div-inv 96.6%

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

      4. pow-to-exp 96.6%

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

   6. Applied egg-rr 96.6%

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

   if -3.99999999999999981e-44 < (/.f64 1 n) < 9.9999999999999998e-86

   1. Initial program 31.8%

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

   2. Taylor expanded in n around inf 80.2%

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

      1. log1p-def 80.2%

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

   4. Simplified 80.2%

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

   if 9.9999999999999998e-86 < (/.f64 1 n) < 2.0000000000000001e-13

   1. Initial program 13.9%

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

   2. Taylor expanded in x around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. log-rec 75.3%

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

      3. mul-1-neg 75.3%

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

      4. distribute-neg-frac 75.3%

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

      5. mul-1-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-commutative 75.3%

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

   4. Simplified 75.3%

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

      1. associate-/r* 75.6%

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

      2. div-inv 75.6%

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

      3. div-inv 75.6%

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

      4. pow-to-exp 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. un-div-inv 75.6%

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

      2. associate-/l/ 75.3%

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

      3. associate-/r* 75.7%

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

   8. Applied egg-rr 75.7%

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

   if 2.0000000000000001e-13 < (/.f64 1 n)

   1. Initial program 65.4%

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

   2. Taylor expanded in n around 0 65.4%

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

      1. log1p-def 96.8%

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

   4. Simplified 96.8%

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

   5. Taylor expanded in x around 0 96.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-85}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{t_0}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-85}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+234}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-44)
     (* (/ 1.0 n) (/ t_0 x))
     (if (<= (/ 1.0 n) 1e-85)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e-13)
         (/ (/ t_0 n) x)
         (if (<= (/ 1.0 n) 1e+234)
           (- (+ 1.0 (/ x n)) t_0)
           (/ (/ 1.0 t_0) (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-44) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-85) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e+234) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 / t_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) <= -4e-44) {
		tmp = (1.0 / n) * (t_0 / x);
	} else if ((1.0 / n) <= 1e-85) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e+234) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 / t_0) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-44:
		tmp = (1.0 / n) * (t_0 / x)
	elif (1.0 / n) <= 1e-85:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-13:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e+234:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (1.0 / t_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) <= -4e-44)
		tmp = Float64(Float64(1.0 / n) * Float64(t_0 / x));
	elseif (Float64(1.0 / n) <= 1e-85)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e+234)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(1.0 / t_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], -4e-44], N[(N[(1.0 / n), $MachinePrecision] * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-85], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+234], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -3.99999999999999981e-44

   1. Initial program 89.1%

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

   2. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. log-rec 96.6%

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

      3. mul-1-neg 96.6%

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

      4. distribute-neg-frac 96.6%

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

      5. mul-1-neg 96.6%

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

      6. remove-double-neg 96.6%

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

      7. *-commutative 96.6%

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

   4. Simplified 96.6%

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

      1. associate-/r* 96.6%

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

      2. div-inv 96.6%

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

      3. div-inv 96.6%

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

      4. pow-to-exp 96.6%

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

   6. Applied egg-rr 96.6%

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

   if -3.99999999999999981e-44 < (/.f64 1 n) < 9.9999999999999998e-86

   1. Initial program 31.8%

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

   2. Taylor expanded in n around inf 80.2%

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

      1. log1p-def 80.2%

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

   4. Simplified 80.2%

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

   if 9.9999999999999998e-86 < (/.f64 1 n) < 2.0000000000000001e-13

   1. Initial program 13.9%

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

   2. Taylor expanded in x around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. log-rec 75.3%

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

      3. mul-1-neg 75.3%

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

      4. distribute-neg-frac 75.3%

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

      5. mul-1-neg 75.3%

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

      6. remove-double-neg 75.3%

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

      7. *-commutative 75.3%

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

   4. Simplified 75.3%

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

      1. associate-/r* 75.6%

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

      2. div-inv 75.6%

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

      3. div-inv 75.6%

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

      4. pow-to-exp 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. un-div-inv 75.6%

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

      2. associate-/l/ 75.3%

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

      3. associate-/r* 75.7%

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

   8. Applied egg-rr 75.7%

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

   if 2.0000000000000001e-13 < (/.f64 1 n) < 1.00000000000000002e234

   1. Initial program 75.1%

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

   2. Taylor expanded in x around 0 77.0%

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

   if 1.00000000000000002e234 < (/.f64 1 n)

