2nthrt (problem 3.4.6)

Percentage Accurate: 53.1% → 84.2%

Time: 20.5s

Alternatives: 12

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: 53.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-151)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 1e-21)
       (/ (- (log1p x) (log 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-151) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-21) {
		tmp = (log1p(x) - log(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-151) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 1e-21) {
		tmp = (Math.log1p(x) - Math.log(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-151:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 1e-21:
		tmp = (math.log1p(x) - math.log(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-151)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e-21)
		tmp = Float64(Float64(log1p(x) - log(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-151], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-21], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[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) < -5.00000000000000003e-151

   1. Initial program 75.0%

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

   2. Taylor expanded in x around inf 92.0%

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

      1. mul-1-neg 92.0%

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

      2. log-rec 92.0%

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

      3. mul-1-neg 92.0%

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

      4. distribute-neg-frac 92.0%

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

      5. mul-1-neg 92.0%

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

      6. remove-double-neg 92.0%

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

      7. *-commutative 92.0%

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

   4. Simplified 92.0%

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

      1. associate-/r* 92.1%

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

      2. div-inv 92.1%

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

      3. div-inv 92.1%

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

      4. pow-to-exp 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. un-div-inv 92.1%

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

   8. Applied egg-rr 92.1%

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

   if -5.00000000000000003e-151 < (/.f64 1 n) < 9.99999999999999908e-22

   1. Initial program 33.8%

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

   2. Taylor expanded in n around inf 88.6%

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

      1. log1p-def 88.6%

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

   4. Simplified 88.6%

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

   if 9.99999999999999908e-22 < (/.f64 1 n)

   1. Initial program 52.9%

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

   2. Taylor expanded in n around 0 52.9%

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

      1. log1p-def 93.3%

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

   4. Simplified 93.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-151)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 5e-9)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e+187) (- (+ 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-151) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 5e-9) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e+187) {
		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-151) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 5e-9) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e+187) {
		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-151:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 5e-9:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e+187:
		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-151)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-9)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e+187)
		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-151], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-9], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+187], 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) < -5.00000000000000003e-151

   1. Initial program 75.0%

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

   2. Taylor expanded in x around inf 92.0%

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

      1. mul-1-neg 92.0%

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

      2. log-rec 92.0%

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

      3. mul-1-neg 92.0%

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

      4. distribute-neg-frac 92.0%

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

      5. mul-1-neg 92.0%

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

      6. remove-double-neg 92.0%

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

      7. *-commutative 92.0%

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

   4. Simplified 92.0%

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

      1. associate-/r* 92.1%

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

      2. div-inv 92.1%

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

      3. div-inv 92.1%

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

      4. pow-to-exp 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. un-div-inv 92.1%

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

   8. Applied egg-rr 92.1%

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

   if -5.00000000000000003e-151 < (/.f64 1 n) < 5.0000000000000001e-9

   1. Initial program 33.4%

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

   2. Taylor expanded in n around inf 86.7%

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

      1. log1p-def 86.7%

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

   4. Simplified 86.7%

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

   if 5.0000000000000001e-9 < (/.f64 1 n) < 1.99999999999999981e187

   1. Initial program 78.9%

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

   2. Taylor expanded in x around 0 79.0%

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

   if 1.99999999999999981e187 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 7.4%

