2nthrt (problem 3.4.6)

Percentage Accurate: 53.0% → 97.8%

Time: 39.6s

Alternatives: 18

Speedup: 1.9×

• 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: 53.0% 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: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-10)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2000000.0)
       (/ (log1p (/ 1.0 x)) n)
       (pow (cbrt (- (exp (/ (log1p x) n)) t_0)) 3.0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2000000.0) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = pow(cbrt((exp((log1p(x) / n)) - t_0)), 3.0);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2000000.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.pow(Math.cbrt((Math.exp((Math.log1p(x) / n)) - t_0)), 3.0);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-10)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2000000.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = cbrt(Float64(exp(Float64(log1p(x) / n)) - t_0)) ^ 3.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-10], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[Power[N[Power[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000031e-10

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. log-rec 98.3%

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

      3. mul-1-neg 98.3%

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

      4. distribute-neg-frac 98.3%

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

      5. mul-1-neg 98.3%

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

      6. remove-double-neg 98.3%

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

      7. *-rgt-identity 98.3%

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

      8. associate-/l* 98.3%

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

      9. exp-to-pow 98.4%

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

      10. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   if -5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 2e6

   1. Initial program 28.4%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. log1p-define 80.0%

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

   5. Simplified 80.0%

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

      1. log1p-expm1-u 80.0%

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

      2. expm1-undefine 80.0%

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

      3. exp-diff 80.0%

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

      4. add-exp-log 59.1%

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

      5. log1p-undefine 59.1%

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

      6. rem-exp-log 80.2%

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

   7. Applied egg-rr 80.2%

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

   8. Taylor expanded in x around 0 97.8%

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

   if 2e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 59.8%

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

      1. add-cube-cbrt 59.8%

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

      2. pow3 59.8%

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

      3. pow-to-exp 59.8%

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

      4. un-div-inv 59.8%

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

      5. +-commutative 59.8%

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

      6. log1p-define 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-10)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2000000.0)
       (/ (log1p (/ 1.0 x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2000000.0) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2000000.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-10:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2000000.0:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-10)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2000000.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-10], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000031e-10

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. log-rec 98.3%

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

      3. mul-1-neg 98.3%

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

      4. distribute-neg-frac 98.3%

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

      5. mul-1-neg 98.3%

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

      6. remove-double-neg 98.3%

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

      7. *-rgt-identity 98.3%

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

      8. associate-/l* 98.3%

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

      9. exp-to-pow 98.4%

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

      10. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   if -5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 2e6

   1. Initial program 28.4%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. log1p-define 80.0%

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

   5. Simplified 80.0%

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

      1. log1p-expm1-u 80.0%

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

      2. expm1-undefine 80.0%

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

      3. exp-diff 80.0%

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

      4. add-exp-log 59.1%

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

      5. log1p-undefine 59.1%

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

      6. rem-exp-log 80.2%

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

   7. Applied egg-rr 80.2%

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

   8. Taylor expanded in x around 0 97.8%

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

   if 2e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 59.8%

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

   3. Taylor expanded in n around 0 59.8%

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

      1. log1p-define 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. associate-/l* 99.8%

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

      4. exp-to-pow 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \frac{1 + \left(x \cdot -0.5 + 0.5 \cdot \frac{x}{n}\right)}{n}\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-10)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2000000.0)
       (/ (log1p (/ 1.0 x)) n)
       (- (+ 1.0 (* x (/ (+ 1.0 (+ (* x -0.5) (* 0.5 (/ x n)))) n))) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2000000.0) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 + ((x * -0.5) + (0.5 * (x / n)))) / n))) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2000000.0) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 + ((x * -0.5) + (0.5 * (x / n)))) / n))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-10:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2000000.0:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = (1.0 + (x * ((1.0 + ((x * -0.5) + (0.5 * (x / n)))) / n))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-10)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2000000.0)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 + Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n)))) / n))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-10], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000.0], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 + N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000031e-10

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. log-rec 98.3%

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

      3. mul-1-neg 98.3%

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

      4. distribute-neg-frac 98.3%

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

      5. mul-1-neg 98.3%

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

      6. remove-double-neg 98.3%

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

      7. *-rgt-identity 98.3%

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

      8. associate-/l* 98.3%

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

      9. exp-to-pow 98.4%

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

      10. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   if -5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 2e6

