2nthrt (problem 3.4.6)

Percentage Accurate: 53.3% → 85.9%

Time: 26.5s

Alternatives: 14

Speedup: 1.8×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-16)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-69)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 5e-21)
       (/ (/ 1.0 x) n)
       (exp (log (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = exp(log((exp((log1p(x) / n)) - pow(x, (1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = Math.exp(Math.log((Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n)))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	else:
		tmp = math.exp(math.log((math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n)))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = exp(log(Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)))));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[Exp[N[Log[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 94.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-def 86.1%

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

   5. Simplified 86.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

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

   3. Taylor expanded in n around inf 25.6%

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

      1. log1p-def 25.6%

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

   5. Simplified 25.6%

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

   6. Taylor expanded in x around inf 80.7%

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

   if 4.99999999999999973e-21 < (/.f64 1 n)

   1. Initial program 58.4%

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

      1. add-exp-log 58.5%

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

      2. add-exp-log 58.5%

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

      3. log-pow 58.5%

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

      4. +-commutative 58.5%

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

      5. log1p-udef 95.4%

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

      6. *-commutative 95.4%

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

      7. un-div-inv 95.4%

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

   4. Applied egg-rr 95.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-16)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-69)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 5e-21)
       (/ (/ 1.0 x) n)
       (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 94.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-def 86.1%

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

   5. Simplified 86.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

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

   3. Taylor expanded in n around inf 25.6%

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

      1. log1p-def 25.6%

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

   5. Simplified 25.6%

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

   6. Taylor expanded in x around inf 80.7%

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

   if 4.99999999999999973e-21 < (/.f64 1 n)

   1. Initial program 58.4%

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

   3. Taylor expanded in n around 0 58.4%

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

      1. log1p-def 95.4%

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

   5. Simplified 95.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-16)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-69)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 5e-21)
       (/ (/ 1.0 x) n)
       (- (exp (/ x n)) (pow x (/ 1.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 94.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-def 86.1%

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

   5. Simplified 86.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

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

   3. Taylor expanded in n around inf 25.6%

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

      1. log1p-def 25.6%

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

   5. Simplified 25.6%

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

   6. Taylor expanded in x around inf 80.7%

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

   if 4.99999999999999973e-21 < (/.f64 1 n)

   1. Initial program 58.4%

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

   3. Taylor expanded in n around 0 58.4%

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

      1. log1p-def 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in x around 0 95.3%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e+214)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) -5e+29)
       (- 1.0 t_0)
       (if (<= (/ 1.0 n) 2e-69)
         (/ (- (log1p x) (log x)) n)
         (if (<= (/ 1.0 n) 5e-21)
           (/ (/ 1.0 x) n)
           (if (<= (/ 1.0 n) 5e+193)
             (- (+ 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+214) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= -5e+29) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		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+214) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= -5e+29) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		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+214:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= -5e+29:
		tmp = 1.0 - t_0
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e+193:
		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+214)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= -5e+29)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e+193)
		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+214], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+29], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+193], 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 6 regimes

2. if (/.f64 1 n) < -4.99999999999999953e214

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 71.2%

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

      1. log1p-def 71.2%

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

   5. Simplified 71.2%

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

      1. log1p-udef 71.2%

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

      2. diff-log 71.2%

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

      3. +-commutative 71.2%

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

   7. Applied egg-rr 71.2%

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

   if -4.99999999999999953e214 < (/.f64 1 n) < -5.0000000000000001e29

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 67.7%

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

   if -5.0000000000000001e29 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 37.0%

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

   3. Taylor expanded in n around inf 81.1%

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

      1. log1p-def 81.1%

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

   5. Simplified 81.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

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

   3. Taylor expanded in n around inf 25.6%

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

      1. log1p-def 25.6%

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

   5. Simplified 25.6%

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

   6. Taylor expanded in x around inf 80.7%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 76.7%

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

   3. Taylor expanded in x around 0 70.9%

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

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

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

   3. Taylor expanded in n around inf 6.3%

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

      1. log1p-def 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in x around inf 74.6%

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

      1. *-commutative 74.6%

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

   8. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-16)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-69)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 5e-21)
       (/ (/ 1.0 x) n)
       (if (<= (/ 1.0 n) 5e+193)
         (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
         (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-16) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-69) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-16:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-69:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e+193:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-16)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e+193)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-16], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+193], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 94.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -2e-16 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 33.5%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-def 86.1%

