2nthrt (problem 3.4.6)

Percentage Accurate: 53.0% → 85.6%

Time: 43.3s

Alternatives: 15

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: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0}\right)}^{3}\\ \end{array} \end{array} \]

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 x)
         (- (/ (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))) n) (log 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(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) / n) - log(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(x) + (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) / n) - Math.log(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(Float64(log1p(x) + Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) / n) - log(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[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $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. distribute-frac-neg 98.3%

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

      4. remove-double-neg 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. associate-/l* 98.3%

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

      7. exp-to-pow 98.4%

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

      8. *-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 80.6%

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

   5. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} - \log x\right)}{n}} \]

   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 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.4× speedup

Math

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

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)
       (/ (log (/ (+ x 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 = log(((x + 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.log(((x + 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(log(Float64(Float64(x + 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[N[(N[(x + 1.0), $MachinePrecision] / 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. distribute-frac-neg 98.3%

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

      4. remove-double-neg 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. associate-/l* 98.3%

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

      7. exp-to-pow 98.4%

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

      8. *-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-undefine 80.0%

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

      2. diff-log 80.2%

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

   7. Applied egg-rr 80.2%

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

   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 87.6%

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

Alternative 3: 85.5% accurate, 0.7× speedup

Math

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

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)
       (/ (log (/ (+ x 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 = log(((x + 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.log(((x + 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.log(((x + 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(log(Float64(Float64(x + 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[N[(N[(x + 1.0), $MachinePrecision] / 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. distribute-frac-neg 98.3%

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

      4. remove-double-neg 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. associate-/l* 98.3%

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

      7. exp-to-pow 98.4%

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

      8. *-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-undefine 80.0%

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

      2. diff-log 80.2%

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

   7. Applied egg-rr 80.2%

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

   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 87.6%

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

Alternative 4: 82.1% accurate, 1.6× speedup

Math

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

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)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (+ (* x (/ (+ (/ (* x 0.5) n) 1.0) n)) 1.0) 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 = log(((x + 1.0) / x)) / n;
	} else {
		tmp = ((x * ((((x * 0.5) / n) + 1.0) / n)) + 1.0) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-10)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2000000.0d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = ((x * ((((x * 0.5d0) / n) + 1.0d0) / n)) + 1.0d0) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2000000.0) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = ((x * ((((x * 0.5) / n) + 1.0) / n)) + 1.0) - 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.log(((x + 1.0) / x)) / n
	else:
		tmp = ((x * ((((x * 0.5) / n) + 1.0) / n)) + 1.0) - 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(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(Float64(Float64(x * 0.5) / n) + 1.0) / n)) + 1.0) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-10)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2000000.0)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = ((x * ((((x * 0.5) / n) + 1.0) / n)) + 1.0) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-10], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2000000.0], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x * N[(N[(N[(N[(x * 0.5), $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + 1.0), $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. distribute-frac-neg 98.3%

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

      4. remove-double-neg 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. associate-/l* 98.3%

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

      7. exp-to-pow 98.4%

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

      8. *-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-undefine 80.0%

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

      2. diff-log 80.2%

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

   7. Applied egg-rr 80.2%

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

   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%

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

   5. Taylor expanded in n around 0 79.4%

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

      1. associate-*r/ 79.4%

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

      2. *-commutative 79.4%

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

   7. Simplified 79.4%

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

4. Final simplification 84.7%

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

Alternative 5: 81.0% accurate, 1.6× speedup

Math

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

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)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+195)
         (+ (/ x n) (- 1.0 t_0))
         (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 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 = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+195) {
		tmp = (x / n) + (1.0 - t_0);
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-10)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 5d+17) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+195) then
        tmp = (x / n) + (1.0d0 - t_0)
    else
        tmp = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+17) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+195) {
		tmp = (x / n) + (1.0 - t_0);
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / 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.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+195:
		tmp = (x / n) + (1.0 - t_0)
	else:
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / 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(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+195)
		tmp = Float64(Float64(x / n) + Float64(1.0 - t_0));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-10)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 5e+17)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+195)
		tmp = (x / n) + (1.0 - t_0);
	else
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+195], N[(N[(x / n), $MachinePrecision] + N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $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. distribute-frac-neg 98.3%

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

      4. remove-double-neg 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. associate-/l* 98.3%

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

      7. exp-to-pow 98.4%

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

      8. *-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-undefine 79.5%

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

      2. diff-log 79.7%

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

   7. Applied egg-rr 79.7%

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

   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%

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

      1. sub-neg 80.9%

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

      2. +-commutative 80.9%

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

      3. associate-+l+ 80.9%

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

      4. sub-neg 80.9%

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

      5. *-rgt-identity 80.9%

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

      6. associate-/l* 80.9%

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

      7. exp-to-pow 80.9%

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

   5. Simplified 80.9%

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

   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. Taylor expanded in x around -inf 0.2%

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

   7. Taylor expanded in x around -inf 70.0%

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

      1. mul-1-neg 70.0%

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

      2. distribute-neg-frac2 70.0%

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

      3. sub-neg 70.0%

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

      4. associate-*r/ 70.0%

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

      5. sub-neg 70.0%

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

      6. metadata-eval 70.0%

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

      7. distribute-lft-in 70.0%

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

      8. neg-mul-1 70.0%

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

      9. associate-*r/ 70.0%

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

      10. metadata-eval 70.0%

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

      11. distribute-neg-frac 70.0%

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

      12. metadata-eval 70.0%

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

      13. metadata-eval 70.0%

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

      14. metadata-eval 70.0%

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

   9. Simplified 70.0%

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

4. Final simplification 83.9%

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

Alternative 6: 81.1% accurate, 1.7× speedup

Math

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

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)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e+178)
         (- 1.0 t_0)
         (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 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 = log(((x + 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)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-10)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 5d+17) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+178) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-10) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e+17) {
		tmp = Math.log(((x + 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)) / 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.log(((x + 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)) / 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(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+178)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-10)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 5e+17)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+178)
		tmp = 1.0 - t_0;
	else
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+178], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $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. distribute-frac-neg 98.3%