   1. Initial program 17.0%

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

   2. Taylor expanded in x around inf 0.2%

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

      1. mul-1-neg 0.2%

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

      2. log-rec 0.2%

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

      3. mul-1-neg 0.2%

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

      4. distribute-neg-frac 0.2%

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

      5. mul-1-neg 0.2%

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

      6. remove-double-neg 0.2%

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

      7. *-commutative 0.2%

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

   4. Simplified 0.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. add-sqr-sqrt 0.2%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 86.2%

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

      5. sqr-neg 86.2%

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

      6. sqrt-unprod 86.2%

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

      7. add-sqr-sqrt 86.2%

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

      8. distribute-frac-neg 86.2%

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

      9. exp-neg 86.2%

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

      10. div-inv 86.2%

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

      11. pow-to-exp 86.2%

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

   6. Applied egg-rr 86.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-85}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+234}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x}\\ \end{array} \]

Alternative 4: 68.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.14 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-155}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-125} \lor \neg \left(x \leq 1.25 \cdot 10^{-91}\right) \land x \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{0.5 - \frac{n}{\log x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 1.14e-303)
     (/ (- (log x)) n)
     (if (<= x 6.2e-155)
       (- 1.0 t_0)
       (if (or (<= x 3.9e-125) (and (not (<= x 1.25e-91)) (<= x 6.6e-9)))
         (/ 1.0 (- 0.5 (/ n (log x))))
         (/ (/ t_0 n) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.14e-303) {
		tmp = -log(x) / n;
	} else if (x <= 6.2e-155) {
		tmp = 1.0 - t_0;
	} else if ((x <= 3.9e-125) || (!(x <= 1.25e-91) && (x <= 6.6e-9))) {
		tmp = 1.0 / (0.5 - (n / log(x)));
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 1.14d-303) then
        tmp = -log(x) / n
    else if (x <= 6.2d-155) then
        tmp = 1.0d0 - t_0
    else if ((x <= 3.9d-125) .or. (.not. (x <= 1.25d-91)) .and. (x <= 6.6d-9)) then
        tmp = 1.0d0 / (0.5d0 - (n / log(x)))
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.14e-303) {
		tmp = -Math.log(x) / n;
	} else if (x <= 6.2e-155) {
		tmp = 1.0 - t_0;
	} else if ((x <= 3.9e-125) || (!(x <= 1.25e-91) && (x <= 6.6e-9))) {
		tmp = 1.0 / (0.5 - (n / Math.log(x)));
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.14e-303:
		tmp = -math.log(x) / n
	elif x <= 6.2e-155:
		tmp = 1.0 - t_0
	elif (x <= 3.9e-125) or (not (x <= 1.25e-91) and (x <= 6.6e-9)):
		tmp = 1.0 / (0.5 - (n / math.log(x)))
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.14e-303)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 6.2e-155)
		tmp = Float64(1.0 - t_0);
	elseif ((x <= 3.9e-125) || (!(x <= 1.25e-91) && (x <= 6.6e-9)))
		tmp = Float64(1.0 / Float64(0.5 - Float64(n / log(x))));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 1.14e-303)
		tmp = -log(x) / n;
	elseif (x <= 6.2e-155)
		tmp = 1.0 - t_0;
	elseif ((x <= 3.9e-125) || (~((x <= 1.25e-91)) && (x <= 6.6e-9)))
		tmp = 1.0 / (0.5 - (n / log(x)));
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.14e-303], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 6.2e-155], N[(1.0 - t$95$0), $MachinePrecision], If[Or[LessEqual[x, 3.9e-125], And[N[Not[LessEqual[x, 1.25e-91]], $MachinePrecision], LessEqual[x, 6.6e-9]]], N[(1.0 / N[(0.5 - N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.14e-303

   1. Initial program 5.2%

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

   2. Taylor expanded in x around 0 5.2%

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

   3. Taylor expanded in n around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   if 1.14e-303 < x < 6.2e-155

   1. Initial program 59.2%

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

   2. Taylor expanded in x around 0 59.2%

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

   if 6.2e-155 < x < 3.89999999999999982e-125 or 1.24999999999999999e-91 < x < 6.60000000000000037e-9