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

      1. log1p-def 7.4%

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

   4. Simplified 7.4%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 3: 79.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{n} \cdot \log \left(\frac{1 + x}{x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-151)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 5e-9)
       (* (/ 1.0 n) (log (/ (+ 1.0 x) x)))
       (if (<= (/ 1.0 n) 2e+187) (- (+ 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-151) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 5e-9) {
		tmp = (1.0 / n) * log(((1.0 + x) / x));
	} else if ((1.0 / n) <= 2e+187) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-151)) then
        tmp = (t_0 / x) / n
    else if ((1.0d0 / n) <= 5d-9) then
        tmp = (1.0d0 / n) * log(((1.0d0 + x) / x))
    else if ((1.0d0 / n) <= 2d+187) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (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 ((1.0 / n) <= -5e-151) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 5e-9) {
		tmp = (1.0 / n) * Math.log(((1.0 + x) / x));
	} else if ((1.0 / n) <= 2e+187) {
		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-151:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 5e-9:
		tmp = (1.0 / n) * math.log(((1.0 + x) / x))
	elif (1.0 / n) <= 2e+187:
		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-151)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-9)
		tmp = Float64(Float64(1.0 / n) * log(Float64(Float64(1.0 + x) / x)));
	elseif (Float64(1.0 / n) <= 2e+187)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-151)
		tmp = (t_0 / x) / n;
	elseif ((1.0 / n) <= 5e-9)
		tmp = (1.0 / n) * log(((1.0 + x) / x));
	elseif ((1.0 / n) <= 2e+187)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (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[N[(1.0 / n), $MachinePrecision], -5e-151], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-9], N[(N[(1.0 / n), $MachinePrecision] * N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+187], 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) < -5.00000000000000003e-151

   1. Initial program 75.0%

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

   2. Taylor expanded in x around inf 92.0%

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

      1. mul-1-neg 92.0%

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

      2. log-rec 92.0%

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

      3. mul-1-neg 92.0%

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

      4. distribute-neg-frac 92.0%

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

      5. mul-1-neg 92.0%

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

      6. remove-double-neg 92.0%

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

      7. *-commutative 92.0%

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

   4. Simplified 92.0%

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

      1. associate-/r* 92.1%

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

      2. div-inv 92.1%

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

      3. div-inv 92.1%

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

      4. pow-to-exp 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. un-div-inv 92.1%

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

   8. Applied egg-rr 92.1%

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

   if -5.00000000000000003e-151 < (/.f64 1 n) < 5.0000000000000001e-9

   1. Initial program 33.4%

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

   2. Taylor expanded in n around inf 86.7%

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

      1. log1p-def 86.7%

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

   4. Simplified 86.7%

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

      1. div-inv 86.7%

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

      2. *-commutative 86.7%

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

   6. Applied egg-rr 86.7%

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

      1. add-log-exp 33.4%

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

      2. *-commutative 33.4%

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

      3. exp-prod 33.4%

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

      4. exp-diff 33.4%

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

      5. log1p-udef 33.4%

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

      6. add-exp-log 33.4%

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

      7. add-exp-log 33.4%

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

   8. Applied egg-rr 33.4%

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

      1. log-pow 86.7%

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

   10. Simplified 86.7%

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

   if 5.0000000000000001e-9 < (/.f64 1 n) < 1.99999999999999981e187

   1. Initial program 78.9%

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

   2. Taylor expanded in x around 0 79.0%

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

   if 1.99999999999999981e187 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 7.4%

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

      1. log1p-def 7.4%

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

   4. Simplified 7.4%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{n} \cdot \log \left(\frac{1 + x}{x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 4: 79.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{n} \cdot \log \left(\frac{1 + x}{x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-151)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 5e-9)
       (* (/ 1.0 n) (log (/ (+ 1.0 x) x)))
       (if (<= (/ 1.0 n) 2e+187) (- 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-151) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 5e-9) {
		tmp = (1.0 / n) * log(((1.0 + x) / x));
	} else if ((1.0 / n) <= 2e+187) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-151)) then
        tmp = (t_0 / x) / n
    else if ((1.0d0 / n) <= 5d-9) then
        tmp = (1.0d0 / n) * log(((1.0d0 + x) / x))
    else if ((1.0d0 / n) <= 2d+187) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (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 ((1.0 / n) <= -5e-151) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 5e-9) {
		tmp = (1.0 / n) * Math.log(((1.0 + x) / x));
	} else if ((1.0 / n) <= 2e+187) {
		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-151:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 5e-9:
		tmp = (1.0 / n) * math.log(((1.0 + x) / x))
	elif (1.0 / n) <= 2e+187:
		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-151)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-9)
		tmp = Float64(Float64(1.0 / n) * log(Float64(Float64(1.0 + x) / x)));
	elseif (Float64(1.0 / n) <= 2e+187)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-151)
		tmp = (t_0 / x) / n;
	elseif ((1.0 / n) <= 5e-9)
		tmp = (1.0 / n) * log(((1.0 + x) / x));
	elseif ((1.0 / n) <= 2e+187)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (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[N[(1.0 / n), $MachinePrecision], -5e-151], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-9], N[(N[(1.0 / n), $MachinePrecision] * N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+187], 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) < -5.00000000000000003e-151