   1. Initial program 28.4%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. log1p-define 80.0%

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

   5. Simplified 80.0%

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

      1. log1p-expm1-u 80.0%

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

      2. expm1-undefine 80.0%

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

      3. exp-diff 80.0%

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

      4. add-exp-log 59.1%

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

      5. log1p-undefine 59.1%

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

      6. rem-exp-log 80.2%

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

   7. Applied egg-rr 80.2%

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

   8. Taylor expanded in x around 0 97.8%

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

   if 2e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 59.8%

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

   3. Taylor expanded in x around 0 71.4%

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

   4. Taylor expanded in n around inf 79.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \frac{1 + \left(x \cdot -0.5 + 0.5 \cdot \frac{x}{n}\right)}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+227}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -0.4:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+227)
     t_0
     (if (<= (/ 1.0 n) -0.4)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 5e+17)
         (/ (log1p (/ 1.0 x)) n)
         (if (<= (/ 1.0 n) 2e+178)
           t_0
           (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* n x))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+227) {
		tmp = t_0;
	} else if ((1.0 / n) <= -0.4) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+17) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = t_0;
	} else {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+227) {
		tmp = t_0;
	} else if ((1.0 / n) <= -0.4) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+17) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = t_0;
	} else {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+227:
		tmp = t_0
	elif (1.0 / n) <= -0.4:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+17:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+178:
		tmp = t_0
	else:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+227)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -0.4)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+17)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+178)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+227], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.4], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+17], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+178], t$95$0, N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000002e227 or 5e17 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e178

   1. Initial program 90.4%

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

   3. Taylor expanded in x around 0 76.2%

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

      1. *-rgt-identity 76.2%

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

      2. associate-/l* 76.2%

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

      3. exp-to-pow 76.2%

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

   5. Simplified 76.2%

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

   if -2.0000000000000002e227 < (/.f64 #s(literal 1 binary64) n) < -0.40000000000000002

   1. Initial program 98.0%

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

   3. Taylor expanded in n around inf 63.9%

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

      1. log1p-define 63.9%

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

   5. Simplified 63.9%

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

      1. log1p-undefine 63.9%

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

      2. diff-log 63.9%

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

   7. Applied egg-rr 63.9%

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

      1. +-commutative 63.9%

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

   9. Simplified 63.9%

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

   if -0.40000000000000002 < (/.f64 #s(literal 1 binary64) n) < 5e17

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf 77.8%

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

      1. log1p-define 77.8%

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

   5. Simplified 77.8%

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

      1. log1p-expm1-u 77.8%

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

      2. expm1-undefine 77.8%

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

      3. exp-diff 77.8%

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

      4. add-exp-log 57.5%

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

      5. log1p-undefine 57.5%

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

      6. rem-exp-log 77.9%

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

   7. Applied egg-rr 77.9%

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

   8. Taylor expanded in x around 0 95.5%

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

   if 2.0000000000000001e178 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

      1. log1p-expm1-u 6.6%

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

      2. expm1-undefine 6.6%

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

      3. exp-diff 6.6%

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

      4. add-exp-log 6.6%

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

      5. log1p-undefine 6.6%

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

      6. rem-exp-log 6.6%

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

   7. Applied egg-rr 6.6%

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

   8. Taylor expanded in x around inf 11.5%

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

   10. Simplified 68.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+227}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -0.4:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+227}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -0.4:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+227)
     t_0
     (if (<= (/ 1.0 n) -0.4)
       0.0
       (if (<= (/ 1.0 n) 5e+17)
         (/ (log1p (/ 1.0 x)) n)
         (if (<= (/ 1.0 n) 2e+178)
           t_0
           (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* n x))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+227) {
		tmp = t_0;
	} else if ((1.0 / n) <= -0.4) {
		tmp = 0.0;
	} else if ((1.0 / n) <= 5e+17) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = t_0;
	} else {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+227) {
		tmp = t_0;
	} else if ((1.0 / n) <= -0.4) {
		tmp = 0.0;
	} else if ((1.0 / n) <= 5e+17) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = t_0;
	} else {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+227:
		tmp = t_0
	elif (1.0 / n) <= -0.4:
		tmp = 0.0
	elif (1.0 / n) <= 5e+17:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+178:
		tmp = t_0
	else:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+227)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -0.4)
		tmp = 0.0;
	elseif (Float64(1.0 / n) <= 5e+17)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+178)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+227], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.4], 0.0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+17], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+178], t$95$0, N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000002e227 or 5e17 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e178