   1. Initial program 28.3%

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

   2. Taylor expanded in x around 0 28.3%

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

      1. flip-- 14.3%

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

      2. clear-num 14.3%

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

      3. metadata-eval 14.3%

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

      4. pow-sqr 14.6%

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

      5. div-inv 14.6%

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

   4. Applied egg-rr 14.6%

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

   5. Taylor expanded in n around inf 58.9%

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

      1. associate-*r/ 58.9%

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

      2. neg-mul-1 58.9%

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

   7. Simplified 58.9%

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

   if 3.89999999999999982e-125 < x < 1.24999999999999999e-91 or 6.60000000000000037e-9 < x

   1. Initial program 66.2%

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

   2. Taylor expanded in x around inf 94.3%

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

      1. mul-1-neg 94.3%

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

      2. log-rec 94.3%

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

      3. mul-1-neg 94.3%

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

      4. distribute-neg-frac 94.3%

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

      5. mul-1-neg 94.3%

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

      6. remove-double-neg 94.3%

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

      7. *-commutative 94.3%

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

   4. Simplified 94.3%

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

      1. associate-/r* 95.4%

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

      2. div-inv 95.3%

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

      3. div-inv 95.3%

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

      4. pow-to-exp 95.3%

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

   6. Applied egg-rr 95.3%

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

      1. un-div-inv 95.4%

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

      2. associate-/l/ 94.3%

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

      3. associate-/r* 95.3%

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

   8. Applied egg-rr 95.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.14 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-155}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-125} \lor \neg \left(x \leq 1.25 \cdot 10^{-91}\right) \land x \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{0.5 - \frac{n}{\log x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 5: 68.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 4.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-150}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-125} \lor \neg \left(x \leq 2.1 \cdot 10^{-91}\right) \land x \leq 0.000122:\\ \;\;\;\;\frac{1}{0.5 - \frac{n}{\log x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 4.2e-303)
     (/ (- (log x)) n)
     (if (<= x 4.7e-150)
       (- (+ 1.0 (/ x n)) t_0)
       (if (or (<= x 3.4e-125) (and (not (<= x 2.1e-91)) (<= x 0.000122)))
         (/ 1.0 (- 0.5 (/ n (log x))))
         (/ (/ t_0 n) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 4.2e-303) {
		tmp = -log(x) / n;
	} else if (x <= 4.7e-150) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if ((x <= 3.4e-125) || (!(x <= 2.1e-91) && (x <= 0.000122))) {
		tmp = 1.0 / (0.5 - (n / log(x)));
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 4.2d-303) then
        tmp = -log(x) / n
    else if (x <= 4.7d-150) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if ((x <= 3.4d-125) .or. (.not. (x <= 2.1d-91)) .and. (x <= 0.000122d0)) then
        tmp = 1.0d0 / (0.5d0 - (n / log(x)))
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 4.2e-303) {
		tmp = -Math.log(x) / n;
	} else if (x <= 4.7e-150) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if ((x <= 3.4e-125) || (!(x <= 2.1e-91) && (x <= 0.000122))) {
		tmp = 1.0 / (0.5 - (n / Math.log(x)));
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 4.2e-303:
		tmp = -math.log(x) / n
	elif x <= 4.7e-150:
		tmp = (1.0 + (x / n)) - t_0
	elif (x <= 3.4e-125) or (not (x <= 2.1e-91) and (x <= 0.000122)):
		tmp = 1.0 / (0.5 - (n / math.log(x)))
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 4.2e-303)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 4.7e-150)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif ((x <= 3.4e-125) || (!(x <= 2.1e-91) && (x <= 0.000122)))
		tmp = Float64(1.0 / Float64(0.5 - Float64(n / log(x))));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 4.2e-303)
		tmp = -log(x) / n;
	elseif (x <= 4.7e-150)
		tmp = (1.0 + (x / n)) - t_0;
	elseif ((x <= 3.4e-125) || (~((x <= 2.1e-91)) && (x <= 0.000122)))
		tmp = 1.0 / (0.5 - (n / log(x)));
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4.2e-303], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 4.7e-150], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[Or[LessEqual[x, 3.4e-125], And[N[Not[LessEqual[x, 2.1e-91]], $MachinePrecision], LessEqual[x, 0.000122]]], N[(1.0 / N[(0.5 - N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.2e-303