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

   5. Simplified 86.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

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

   3. Taylor expanded in n around inf 25.6%

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

      1. log1p-def 25.6%

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

   5. Simplified 25.6%

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

   6. Taylor expanded in x around inf 80.7%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 76.7%

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

   3. Taylor expanded in x around 0 70.9%

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

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

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

   3. Taylor expanded in n around inf 6.3%

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

      1. log1p-def 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in x around inf 74.6%

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

      1. *-commutative 74.6%

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

   8. Simplified 74.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;1 + \sqrt[3]{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -5e+214)
     (+ 1.0 (cbrt -1.0))
     (if (<= (/ 1.0 n) -5000.0)
       t_1
       (if (<= (/ 1.0 n) -2e-134)
         (/ (- x (log x)) n)
         (if (<= (/ 1.0 n) -5e-251)
           t_0
           (if (<= (/ 1.0 n) 2e-69)
             (/ (- (log x)) n)
             (if (<= (/ 1.0 n) 5e-21)
               t_0
               (if (<= (/ 1.0 n) 5e+193) t_1 (/ 1.0 (* n x)))))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e+214) {
		tmp = 1.0 + cbrt(-1.0);
	} else if ((1.0 / n) <= -5000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e-134) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= -5e-251) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e+214) {
		tmp = 1.0 + Math.cbrt(-1.0);
	} else if ((1.0 / n) <= -5000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e-134) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= -5e-251) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+214)
		tmp = Float64(1.0 + cbrt(-1.0));
	elseif (Float64(1.0 / n) <= -5000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -2e-134)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= -5e-251)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+193)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+214], N[(1.0 + N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-134], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-251], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+193], t$95$1, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -4.99999999999999953e214

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. add-cube-cbrt 100.0%

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

      4. fma-def 100.0%

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

      5. add-exp-log 100.0%

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

      6. log-pow 100.0%

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

      7. +-commutative 100.0%

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

      8. log1p-udef 100.0%

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

      9. *-commutative 100.0%

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

      10. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}\right)\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. expm1-udef 100.0%

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

      3. cbrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. pow-sqr 100.0%

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

      6. div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{x}^{\left(\frac{2}{n}\right)}}\right)\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. expm1-log1p 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in n around inf 69.2%

      \[\leadsto \color{blue}{1 + \sqrt[3]{-1}} \]

   if -4.99999999999999953e214 < (/.f64 1 n) < -5e3 or 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0 66.7%

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

   if -5e3 < (/.f64 1 n) < -2.00000000000000008e-134

   1. Initial program 14.3%

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

   3. Taylor expanded in n around 0 14.3%

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

      1. log1p-def 14.3%

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

   5. Simplified 14.3%

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

   6. Taylor expanded in x around 0 4.0%

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

   7. Taylor expanded in n around inf 59.7%

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

   if -2.00000000000000008e-134 < (/.f64 1 n) < -5.0000000000000003e-251 or 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 65.8%

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

      1. log1p-def 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf 73.1%

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

   if -5.0000000000000003e-251 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 30.3%

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

   3. Taylor expanded in x around 0 30.3%

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

   4. Taylor expanded in n around inf 64.7%

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

      1. associate-*r/ 64.7%

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

      2. mul-1-neg 64.7%

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

   6. Simplified 64.7%

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

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

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

   3. Taylor expanded in n around inf 6.3%

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

      1. log1p-def 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in x around inf 74.6%

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

      1. *-commutative 74.6%

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

   8. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;1 + \sqrt[3]{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq -5000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e+214)
     t_1
     (if (<= (/ 1.0 n) -5e+29)
       (- 1.0 t_0)
       (if (<= (/ 1.0 n) 2e-69)
         t_1
         (if (<= (/ 1.0 n) 5e-21)
           (/ (/ 1.0 x) n)
           (if (<= (/ 1.0 n) 5e+193)
             (- (+ 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 t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+214) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e+29) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		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) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d+214)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d+29)) then
        tmp = 1.0d0 - t_0
    else if ((1.0d0 / n) <= 2d-69) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-21) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d+193) 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 t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+214) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e+29) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		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))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e+214:
		tmp = t_1
	elif (1.0 / n) <= -5e+29:
		tmp = 1.0 - t_0
	elif (1.0 / n) <= 2e-69:
		tmp = t_1
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e+193:
		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)
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+214)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e+29)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e+193)
		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);
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e+214)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e+29)
		tmp = 1.0 - t_0;
	elseif ((1.0 / n) <= 2e-69)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-21)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e+193)
		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]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+214], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+29], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+193], 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 5 regimes