   1. Initial program 90.4%

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

   3. Taylor expanded in x around 0 76.2%

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

      1. *-rgt-identity 76.2%

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

      2. associate-/l* 76.2%

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

      3. exp-to-pow 76.2%

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

   5. Simplified 76.2%

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

   if -2.0000000000000002e227 < (/.f64 #s(literal 1 binary64) n) < -0.40000000000000002

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0 37.6%

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

   4. Taylor expanded in n around inf 63.0%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 63.0%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{0} \]

   if -0.40000000000000002 < (/.f64 #s(literal 1 binary64) n) < 5e17

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf 77.8%

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

      1. log1p-define 77.8%

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

   5. Simplified 77.8%

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

      1. log1p-expm1-u 77.8%

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

      2. expm1-undefine 77.8%

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

      3. exp-diff 77.8%

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

      4. add-exp-log 57.5%

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

      5. log1p-undefine 57.5%

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

      6. rem-exp-log 77.9%

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

   7. Applied egg-rr 77.9%

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

   8. Taylor expanded in x around 0 95.5%

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

   if 2.0000000000000001e178 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

      1. log1p-expm1-u 6.6%

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

      2. expm1-undefine 6.6%

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

      3. exp-diff 6.6%

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

      4. add-exp-log 6.6%

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

      5. log1p-undefine 6.6%

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

      6. rem-exp-log 6.6%

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

   7. Applied egg-rr 6.6%

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

   8. Taylor expanded in x around inf 11.5%

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

   10. Simplified 68.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+227}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -0.4:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\frac{1 + \frac{-1 - \frac{-1.1666666666666667}{x}}{x}}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-10)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e+17)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+195)
         (- (+ 1.0 (/ x n)) t_0)
         (/
          (expm1 (/ (+ 1.0 (/ (- -1.0 (/ -1.1666666666666667 x)) x)) x))
          n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+17) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+195) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = expm1(((1.0 + ((-1.0 - (-1.1666666666666667 / x)) / x)) / x)) / n;
	}
	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-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+17) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+195) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.expm1(((1.0 + ((-1.0 - (-1.1666666666666667 / x)) / x)) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-10:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e+17:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+195:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.expm1(((1.0 + ((-1.0 - (-1.1666666666666667 / x)) / x)) / x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-10)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e+17)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+195)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(expm1(Float64(Float64(1.0 + Float64(Float64(-1.0 - Float64(-1.1666666666666667 / x)) / x)) / x)) / n);
	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-10], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+17], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+195], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(Exp[N[(N[(1.0 + N[(N[(-1.0 - N[(-1.1666666666666667 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]] - 1), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000031e-10

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. log-rec 98.3%

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

      3. mul-1-neg 98.3%

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

      4. distribute-neg-frac 98.3%

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

      5. mul-1-neg 98.3%

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

      6. remove-double-neg 98.3%

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

      7. *-rgt-identity 98.3%

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

      8. associate-/l* 98.3%

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

      9. exp-to-pow 98.4%

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

      10. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   if -5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 5e17

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

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

      1. log1p-expm1-u 79.5%

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

      2. expm1-undefine 79.5%

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

      3. exp-diff 79.5%

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

      4. add-exp-log 58.8%

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

      5. log1p-undefine 58.8%

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

      6. rem-exp-log 79.7%

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

   7. Applied egg-rr 79.7%

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

   8. Taylor expanded in x around 0 97.2%

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

   if 5e17 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e195

   1. Initial program 78.9%

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

   3. Taylor expanded in x around 0 80.9%

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

   if 4.9999999999999998e195 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 6.8%

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

      1. log1p-define 6.8%

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

   5. Simplified 6.8%

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

      1. expm1-log1p-u 6.8%

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

   7. Applied egg-rr 6.8%

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

   8. Taylor expanded in x around -inf 75.8%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{1.1666666666666667 \cdot \frac{1}{x} - 1}{x} - 1}{x}}\right)}{n} \]