   1. Initial program 5.2%

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

   2. Taylor expanded in x around 0 5.2%

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

   3. Taylor expanded in n around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   if 4.2e-303 < x < 4.6999999999999999e-150

   1. Initial program 59.2%

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

   2. Taylor expanded in x around 0 59.6%

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

   if 4.6999999999999999e-150 < x < 3.39999999999999975e-125 or 2.0999999999999999e-91 < x < 1.21999999999999997e-4

   1. Initial program 28.3%

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

   2. Taylor expanded in x around 0 28.3%

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

      1. flip-- 14.3%

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

      2. clear-num 14.3%

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

      3. metadata-eval 14.3%

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

      4. pow-sqr 14.6%

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

      5. div-inv 14.6%

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

   4. Applied egg-rr 14.6%

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

   5. Taylor expanded in n around inf 58.9%

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

      1. associate-*r/ 58.9%

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

      2. neg-mul-1 58.9%

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

   7. Simplified 58.9%

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

   if 3.39999999999999975e-125 < x < 2.0999999999999999e-91 or 1.21999999999999997e-4 < x

   1. Initial program 66.2%

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

   2. Taylor expanded in x around inf 94.3%

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

      1. mul-1-neg 94.3%

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

      2. log-rec 94.3%

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

      3. mul-1-neg 94.3%

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

      4. distribute-neg-frac 94.3%

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

      5. mul-1-neg 94.3%

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

      6. remove-double-neg 94.3%

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

      7. *-commutative 94.3%

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

   4. Simplified 94.3%

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

      1. associate-/r* 95.4%

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

      2. div-inv 95.3%

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

      3. div-inv 95.3%

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

      4. pow-to-exp 95.3%

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

   6. Applied egg-rr 95.3%

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

      1. un-div-inv 95.4%

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

      2. associate-/l/ 94.3%

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

      3. associate-/r* 95.3%

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

   8. Applied egg-rr 95.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-150}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-125} \lor \neg \left(x \leq 2.1 \cdot 10^{-91}\right) \land x \leq 0.000122:\\ \;\;\;\;\frac{1}{0.5 - \frac{n}{\log x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 6: 55.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{0.5 - \frac{n}{\log x}}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2.8 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.03:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- 0.5 (/ n (log x))))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 2.8e-303)
     (/ (- (log x)) n)
     (if (<= x 2.05e-154)
       t_1
       (if (<= x 3.9e-125)
         t_0
         (if (<= x 1.35e-91)
           t_1
           (if (<= x 0.03) t_0 (* (/ 1.0 n) (/ 1.0 x)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (0.5 - (n / log(x)));
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.8e-303) {
		tmp = -log(x) / n;
	} else if (x <= 2.05e-154) {
		tmp = t_1;
	} else if (x <= 3.9e-125) {
		tmp = t_0;
	} else if (x <= 1.35e-91) {
		tmp = t_1;
	} else if (x <= 0.03) {
		tmp = t_0;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (0.5d0 - (n / log(x)))
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 2.8d-303) then
        tmp = -log(x) / n
    else if (x <= 2.05d-154) then
        tmp = t_1
    else if (x <= 3.9d-125) then
        tmp = t_0
    else if (x <= 1.35d-91) then
        tmp = t_1
    else if (x <= 0.03d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / n) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (0.5 - (n / Math.log(x)));
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.8e-303) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.05e-154) {
		tmp = t_1;
	} else if (x <= 3.9e-125) {
		tmp = t_0;
	} else if (x <= 1.35e-91) {
		tmp = t_1;
	} else if (x <= 0.03) {
		tmp = t_0;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (0.5 - (n / math.log(x)))
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 2.8e-303:
		tmp = -math.log(x) / n
	elif x <= 2.05e-154:
		tmp = t_1
	elif x <= 3.9e-125:
		tmp = t_0
	elif x <= 1.35e-91:
		tmp = t_1
	elif x <= 0.03:
		tmp = t_0
	else:
		tmp = (1.0 / n) * (1.0 / x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(0.5 - Float64(n / log(x))))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 2.8e-303)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.05e-154)
		tmp = t_1;
	elseif (x <= 3.9e-125)
		tmp = t_0;
	elseif (x <= 1.35e-91)
		tmp = t_1;
	elseif (x <= 0.03)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (0.5 - (n / log(x)));
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 2.8e-303)
		tmp = -log(x) / n;
	elseif (x <= 2.05e-154)
		tmp = t_1;
	elseif (x <= 3.9e-125)
		tmp = t_0;
	elseif (x <= 1.35e-91)
		tmp = t_1;
	elseif (x <= 0.03)
		tmp = t_0;
	else
		tmp = (1.0 / n) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(0.5 - N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.8e-303], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.05e-154], t$95$1, If[LessEqual[x, 3.9e-125], t$95$0, If[LessEqual[x, 1.35e-91], t$95$1, If[LessEqual[x, 0.03], t$95$0, N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.8e-303