   1. Initial program 75.0%

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

   2. Taylor expanded in x around inf 92.0%

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

      1. mul-1-neg 92.0%

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

      2. log-rec 92.0%

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

      3. mul-1-neg 92.0%

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

      4. distribute-neg-frac 92.0%

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

      5. mul-1-neg 92.0%

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

      6. remove-double-neg 92.0%

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

      7. *-commutative 92.0%

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

   4. Simplified 92.0%

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

      1. associate-/r* 92.1%

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

      2. div-inv 92.1%

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

      3. div-inv 92.1%

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

      4. pow-to-exp 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. un-div-inv 92.1%

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

   8. Applied egg-rr 92.1%

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

   if -5.00000000000000003e-151 < (/.f64 1 n) < 5.0000000000000001e-9

   1. Initial program 33.4%

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

   2. Taylor expanded in n around inf 86.7%

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

      1. log1p-def 86.7%

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

   4. Simplified 86.7%

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

      1. div-inv 86.7%

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

      2. *-commutative 86.7%

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

   6. Applied egg-rr 86.7%

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

      1. add-log-exp 33.4%

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

      2. *-commutative 33.4%

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

      3. exp-prod 33.4%

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

      4. exp-diff 33.4%

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

      5. log1p-udef 33.4%

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

      6. add-exp-log 33.4%

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

      7. add-exp-log 33.4%

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

   8. Applied egg-rr 33.4%

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

      1. log-pow 86.7%

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

   10. Simplified 86.7%

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

   if 5.0000000000000001e-9 < (/.f64 1 n) < 1.99999999999999981e187

   1. Initial program 78.9%

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

   2. Taylor expanded in x around 0 78.9%

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

   if 1.99999999999999981e187 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 7.4%

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

      1. log1p-def 7.4%

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

   4. Simplified 7.4%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{n} \cdot \log \left(\frac{1 + x}{x}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 5: 61.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{+197}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;n \leq -10.5:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 230000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -1.45e+197)
   (* (log x) (/ -1.0 n))
   (if (<= n -10.5)
     (* (/ 1.0 n) (/ 1.0 x))
     (if (<= n 2.5e-188)
       (/ 0.3333333333333333 (* n (pow x 3.0)))
       (if (<= n 230000000.0) (- 1.0 (pow x (/ 1.0 n))) (/ (pow n -1.0) x))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -1.45e+197) {
		tmp = log(x) * (-1.0 / n);
	} else if (n <= -10.5) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 2.5e-188) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 230000000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = pow(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 (n <= (-1.45d+197)) then
        tmp = log(x) * ((-1.0d0) / n)
    else if (n <= (-10.5d0)) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if (n <= 2.5d-188) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 230000000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (n ** (-1.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -1.45e+197) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if (n <= -10.5) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 2.5e-188) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 230000000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.pow(n, -1.0) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -1.45e+197:
		tmp = math.log(x) * (-1.0 / n)
	elif n <= -10.5:
		tmp = (1.0 / n) * (1.0 / x)
	elif n <= 2.5e-188:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 230000000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.pow(n, -1.0) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -1.45e+197)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (n <= -10.5)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (n <= 2.5e-188)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 230000000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64((n ^ -1.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -1.45e+197)
		tmp = log(x) * (-1.0 / n);
	elseif (n <= -10.5)
		tmp = (1.0 / n) * (1.0 / x);
	elseif (n <= 2.5e-188)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 230000000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (n ^ -1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -1.45e+197], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -10.5], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.5e-188], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 230000000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.45000000000000001e197