   9. Taylor expanded in x around 0 75.8%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{-1 \cdot \frac{1.1666666666666667 \cdot \frac{1}{x} - 1}{x} - 1}{x}} - 1}{n}} \]

   11. Simplified 75.8%

       \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\frac{1 - \frac{1 + \frac{-1.1666666666666667}{x}}{x}}{x}\right)}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\frac{1 + \frac{-1 - \frac{-1.1666666666666667}{x}}{x}}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-10)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e+17)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+195)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+17) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+195) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (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-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+17) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+195) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-10:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e+17:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+195:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (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-10)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e+17)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+195)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / 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-10], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+17], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+195], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000031e-10

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. log-rec 98.3%

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

      3. mul-1-neg 98.3%

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

      4. distribute-neg-frac 98.3%

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

      5. mul-1-neg 98.3%

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

      6. remove-double-neg 98.3%

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

      7. *-rgt-identity 98.3%

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

      8. associate-/l* 98.3%

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

      9. exp-to-pow 98.4%

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

      10. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   if -5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 5e17

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

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

      1. log1p-expm1-u 79.5%

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

      2. expm1-undefine 79.5%

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

      3. exp-diff 79.5%

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

      4. add-exp-log 58.8%

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

      5. log1p-undefine 58.8%

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

      6. rem-exp-log 79.7%

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

   7. Applied egg-rr 79.7%

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

   8. Taylor expanded in x around 0 97.2%

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

   if 5e17 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e195

   1. Initial program 78.9%

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

   3. Taylor expanded in x around 0 80.9%

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

   if 4.9999999999999998e195 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 6.8%

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

      1. log1p-define 6.8%

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

   5. Simplified 6.8%

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

      1. log1p-expm1-u 6.8%

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

      2. expm1-undefine 6.8%

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

      3. exp-diff 6.8%

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

      4. add-exp-log 6.8%

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

      5. log1p-undefine 6.8%

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

      6. rem-exp-log 6.8%

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

   7. Applied egg-rr 6.8%

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

   8. Taylor expanded in x around inf 6.7%

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

   10. Simplified 70.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-10)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e+17)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 2e+178)
         (- 1.0 t_0)
         (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+17) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (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-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+17) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-10:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e+17:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 2e+178:
		tmp = 1.0 - t_0
	else:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (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-10)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e+17)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+178)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / 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-10], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+17], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+178], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000031e-10

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. log-rec 98.3%

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

      3. mul-1-neg 98.3%

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

      4. distribute-neg-frac 98.3%

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

      5. mul-1-neg 98.3%

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

      6. remove-double-neg 98.3%

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

      7. *-rgt-identity 98.3%

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

      8. associate-/l* 98.3%

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

      9. exp-to-pow 98.4%

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

      10. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   if -5.00000000000000031e-10 < (/.f64 #s(literal 1 binary64) n) < 5e17

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

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

      1. log1p-expm1-u 79.5%

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

      2. expm1-undefine 79.5%

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

      3. exp-diff 79.5%

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

      4. add-exp-log 58.8%

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

      5. log1p-undefine 58.8%

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

      6. rem-exp-log 79.7%

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

   7. Applied egg-rr 79.7%

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

   8. Taylor expanded in x around 0 97.2%

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

   if 5e17 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e178

   1. Initial program 81.9%

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

   3. Taylor expanded in x around 0 81.9%

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

      1. *-rgt-identity 81.9%

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

      2. associate-/l* 81.9%

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

      3. exp-to-pow 81.9%

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

   5. Simplified 81.9%

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

   if 2.0000000000000001e178 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

      1. log1p-expm1-u 6.6%

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

      2. expm1-undefine 6.6%

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

      3. exp-diff 6.6%

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

      4. add-exp-log 6.6%

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

      5. log1p-undefine 6.6%

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

      6. rem-exp-log 6.6%

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

   7. Applied egg-rr 6.6%

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

   8. Taylor expanded in x around inf 11.5%

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

   10. Simplified 68.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+227}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -0.4:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* n x))))
   (if (<= (/ 1.0 n) -2e+227)
     t_0
     (if (<= (/ 1.0 n) -0.4)
       0.0
       (if (<= (/ 1.0 n) 2e+178) (/ (log1p (/ 1.0 x)) n) t_0)))))