   1. Initial program 5.2%

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

   2. Taylor expanded in x around 0 5.2%

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

   3. Taylor expanded in n around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   if 2.8e-303 < x < 2.05e-154 or 3.89999999999999982e-125 < x < 1.3499999999999999e-91

   1. Initial program 60.4%

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

   2. Taylor expanded in x around 0 60.4%

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

   if 2.05e-154 < x < 3.89999999999999982e-125 or 1.3499999999999999e-91 < x < 0.029999999999999999

   1. Initial program 28.3%

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

   2. Taylor expanded in x around 0 28.3%

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

      1. flip-- 14.3%

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

      2. clear-num 14.3%

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

      3. metadata-eval 14.3%

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

      4. pow-sqr 14.6%

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

      5. div-inv 14.6%

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

   4. Applied egg-rr 14.6%

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

   5. Taylor expanded in n around inf 58.9%

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

      1. associate-*r/ 58.9%

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

      2. neg-mul-1 58.9%

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

   7. Simplified 58.9%

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

   if 0.029999999999999999 < x

   1. Initial program 66.2%

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

   2. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   4. Simplified 97.7%

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

      1. associate-/r* 98.8%

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

      2. div-inv 98.8%

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

      3. div-inv 98.8%

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

      4. pow-to-exp 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in n around inf 61.5%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-154}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{0.5 - \frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-91}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.03:\\ \;\;\;\;\frac{1}{0.5 - \frac{n}{\log x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 7: 55.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.9 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.03:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.9e-303)
     t_0
     (if (<= x 6.2e-155)
       t_1
       (if (<= x 2.6e-125)
         t_0
         (if (<= x 2.6e-81)
           t_1
           (if (<= x 0.03) t_0 (* (/ 1.0 n) (/ 1.0 x)))))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.9e-303) {
		tmp = t_0;
	} else if (x <= 6.2e-155) {
		tmp = t_1;
	} else if (x <= 2.6e-125) {
		tmp = t_0;
	} else if (x <= 2.6e-81) {
		tmp = t_1;
	} else if (x <= 0.03) {
		tmp = t_0;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -log(x) / n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1.9d-303) then
        tmp = t_0
    else if (x <= 6.2d-155) then
        tmp = t_1
    else if (x <= 2.6d-125) then
        tmp = t_0
    else if (x <= 2.6d-81) then
        tmp = t_1
    else if (x <= 0.03d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / n) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.9e-303) {
		tmp = t_0;
	} else if (x <= 6.2e-155) {
		tmp = t_1;
	} else if (x <= 2.6e-125) {
		tmp = t_0;
	} else if (x <= 2.6e-81) {
		tmp = t_1;
	} else if (x <= 0.03) {
		tmp = t_0;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.9e-303:
		tmp = t_0
	elif x <= 6.2e-155:
		tmp = t_1
	elif x <= 2.6e-125:
		tmp = t_0
	elif x <= 2.6e-81:
		tmp = t_1
	elif x <= 0.03:
		tmp = t_0
	else:
		tmp = (1.0 / n) * (1.0 / x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.9e-303)
		tmp = t_0;
	elseif (x <= 6.2e-155)
		tmp = t_1;
	elseif (x <= 2.6e-125)
		tmp = t_0;
	elseif (x <= 2.6e-81)
		tmp = t_1;
	elseif (x <= 0.03)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1.9e-303)
		tmp = t_0;
	elseif (x <= 6.2e-155)
		tmp = t_1;
	elseif (x <= 2.6e-125)
		tmp = t_0;
	elseif (x <= 2.6e-81)
		tmp = t_1;
	elseif (x <= 0.03)
		tmp = t_0;
	else
		tmp = (1.0 / n) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.9e-303], t$95$0, If[LessEqual[x, 6.2e-155], t$95$1, If[LessEqual[x, 2.6e-125], t$95$0, If[LessEqual[x, 2.6e-81], t$95$1, If[LessEqual[x, 0.03], t$95$0, N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.90000000000000005e-303 or 6.2e-155 < x < 2.60000000000000006e-125 or 2.5999999999999999e-81 < x < 0.029999999999999999