   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 98.8%

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

      1. log1p-def 98.8%

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

   4. Simplified 98.8%

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

      1. div-inv 98.9%

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

      2. *-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in x around 0 68.0%

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

      1. mul-1-neg 67.9%

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

   9. Simplified 68.0%

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

   if -1.45000000000000001e197 < n < -10.5

   1. Initial program 16.1%

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

   2. Taylor expanded in n around inf 47.5%

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

      1. log1p-def 47.5%

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

   4. Simplified 47.5%

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

   5. Taylor expanded in x around inf 65.3%

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

      1. *-commutative 65.3%

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

   7. Simplified 65.3%

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

      1. associate-/r* 65.4%

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

      2. div-inv 65.4%

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

   9. Applied egg-rr 65.4%

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

   if -10.5 < n < 2.5e-188

   1. Initial program 88.3%

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

   2. Taylor expanded in n around inf 51.7%

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

      1. log1p-def 51.7%

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

   4. Simplified 51.7%

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

   5. Taylor expanded in x around inf 6.4%

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

   6. Taylor expanded in x around 0 80.8%

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

   if 2.5e-188 < n < 2.3e8

   1. Initial program 78.9%

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

   2. Taylor expanded in x around 0 78.9%

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

   if 2.3e8 < n

   1. Initial program 33.1%

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

   2. Taylor expanded in n around inf 81.8%

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

      1. log1p-def 81.8%

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

   4. Simplified 81.8%

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

   5. Taylor expanded in x around inf 51.4%

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

      1. div-inv 51.4%

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

      2. add-sqr-sqrt 51.3%

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

      3. associate-*r* 51.3%

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

      4. inv-pow 51.3%

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

      5. metadata-eval 51.3%

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

      6. sqrt-pow1 51.3%

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

      7. metadata-eval 51.3%

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

      8. metadata-eval 51.3%

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

      9. inv-pow 51.3%

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

      10. metadata-eval 51.3%

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

      11. sqrt-pow1 51.3%

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

      12. metadata-eval 51.3%

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

      13. metadata-eval 51.3%

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

   7. Applied egg-rr 51.3%

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

      1. associate-*l/ 51.3%

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

      2. associate-*l/ 51.3%

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

      3. *-lft-identity 51.3%

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

      4. pow-sqr 51.4%

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

      5. metadata-eval 51.4%

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

   9. Simplified 51.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{+197}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;n \leq -10.5:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 230000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \]

Alternative 6: 71.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 6.2 \cdot 10^{-164}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 6.2e-164)
     (- 1.0 t_0)
     (if (<= x 0.095) (- (/ x n) (/ (log x) n)) (/ (/ t_0 x) n)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 6.2e-164) {
		tmp = 1.0 - t_0;
	} else if (x <= 0.095) {
		tmp = (x / n) - (log(x) / n);
	} else {
		tmp = (t_0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 6.2d-164) then
        tmp = 1.0d0 - t_0
    else if (x <= 0.095d0) then
        tmp = (x / n) - (log(x) / n)
    else
        tmp = (t_0 / x) / n
    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 <= 6.2e-164) {
		tmp = 1.0 - t_0;
	} else if (x <= 0.095) {
		tmp = (x / n) - (Math.log(x) / n);
	} else {
		tmp = (t_0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 6.2e-164:
		tmp = 1.0 - t_0
	elif x <= 0.095:
		tmp = (x / n) - (math.log(x) / n)
	else:
		tmp = (t_0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 6.2e-164)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 0.095)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(t_0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 6.2e-164)
		tmp = 1.0 - t_0;
	elseif (x <= 0.095)
		tmp = (x / n) - (log(x) / n);
	else
		tmp = (t_0 / x) / n;
	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, 6.2e-164], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 0.095], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.2000000000000001e-164