C

double code(double x, double n) {
	double t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e+227) {
		tmp = t_0;
	} else if ((1.0 / n) <= -0.4) {
		tmp = 0.0;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e+227) {
		tmp = t_0;
	} else if ((1.0 / n) <= -0.4) {
		tmp = 0.0;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x)
	tmp = 0
	if (1.0 / n) <= -2e+227:
		tmp = t_0
	elif (1.0 / n) <= -0.4:
		tmp = 0.0
	elif (1.0 / n) <= 2e+178:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+227)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -0.4)
		tmp = 0.0;
	elseif (Float64(1.0 / n) <= 2e+178)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+227], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.4], 0.0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+178], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000002e227 or 2.0000000000000001e178 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 65.8%

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

   3. Taylor expanded in n around inf 20.8%

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

      1. log1p-define 20.8%

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

   5. Simplified 20.8%

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

      1. log1p-expm1-u 20.8%

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

      2. expm1-undefine 20.8%

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

      3. exp-diff 20.8%

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

      4. add-exp-log 6.3%

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

      5. log1p-undefine 6.3%

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

      6. rem-exp-log 20.8%

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

   7. Applied egg-rr 20.8%

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

   8. Taylor expanded in x around inf 9.7%

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

   10. Simplified 66.0%

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

   if -2.0000000000000002e227 < (/.f64 #s(literal 1 binary64) n) < -0.40000000000000002

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0 37.6%

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

   4. Taylor expanded in n around inf 63.0%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 63.0%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{0} \]

   if -0.40000000000000002 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e178

   1. Initial program 34.3%

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

   3. Taylor expanded in n around inf 70.5%

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

      1. log1p-define 70.5%

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

   5. Simplified 70.5%

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

      1. log1p-expm1-u 70.5%

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

      2. expm1-undefine 70.5%

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

      3. exp-diff 70.5%

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

      4. add-exp-log 52.3%

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

      5. log1p-undefine 52.3%

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

      6. rem-exp-log 70.7%

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

   7. Applied egg-rr 70.7%

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

   8. Taylor expanded in x around 0 86.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+227}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -0.4:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.7)
   (/ (log x) (- n))
   (if (<= x 2.5e+132)
     (/
      (/
       (+ 1.0 (/ (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5) x))
       x)
      n)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = log(x) / -n;
	} else if (x <= 2.5e+132) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = log(x) / -n
    else if (x <= 2.5d+132) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = Math.log(x) / -n;
	} else if (x <= 2.5e+132) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.7:
		tmp = math.log(x) / -n
	elif x <= 2.5e+132:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 2.5e+132)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = log(x) / -n;
	elseif (x <= 2.5e+132)
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.7], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 2.5e+132], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.69999999999999996