2. if (/.f64 1 n) < -4.99999999999999953e214 or -5.0000000000000001e29 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 43.5%

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

   3. Taylor expanded in n around inf 80.1%

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

      1. log1p-def 80.1%

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

   5. Simplified 80.1%

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

      1. log1p-udef 80.1%

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

      2. diff-log 80.1%

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

      3. +-commutative 80.1%

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

   7. Applied egg-rr 80.1%

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

   if -4.99999999999999953e214 < (/.f64 1 n) < -5.0000000000000001e29

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 67.7%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

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

   3. Taylor expanded in n around inf 25.6%

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

      1. log1p-def 25.6%

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

   5. Simplified 25.6%

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

   6. Taylor expanded in x around inf 80.7%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 76.7%

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

   3. Taylor expanded in x around 0 70.9%

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

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

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

   3. Taylor expanded in n around inf 6.3%

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

      1. log1p-def 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in x around inf 74.6%

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

      1. *-commutative 74.6%

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

   8. Simplified 74.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e+214)
     t_1
     (if (<= (/ 1.0 n) -5e+29)
       t_0
       (if (<= (/ 1.0 n) 2e-69)
         t_1
         (if (<= (/ 1.0 n) 5e-21)
           (/ (/ 1.0 x) n)
           (if (<= (/ 1.0 n) 5e+193) t_0 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+214) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e+29) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d+214)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d+29)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-69) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-21) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d+193) then
        tmp = 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 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+214) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e+29) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-69) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e+193) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e+214:
		tmp = t_1
	elif (1.0 / n) <= -5e+29:
		tmp = t_0
	elif (1.0 / n) <= 2e-69:
		tmp = t_1
	elif (1.0 / n) <= 5e-21:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e+193:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+214)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e+29)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-69)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e+193)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e+214)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e+29)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-69)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-21)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e+193)
		tmp = t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+214], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+29], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-69], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+193], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.99999999999999953e214 or -5.0000000000000001e29 < (/.f64 1 n) < 1.9999999999999999e-69

   1. Initial program 43.5%

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

   3. Taylor expanded in n around inf 80.1%

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

      1. log1p-def 80.1%

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

   5. Simplified 80.1%

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

      1. log1p-udef 80.1%

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

      2. diff-log 80.1%

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

      3. +-commutative 80.1%

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

   7. Applied egg-rr 80.1%

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

   if -4.99999999999999953e214 < (/.f64 1 n) < -5.0000000000000001e29 or 4.99999999999999973e-21 < (/.f64 1 n) < 4.99999999999999972e193

   1. Initial program 89.6%

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

   3. Taylor expanded in x around 0 69.1%

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

   if 1.9999999999999999e-69 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 6.4%

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

   3. Taylor expanded in n around inf 25.6%

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

      1. log1p-def 25.6%

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

   5. Simplified 25.6%

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

   6. Taylor expanded in x around inf 80.7%

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

   if 4.99999999999999972e193 < (/.f64 1 n)

   1. Initial program 17.1%

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

   3. Taylor expanded in n around inf 6.3%

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

      1. log1p-def 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in x around inf 74.6%

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

      1. *-commutative 74.6%

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

   8. Simplified 74.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+29}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+193}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+59} \lor \neg \left(x \leq 1.25 \cdot 10^{+181}\right):\\ \;\;\;\;1 + \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (- (log x)) n)
   (if (or (<= x 2e+59) (not (<= x 1.25e+181)))
     (+ 1.0 (cbrt -1.0))
     (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = -log(x) / n;
	} else if ((x <= 2e+59) || !(x <= 1.25e+181)) {
		tmp = 1.0 + cbrt(-1.0);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = -Math.log(x) / n;
	} else if ((x <= 2e+59) || !(x <= 1.25e+181)) {
		tmp = 1.0 + Math.cbrt(-1.0);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(-log(x)) / n);
	elseif ((x <= 2e+59) || !(x <= 1.25e+181))
		tmp = Float64(1.0 + cbrt(-1.0));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[Or[LessEqual[x, 2e+59], N[Not[LessEqual[x, 1.25e+181]], $MachinePrecision]], N[(1.0 + N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 44.6%