   1. Initial program 55.1%

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

   2. Taylor expanded in x around 0 55.1%

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

   if 6.2000000000000001e-164 < x < 0.095000000000000001

   1. Initial program 27.5%

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

   2. Taylor expanded in n around inf 64.8%

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

      1. log1p-def 64.8%

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

   4. Simplified 64.8%

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

      1. div-sub 64.8%

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

   6. Applied egg-rr 64.8%

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

   7. Taylor expanded in x around 0 61.9%

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

   if 0.095000000000000001 < x

   1. Initial program 69.6%

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

   2. Taylor expanded in x around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. log-rec 96.4%

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

      3. mul-1-neg 96.4%

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

      4. distribute-neg-frac 96.4%

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

      5. mul-1-neg 96.4%

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

      6. remove-double-neg 96.4%

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

      7. *-commutative 96.4%

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

   4. Simplified 96.4%

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

      1. associate-/r* 97.9%

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

      2. div-inv 97.9%

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

      3. div-inv 97.9%

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

      4. pow-to-exp 97.9%

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

   6. Applied egg-rr 97.9%

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

      1. un-div-inv 97.9%

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

   8. Applied egg-rr 97.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-164}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \end{array} \]

Alternative 7: 56.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-164}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.54:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4e-164)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.54) (/ (- (log x)) n) (/ (pow n -1.0) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4e-164) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.54) {
		tmp = -log(x) / n;
	} else {
		tmp = pow(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 <= 4d-164) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.54d0) then
        tmp = -log(x) / n
    else
        tmp = (n ** (-1.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4e-164) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.54) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = Math.pow(n, -1.0) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4e-164:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.54:
		tmp = -math.log(x) / n
	else:
		tmp = math.pow(n, -1.0) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4e-164)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.54)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64((n ^ -1.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4e-164)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.54)
		tmp = -log(x) / n;
	else
		tmp = (n ^ -1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4e-164], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.54], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.99999999999999985e-164

   1. Initial program 55.1%

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

   2. Taylor expanded in x around 0 55.1%

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

   if 3.99999999999999985e-164 < x < 0.54000000000000004

   1. Initial program 27.5%

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

   2. Taylor expanded in n around inf 64.8%

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

      1. log1p-def 64.8%

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

   4. Simplified 64.8%

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

   5. Taylor expanded in x around 0 60.2%

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

      1. mul-1-neg 60.2%

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

   7. Simplified 60.2%

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

   if 0.54000000000000004 < x

   1. Initial program 69.6%

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

   2. Taylor expanded in n around inf 69.5%

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

      1. log1p-def 69.5%

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

   4. Simplified 69.5%

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

   5. Taylor expanded in x around inf 58.0%

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

      1. div-inv 58.0%

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

      2. add-sqr-sqrt 30.5%

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

      3. associate-*r* 30.5%

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

      4. inv-pow 30.5%

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

      5. metadata-eval 30.5%

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

      6. sqrt-pow1 30.5%

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

      7. metadata-eval 30.5%

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

      8. metadata-eval 30.5%

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

      9. inv-pow 30.5%

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

      10. metadata-eval 30.5%

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

      11. sqrt-pow1 30.5%

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

      12. metadata-eval 30.5%

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

      13. metadata-eval 30.5%

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

   7. Applied egg-rr 30.5%

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

      1. associate-*l/ 30.5%

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

      2. associate-*l/ 30.5%

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

      3. *-lft-identity 30.5%

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

      4. pow-sqr 58.0%

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

      5. metadata-eval 58.0%

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

   9. Simplified 58.0%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-164}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.54:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \]

Alternative 8: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.54:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.54) (/ (- (log x)) n) (* (/ 1.0 n) (/ 1.0 x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.54) {
		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 <= 0.54d0) 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 <= 0.54) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.54:
		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 <= 0.54)
		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 <= 0.54)
		tmp = -log(x) / n;
	else
		tmp = (1.0 / n) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.54], 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 < 0.54000000000000004