   1. Initial program 37.4%

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

   3. Taylor expanded in x around 0 36.1%

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

      1. *-rgt-identity 36.1%

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

      2. associate-/l* 36.1%

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

      3. exp-to-pow 36.1%

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

   5. Simplified 36.1%

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

   6. Taylor expanded in n around inf 58.1%

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

      1. associate-*r/ 58.1%

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

      2. neg-mul-1 58.1%

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

   8. Simplified 58.1%

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

   if 0.69999999999999996 < x < 2.5000000000000001e132

   1. Initial program 44.1%

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

   3. Taylor expanded in n around inf 48.6%

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

      1. log1p-define 48.6%

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

   5. Simplified 48.6%

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

   6. Taylor expanded in x around -inf 63.3%

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

   if 2.5000000000000001e132 < x

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0 47.3%

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

   4. Taylor expanded in n around inf 79.8%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 79.8%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.000116 \lor \neg \left(n \leq -6.5 \cdot 10^{-227}\right):\\ \;\;\;\;\frac{\frac{1}{n} - \frac{0.5 + \frac{-0.3333333333333333}{x}}{n \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -0.000116) (not (<= n -6.5e-227)))
   (/ (- (/ 1.0 n) (/ (+ 0.5 (/ -0.3333333333333333 x)) (* n x))) x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -0.000116) || !(n <= -6.5e-227)) {
		tmp = ((1.0 / n) - ((0.5 + (-0.3333333333333333 / x)) / (n * x))) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-0.000116d0)) .or. (.not. (n <= (-6.5d-227)))) then
        tmp = ((1.0d0 / n) - ((0.5d0 + ((-0.3333333333333333d0) / x)) / (n * x))) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -0.000116) || !(n <= -6.5e-227)) {
		tmp = ((1.0 / n) - ((0.5 + (-0.3333333333333333 / x)) / (n * x))) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -0.000116) or not (n <= -6.5e-227):
		tmp = ((1.0 / n) - ((0.5 + (-0.3333333333333333 / x)) / (n * x))) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -0.000116) || !(n <= -6.5e-227))
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / Float64(n * x))) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -0.000116) || ~((n <= -6.5e-227)))
		tmp = ((1.0 / n) - ((0.5 + (-0.3333333333333333 / x)) / (n * x))) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -0.000116], N[Not[LessEqual[n, -6.5e-227]], $MachinePrecision]], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.16e-4 or -6.4999999999999996e-227 < n

   1. Initial program 39.5%

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

   3. Taylor expanded in n around inf 62.0%

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

      1. log1p-define 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in x around -inf 44.1%

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

   7. Taylor expanded in n around 0 44.1%

      \[\leadsto -1 \cdot \frac{\color{blue}{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{n \cdot x}} - \frac{1}{n}}{x} \]
   8. Step-by-step derivation

      1. associate-*r/ 44.1%

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

      2. sub-neg 44.1%

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

      3. metadata-eval 44.1%

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

      4. distribute-lft-in 44.1%

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

      5. neg-mul-1 44.1%

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

      6. metadata-eval 44.1%

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

      7. +-commutative 44.1%

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

      8. sub-neg 44.1%

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

      9. *-commutative 44.1%

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

      10. sub-neg 44.1%

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

      11. associate-*r/ 44.1%

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

      12. metadata-eval 44.1%

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

      13. distribute-neg-frac 44.1%

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

      14. metadata-eval 44.1%

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

   9. Simplified 44.1%

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

   if -1.16e-4 < n < -6.4999999999999996e-227

   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 36.9%

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

   4. Taylor expanded in n around inf 65.7%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 65.7%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.000116 \lor \neg \left(n \leq -6.5 \cdot 10^{-227}\right):\\ \;\;\;\;\frac{\frac{1}{n} - \frac{0.5 + \frac{-0.3333333333333333}{x}}{n \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.4% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{-6} \lor \neg \left(n \leq -5.5 \cdot 10^{-226}\right):\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -6e-6) (not (<= n -5.5e-226)))
   (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) n) x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -6e-6) || !(n <= -5.5e-226)) {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-6d-6)) .or. (.not. (n <= (-5.5d-226)))) then
        tmp = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -6e-6) || !(n <= -5.5e-226)) {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -6e-6) or not (n <= -5.5e-226):
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -6e-6) || !(n <= -5.5e-226))
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -6e-6) || ~((n <= -5.5e-226)))
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -6e-6], N[Not[LessEqual[n, -5.5e-226]], $MachinePrecision]], N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -6.0000000000000002e-6 or -5.5e-226 < n

   1. Initial program 39.5%

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

   3. Taylor expanded in n around inf 62.0%

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

      1. log1p-define 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in x around -inf 44.1%

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

   7. Taylor expanded in n around 0 44.1%

      \[\leadsto -1 \cdot \frac{\color{blue}{\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{n}}}{x} \]
   8. Step-by-step derivation