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

   3. Taylor expanded in x around 0 42.7%

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

   4. Taylor expanded in n around inf 47.8%

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

      1. associate-*r/ 47.8%

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

      2. mul-1-neg 47.8%

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

   6. Simplified 47.8%

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

   if 1 < x < 1.99999999999999994e59 or 1.2500000000000001e181 < x

   1. Initial program 81.8%

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

      1. sub-neg 81.8%

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

      2. +-commutative 81.8%

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

      3. add-cube-cbrt 81.7%

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

      4. fma-def 81.7%

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

      5. add-exp-log 81.7%

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

      6. log-pow 81.7%

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

      7. +-commutative 81.7%

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

      8. log1p-udef 81.7%

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

      9. *-commutative 81.7%

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

      10. un-div-inv 81.7%

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

   4. Applied egg-rr 81.7%

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

      1. expm1-log1p-u 81.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}\right)\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. expm1-udef 81.7%

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

      3. cbrt-unprod 81.7%

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

      4. sqr-neg 81.7%

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

      5. pow-sqr 81.7%

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

      6. div-inv 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. expm1-def 81.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{x}^{\left(\frac{2}{n}\right)}}\right)\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. expm1-log1p 81.7%

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

   8. Simplified 81.7%

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

   9. Taylor expanded in n around inf 81.8%

      \[\leadsto \color{blue}{1 + \sqrt[3]{-1}} \]

   if 1.99999999999999994e59 < x < 1.2500000000000001e181

   1. Initial program 50.3%

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

   3. Taylor expanded in n around inf 50.3%

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

      1. log1p-def 50.3%

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

   5. Simplified 50.3%

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

   6. Taylor expanded in x around inf 74.5%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+59} \lor \neg \left(x \leq 1.25 \cdot 10^{+181}\right):\\ \;\;\;\;1 + \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+57} \lor \neg \left(x \leq 1.9 \cdot 10^{+183}\right):\\ \;\;\;\;1 + \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (- x (log x)) n)
   (if (or (<= x 7.5e+57) (not (<= x 1.9e+183)))
     (+ 1.0 (cbrt -1.0))
     (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if ((x <= 7.5e+57) || !(x <= 1.9e+183)) {
		tmp = 1.0 + cbrt(-1.0);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if ((x <= 7.5e+57) || !(x <= 1.9e+183)) {
		tmp = 1.0 + Math.cbrt(-1.0);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif ((x <= 7.5e+57) || !(x <= 1.9e+183))
		tmp = Float64(1.0 + cbrt(-1.0));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[Or[LessEqual[x, 7.5e+57], N[Not[LessEqual[x, 1.9e+183]], $MachinePrecision]], N[(1.0 + N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 44.6%

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

   3. Taylor expanded in n around 0 44.6%

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

      1. log1p-def 56.2%

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0 56.2%

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

   7. Taylor expanded in n around inf 48.2%

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

   if 1 < x < 7.5000000000000006e57 or 1.9e183 < x

   1. Initial program 81.8%

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

      1. sub-neg 81.8%

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

      2. +-commutative 81.8%

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

      3. add-cube-cbrt 81.7%

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

      4. fma-def 81.7%

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

      5. add-exp-log 81.7%

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

      6. log-pow 81.7%

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

      7. +-commutative 81.7%

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

      8. log1p-udef 81.7%

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

      9. *-commutative 81.7%

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

      10. un-div-inv 81.7%

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

   4. Applied egg-rr 81.7%

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

      1. expm1-log1p-u 81.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}\right)\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. expm1-udef 81.7%

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

      3. cbrt-unprod 81.7%

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

      4. sqr-neg 81.7%

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

      5. pow-sqr 81.7%

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

      6. div-inv 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. expm1-def 81.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{x}^{\left(\frac{2}{n}\right)}}\right)\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. expm1-log1p 81.7%