   1. Initial program 40.3%

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

   2. Taylor expanded in n around inf 54.6%

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

      1. log1p-def 54.6%

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

   4. Simplified 54.6%

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

   5. Taylor expanded in x around 0 52.2%

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

      1. mul-1-neg 52.2%

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

   7. Simplified 52.2%

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

   if 0.54000000000000004 < x

   1. Initial program 69.6%

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

   2. Taylor expanded in n around inf 69.5%

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

      1. log1p-def 69.5%

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

   4. Simplified 69.5%

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

   5. Taylor expanded in x around inf 56.5%

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

      1. *-commutative 56.5%

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

   7. Simplified 56.5%

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

      1. associate-/r* 58.0%

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

      2. div-inv 58.0%

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

   9. Applied egg-rr 58.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.54:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 9: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55) (/ (- (log x)) n) (/ (pow n -1.0) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -log(x) / n;
	} else {
		tmp = pow(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 <= 0.55d0) then
        tmp = -log(x) / n
    else
        tmp = (n ** (-1.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = Math.pow(n, -1.0) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -math.log(x) / n
	else:
		tmp = math.pow(n, -1.0) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64((n ^ -1.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -log(x) / n;
	else
		tmp = (n ^ -1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.55000000000000004

   1. Initial program 40.3%

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

   2. Taylor expanded in n around inf 54.6%

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

      1. log1p-def 54.6%

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

   4. Simplified 54.6%

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

   5. Taylor expanded in x around 0 52.2%

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

      1. mul-1-neg 52.2%

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

   7. Simplified 52.2%

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

   if 0.55000000000000004 < x

   1. Initial program 69.6%

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

   2. Taylor expanded in n around inf 69.5%

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

      1. log1p-def 69.5%

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

   4. Simplified 69.5%

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

   5. Taylor expanded in x around inf 58.0%

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

      1. div-inv 58.0%

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

      2. add-sqr-sqrt 30.5%

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

      3. associate-*r* 30.5%

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

      4. inv-pow 30.5%

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

      5. metadata-eval 30.5%

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

      6. sqrt-pow1 30.5%

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

      7. metadata-eval 30.5%

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

      8. metadata-eval 30.5%

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

      9. inv-pow 30.5%

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

      10. metadata-eval 30.5%

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

      11. sqrt-pow1 30.5%

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

      12. metadata-eval 30.5%

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

      13. metadata-eval 30.5%

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

   7. Applied egg-rr 30.5%

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

      1. associate-*l/ 30.5%

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

      2. associate-*l/ 30.5%

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

      3. *-lft-identity 30.5%

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

      4. pow-sqr 58.0%

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

      5. metadata-eval 58.0%

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

   9. Simplified 58.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \]

Alternative 10: 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 54.0%

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

2. Taylor expanded in n around inf 61.6%

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

   1. log1p-def 61.6%

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

4. Simplified 61.6%

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

5. Taylor expanded in x around inf 39.4%

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

   1. *-commutative 39.4%

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

7. Simplified 39.4%

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

   1. associate-/r* 40.1%

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

   2. div-inv 40.1%

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

9. Applied egg-rr 40.1%

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

10. Final simplification 40.1%

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

Alternative 11: 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 54.0%

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

2. Taylor expanded in n around inf 61.6%

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

   1. log1p-def 61.6%

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

4. Simplified 61.6%

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

5. Taylor expanded in x around inf 39.4%

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

   1. *-commutative 39.4%

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

7. Simplified 39.4%

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

8. Final simplification 39.4%

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

Alternative 12: 41.1% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in n around inf 61.6%

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

   1. log1p-def 61.6%

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

4. Simplified 61.6%

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

5. Taylor expanded in x around inf 40.1%

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

6. Final simplification 40.1%

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))