      1. sub-neg 44.1%

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

      2. metadata-eval 44.1%

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

      3. +-commutative 44.1%

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

      4. associate-*r/ 44.1%

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

      5. sub-neg 44.1%

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

      6. metadata-eval 44.1%

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

      7. distribute-lft-in 44.1%

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

      8. neg-mul-1 44.1%

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

      9. metadata-eval 44.1%

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

      10. +-commutative 44.1%

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

      11. associate-*r/ 44.1%

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

      12. metadata-eval 44.1%

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

      13. distribute-neg-frac 44.1%

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

      14. metadata-eval 44.1%

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

   9. Simplified 44.1%

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

   if -6.0000000000000002e-6 < n < -5.5e-226

   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 36.9%

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

   4. Taylor expanded in n around inf 65.7%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 65.7%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{-6} \lor \neg \left(n \leq -5.5 \cdot 10^{-226}\right):\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.4% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{-6} \lor \neg \left(n \leq -4.6 \cdot 10^{-226}\right):\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -9.6e-6) (not (<= n -4.6e-226)))
   (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x) n)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -9.6e-6) || !(n <= -4.6e-226)) {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-9.6d-6)) .or. (.not. (n <= (-4.6d-226)))) then
        tmp = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -9.6e-6) || !(n <= -4.6e-226)) {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -9.6e-6) or not (n <= -4.6e-226):
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -9.6e-6) || !(n <= -4.6e-226))
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -9.6e-6) || ~((n <= -4.6e-226)))
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -9.6e-6], N[Not[LessEqual[n, -4.6e-226]], $MachinePrecision]], N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -9.5999999999999996e-6 or -4.6000000000000001e-226 < n

   1. Initial program 39.5%

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

   3. Taylor expanded in n around inf 62.0%

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

      1. log1p-define 62.0%

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

   5. Simplified 62.0%

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

      1. log1p-expm1-u 62.0%

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

      2. expm1-undefine 62.0%

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

      3. exp-diff 62.0%

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

      4. add-exp-log 44.5%

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

      5. log1p-undefine 44.5%

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

      6. rem-exp-log 62.1%

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

   7. Applied egg-rr 62.1%

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

   8. Taylor expanded in x around -inf 44.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
   9. Step-by-step derivation

      1. associate-*r/ 44.0%

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

      2. sub-neg 44.0%

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

      3. metadata-eval 44.0%

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

      4. distribute-lft-in 44.0%

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

      5. associate-*r/ 44.0%

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

      6. sub-neg 44.0%

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

      7. metadata-eval 44.0%

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

      8. distribute-lft-in 44.0%

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

      9. neg-mul-1 44.0%

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

      10. metadata-eval 44.0%

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

      11. +-commutative 44.0%

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

      12. sub-neg 44.0%

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

      13. metadata-eval 44.0%

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

      14. +-commutative 44.0%

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

   10. Simplified 44.0%

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

   if -9.5999999999999996e-6 < n < -4.6000000000000001e-226

   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 36.9%

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

   4. Taylor expanded in n around inf 65.7%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 65.7%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{-6} \lor \neg \left(n \leq -4.6 \cdot 10^{-226}\right):\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.2% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.048:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -5.5 \cdot 10^{-226}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -0.048)
   (/ (/ 1.0 x) n)
   (if (<= n -5.5e-226)
     0.0
     (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* n x)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -0.048) {
		tmp = (1.0 / x) / n;
	} else if (n <= -5.5e-226) {
		tmp = 0.0;
	} else {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-0.048d0)) then
        tmp = (1.0d0 / x) / n
    else if (n <= (-5.5d-226)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -0.048) {
		tmp = (1.0 / x) / n;
	} else if (n <= -5.5e-226) {
		tmp = 0.0;
	} else {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -0.048:
		tmp = (1.0 / x) / n
	elif n <= -5.5e-226:
		tmp = 0.0
	else:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -0.048)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (n <= -5.5e-226)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -0.048)
		tmp = (1.0 / x) / n;
	elseif (n <= -5.5e-226)
		tmp = 0.0;
	else
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -0.048], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -5.5e-226], 0.0, N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.048000000000000001

   1. Initial program 24.9%

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

   3. Taylor expanded in n around inf 78.2%

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

      1. log1p-define 78.2%

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

   5. Simplified 78.2%

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

   6. Taylor expanded in x around inf 35.8%

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

   if -0.048000000000000001 < n < -5.5e-226

   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 36.9%

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

   4. Taylor expanded in n around inf 65.7%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 65.7%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{0} \]

   if -5.5e-226 < n

   1. Initial program 46.5%

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

   3. Taylor expanded in n around inf 54.2%

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

      1. log1p-define 54.2%

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

   5. Simplified 54.2%

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

      1. log1p-expm1-u 54.2%

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

      2. expm1-undefine 54.2%

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

      3. exp-diff 54.2%

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

      4. add-exp-log 36.0%

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

      5. log1p-undefine 36.0%

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

      6. rem-exp-log 54.3%

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

   7. Applied egg-rr 54.3%

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

   8. Taylor expanded in x around inf 34.4%

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

   10. Simplified 47.2%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.048:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -5.5 \cdot 10^{-226}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.7% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.048 \lor \neg \left(n \leq -5.1 \cdot 10^{-226}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -0.048) (not (<= n -5.1e-226))) (/ (/ 1.0 x) n) 0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -0.048) || !(n <= -5.1e-226)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-0.048d0)) .or. (.not. (n <= (-5.1d-226)))) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -0.048) || !(n <= -5.1e-226)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -0.048) or not (n <= -5.1e-226):
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -0.048) || !(n <= -5.1e-226))
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -0.048) || ~((n <= -5.1e-226)))
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -0.048], N[Not[LessEqual[n, -5.1e-226]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -0.048000000000000001 or -5.09999999999999973e-226 < n