   1. Initial program 23.9%

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

   2. Taylor expanded in x around 0 23.9%

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

   3. Taylor expanded in n around inf 65.2%

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

      1. associate-*r/ 65.2%

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

      2. mul-1-neg 65.2%

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

   5. Simplified 65.2%

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

   if 1.90000000000000005e-303 < x < 6.2e-155 or 2.60000000000000006e-125 < x < 2.5999999999999999e-81

   1. Initial program 58.9%

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

   2. Taylor expanded in x around 0 58.9%

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

   if 0.029999999999999999 < x

   1. Initial program 66.2%

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

   2. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   4. Simplified 97.7%

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

      1. associate-/r* 98.8%

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

      2. div-inv 98.8%

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

      3. div-inv 98.8%

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

      4. pow-to-exp 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in n around inf 61.5%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-155}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-125}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-81}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.03:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 8: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.03) (/ (- (log x)) n) (* (/ 1.0 n) (/ 1.0 x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.03) {
		tmp = -log(x) / n;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.03d0) then
        tmp = -log(x) / n
    else
        tmp = (1.0d0 / n) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.03) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.03:
		tmp = -math.log(x) / n
	else:
		tmp = (1.0 / n) * (1.0 / x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.03)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.03)
		tmp = -log(x) / n;
	else
		tmp = (1.0 / n) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.03], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.029999999999999999

   1. Initial program 46.6%

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

   2. Taylor expanded in x around 0 46.6%

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

   3. Taylor expanded in n around inf 49.8%

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

      1. associate-*r/ 49.8%

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

      2. mul-1-neg 49.8%

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

   5. Simplified 49.8%

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

   if 0.029999999999999999 < x

   1. Initial program 66.2%

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

   2. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   4. Simplified 97.7%

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

      1. associate-/r* 98.8%

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

      2. div-inv 98.8%

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

      3. div-inv 98.8%

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

      4. pow-to-exp 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in n around inf 61.5%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{1}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 9: 41.1% accurate, 30.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

2. Taylor expanded in x around inf 59.5%

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

   1. mul-1-neg 59.5%

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

   2. log-rec 59.5%

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

   3. mul-1-neg 59.5%

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

   4. distribute-neg-frac 59.5%

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

   5. mul-1-neg 59.5%

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

   6. remove-double-neg 59.5%

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

   7. *-commutative 59.5%

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

4. Simplified 59.5%

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

   1. associate-/r* 60.2%

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

   2. div-inv 60.2%

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

   3. div-inv 60.2%

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

   4. pow-to-exp 60.2%

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

6. Applied egg-rr 60.2%

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

7. Taylor expanded in n around inf 39.2%

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

8. Final simplification 39.2%

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

Alternative 10: 40.5% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

2. Taylor expanded in x around inf 59.5%

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

   1. mul-1-neg 59.5%

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

   2. log-rec 59.5%

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

   3. mul-1-neg 59.5%

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

   4. distribute-neg-frac 59.5%

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

   5. mul-1-neg 59.5%

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

   6. remove-double-neg 59.5%

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

   7. *-commutative 59.5%

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

4. Simplified 59.5%

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

5. Taylor expanded in n around inf 38.6%

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

6. Final simplification 38.6%

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

Reproduce

herbie shell --seed 2023305 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))