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

   8. Simplified 81.7%

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

   9. Taylor expanded in n around inf 81.8%

      \[\leadsto \color{blue}{1 + \sqrt[3]{-1}} \]

   if 7.5000000000000006e57 < x < 1.9e183

   1. Initial program 50.3%

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

   3. Taylor expanded in n around inf 50.3%

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

      1. log1p-def 50.3%

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

   5. Simplified 50.3%

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

   6. Taylor expanded in x around inf 74.5%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+57} \lor \neg \left(x \leq 1.9 \cdot 10^{+183}\right):\\ \;\;\;\;1 + \sqrt[3]{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-9} \lor \neg \left(n \leq -4.5 \cdot 10^{-308}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt[3]{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -3.3e-9) (not (<= n -4.5e-308)))
   (/ (/ 1.0 x) n)
   (+ 1.0 (cbrt -1.0))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -3.3e-9) || !(n <= -4.5e-308)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 1.0 + cbrt(-1.0);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -3.3e-9) || !(n <= -4.5e-308)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 1.0 + Math.cbrt(-1.0);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -3.3e-9) || !(n <= -4.5e-308))
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(1.0 + cbrt(-1.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -3.3e-9], N[Not[LessEqual[n, -4.5e-308]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(1.0 + N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.30000000000000018e-9 or -4.50000000000000009e-308 < n

   1. Initial program 38.6%

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

   3. Taylor expanded in n around inf 60.3%

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

      1. log1p-def 60.3%

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

   5. Simplified 60.3%

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

   6. Taylor expanded in x around inf 43.5%

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

   if -3.30000000000000018e-9 < n < -4.50000000000000009e-308

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. add-cube-cbrt 100.0%

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

      4. fma-def 100.0%

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

      5. add-exp-log 100.0%

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

      6. log-pow 100.0%

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

      7. +-commutative 100.0%

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

      8. log1p-udef 100.0%

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

      9. *-commutative 100.0%

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

      10. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}\right)\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. expm1-udef 100.0%

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

      3. cbrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. pow-sqr 100.0%

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

      6. div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{x}^{\left(\frac{2}{n}\right)}}\right)\right)}, \sqrt[3]{-{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. expm1-log1p 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in n around inf 47.0%

      \[\leadsto \color{blue}{1 + \sqrt[3]{-1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{-9} \lor \neg \left(n \leq -4.5 \cdot 10^{-308}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt[3]{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.9% 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 54.2%

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

3. Taylor expanded in n around inf 56.2%

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

   1. log1p-def 56.2%

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

5. Simplified 56.2%

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

6. Taylor expanded in x around inf 38.5%

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

   1. *-commutative 38.5%

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

8. Simplified 38.5%

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

9. Final simplification 38.5%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Alternative 13: 40.5% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in n around inf 56.2%

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

   1. log1p-def 56.2%

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

5. Simplified 56.2%

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

6. Taylor expanded in x around inf 39.2%

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

7. Final simplification 39.2%

   \[\leadsto \frac{\frac{1}{x}}{n} \]
8. Add Preprocessing

Alternative 14: 4.1% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 2.0)

C

double code(double x, double n) {
	return 2.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 2.0d0
end function

Java

public static double code(double x, double n) {
	return 2.0;
}

Python

def code(x, n):
	return 2.0

Julia

function code(x, n)
	return 2.0
end

MATLAB

function tmp = code(x, n)
	tmp = 2.0;
end

Wolfram

code[x_, n_] := 2.0

Derivation

1. Initial program 54.2%

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

   1. add-sqr-sqrt 39.7%

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

   2. pow2 39.7%

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

   3. add-exp-log 39.7%

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

   4. log-pow 39.7%

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

   5. +-commutative 39.7%

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

   6. log1p-udef 46.8%

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

   7. *-commutative 46.8%

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

   8. un-div-inv 46.8%

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

4. Applied egg-rr 46.8%

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

   1. add-cbrt-cube 46.8%

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

   2. add-sqr-sqrt 46.8%

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

   3. pow1 46.8%

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

   4. pow1/2 46.8%

      \[\leadsto {\left(\sqrt[3]{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{1} \cdot \color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{0.5}}}\right)}^{2} \]

   5. pow-prod-up 46.8%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{\left(1 + 0.5\right)}}}\right)}^{2} \]

6. Applied egg-rr 30.9%

   \[\leadsto {\color{blue}{\left(\sqrt[3]{{\left({x}^{\left(\frac{1}{n}\right)} + e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{1.5}}\right)}}^{2} \]

7. Taylor expanded in n around inf 4.5%

   \[\leadsto \color{blue}{{\left(\sqrt{2}\right)}^{2}} \]
8. Step-by-step derivation

   1. unpow2 4.5%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{2}} \]

   2. rem-square-sqrt 4.5%

      \[\leadsto \color{blue}{2} \]

9. Simplified 4.5%

   \[\leadsto \color{blue}{2} \]

10. Final simplification 4.5%

    \[\leadsto 2 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))