   1. Initial program 39.5%

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

   3. Taylor expanded in n around inf 62.0%

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

      1. log1p-define 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in x around inf 43.0%

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

   if -0.048000000000000001 < n < -5.09999999999999973e-226

   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 36.9%

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

   4. Taylor expanded in n around inf 65.7%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 65.7%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.048 \lor \neg \left(n \leq -5.1 \cdot 10^{-226}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.2% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.048 \lor \neg \left(n \leq -9.8 \cdot 10^{-226}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -0.048) (not (<= n -9.8e-226))) (/ 1.0 (* n x)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -0.048) || !(n <= -9.8e-226)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-0.048d0)) .or. (.not. (n <= (-9.8d-226)))) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -0.048) || !(n <= -9.8e-226)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -0.048) or not (n <= -9.8e-226):
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -0.048) || !(n <= -9.8e-226))
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -0.048) || ~((n <= -9.8e-226)))
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -0.048], N[Not[LessEqual[n, -9.8e-226]], $MachinePrecision]], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -0.048000000000000001 or -9.79999999999999972e-226 < n

   1. Initial program 39.5%

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

   3. Taylor expanded in n around inf 62.0%

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

      1. log1p-define 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in x around inf 42.4%

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

      1. *-commutative 42.4%

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

   8. Simplified 42.4%

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

   if -0.048000000000000001 < n < -9.79999999999999972e-226

   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 36.9%

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

   4. Taylor expanded in n around inf 65.7%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 65.7%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.048 \lor \neg \left(n \leq -9.8 \cdot 10^{-226}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.8% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.048:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -4.3 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -0.048) (/ (/ 1.0 x) n) (if (<= n -4.3e-228) 0.0 (/ (/ 1.0 n) x))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -0.048) {
		tmp = (1.0 / x) / n;
	} else if (n <= -4.3e-228) {
		tmp = 0.0;
	} 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) :: tmp
    if (n <= (-0.048d0)) then
        tmp = (1.0d0 / x) / n
    else if (n <= (-4.3d-228)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -0.048) {
		tmp = (1.0 / x) / n;
	} else if (n <= -4.3e-228) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -0.048:
		tmp = (1.0 / x) / n
	elif n <= -4.3e-228:
		tmp = 0.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -0.048)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (n <= -4.3e-228)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -0.048)
		tmp = (1.0 / x) / n;
	elseif (n <= -4.3e-228)
		tmp = 0.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -0.048], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -4.3e-228], 0.0, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.048000000000000001

   1. Initial program 24.9%

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

   3. Taylor expanded in n around inf 78.2%

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

      1. log1p-define 78.2%

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

   5. Simplified 78.2%

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

   6. Taylor expanded in x around inf 35.8%

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

   if -0.048000000000000001 < n < -4.3e-228

   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 36.9%

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

   4. Taylor expanded in n around inf 65.7%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 65.7%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{0} \]

   if -4.3e-228 < n

   1. Initial program 46.5%

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

   3. Taylor expanded in n around inf 54.2%

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

      1. log1p-define 54.2%

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

   5. Simplified 54.2%

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

   6. Taylor expanded in x around -inf 47.8%

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

   7. Taylor expanded in x around inf 46.0%

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

      1. associate-/r* 46.5%

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

   9. Simplified 46.5%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.048:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -4.3 \cdot 10^{-228}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.1% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

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

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in x around 0 36.1%

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

4. Taylor expanded in n around inf 30.0%

   \[\leadsto 1 - \color{blue}{1} \]
5. Step-by-step derivation

   1. metadata-eval 30.0%

      \[\leadsto \color{blue}{0} \]

6. Applied egg-rr 30.0%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024117 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))