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

      4. remove-double-neg 98.3%

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

      5. *-rgt-identity 98.3%

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

      6. associate-/l* 98.3%

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

      7. exp-to-pow 98.4%

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

      8. *-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-undefine 79.5%

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

      2. diff-log 79.7%

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

   7. Applied egg-rr 79.7%

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

   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. Taylor expanded in x around -inf 0.2%

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

   7. Taylor expanded in x around -inf 68.0%

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

      1. mul-1-neg 68.0%

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

      2. distribute-neg-frac2 68.0%

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

      3. sub-neg 68.0%

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

      4. associate-*r/ 68.0%

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

      5. sub-neg 68.0%

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

      6. metadata-eval 68.0%

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

      7. distribute-lft-in 68.0%

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

      8. neg-mul-1 68.0%

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

      9. associate-*r/ 68.0%

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

      10. metadata-eval 68.0%

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

      11. distribute-neg-frac 68.0%

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

      12. metadata-eval 68.0%

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

      13. metadata-eval 68.0%

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

      14. metadata-eval 68.0%

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

   9. Simplified 68.0%

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

4. Final simplification 83.7%

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

Alternative 7: 67.1% accurate, 1.7× speedup

Math

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

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) 5e+17)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e+178)
         t_0
         (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x) n))))))

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) <= 5e+17) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-2d+227)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d+17) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+178) then
        tmp = t_0
    else
        tmp = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+227) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+17) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+178) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	}
	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) <= 5e+17:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+178:
		tmp = t_0
	else:
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n
	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) <= 5e+17)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+178)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -2e+227)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e+17)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+178)
		tmp = t_0;
	else
		tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 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) < 5e17

   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 74.7%

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

      1. log1p-define 74.7%

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

   5. Simplified 74.7%

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

      1. log1p-undefine 74.7%

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

      2. diff-log 74.8%

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

   7. Applied egg-rr 74.8%

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

   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. Taylor expanded in x around -inf 0.2%

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

   7. Taylor expanded in x around -inf 68.0%

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

      1. mul-1-neg 68.0%

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

      2. distribute-neg-frac2 68.0%

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

      3. sub-neg 68.0%

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

      4. associate-*r/ 68.0%

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

      5. sub-neg 68.0%

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

      6. metadata-eval 68.0%

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

      7. distribute-lft-in 68.0%

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

      8. neg-mul-1 68.0%

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

      9. associate-*r/ 68.0%

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

      10. metadata-eval 68.0%

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

      11. distribute-neg-frac 68.0%

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

      12. metadata-eval 68.0%

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

      13. metadata-eval 68.0%

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

      14. metadata-eval 68.0%

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

   9. Simplified 68.0%

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

4. Final simplification 74.5%

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

Alternative 8: 61.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{n}}{{x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.88)
   (/ (- x (log x)) n)
   (if (<= x 2.5e+132)
     (*
      (/ 1.0 n)
      (/ (+ (/ (- (/ (+ 0.3333333333333333 (/ -0.25 x)) x) 0.5) x) 1.0) x))
     (/ (/ -0.25 n) (pow x 4.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.5e+132) {
		tmp = (1.0 / n) * ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x);
	} else {
		tmp = (-0.25 / n) / pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.5d+132) then
        tmp = (1.0d0 / n) * ((((((0.3333333333333333d0 + ((-0.25d0) / x)) / x) - 0.5d0) / x) + 1.0d0) / x)
    else
        tmp = ((-0.25d0) / n) / (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.5e+132) {
		tmp = (1.0 / n) * ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x);
	} else {
		tmp = (-0.25 / n) / Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 2.5e+132:
		tmp = (1.0 / n) * ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x)
	else:
		tmp = (-0.25 / n) / math.pow(x, 4.0)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.5e+132)
		tmp = Float64(Float64(1.0 / n) * Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x) - 0.5) / x) + 1.0) / x));
	else
		tmp = Float64(Float64(-0.25 / n) / (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.5e+132)
		tmp = (1.0 / n) * ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x);
	else
		tmp = (-0.25 / n) / (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.5e+132], N[(N[(1.0 / n), $MachinePrecision] * N[(N[(N[(N[(N[(N[(0.3333333333333333 + N[(-0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 / n), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.880000000000000004

   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 36.5%

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

      1. sub-neg 36.5%

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

      2. +-commutative 36.5%

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

      3. associate-+l+ 37.1%

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

      4. sub-neg 37.1%

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

      5. *-rgt-identity 37.1%

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

      6. associate-/l* 37.0%

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

      7. exp-to-pow 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in n around inf 58.6%

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

   if 0.880000000000000004 < 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%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. clear-num 62.2%

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

      2. inv-pow 62.2%

         \[\leadsto \color{blue}{{\left(\frac{n}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}^{-1}} \]

   8. Applied egg-rr 62.2%

      \[\leadsto \color{blue}{{\left(\frac{n}{-\frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\frac{0.25}{x} + -0.3333333333333333}{x}, -0.5\right)}{x}, -1\right)}{x}}\right)}^{-1}} \]

   10. Simplified 62.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 63.4%

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

   12. Applied egg-rr 63.4%

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

   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 n around inf 79.8%

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

      1. log1p-define 79.8%

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

   5. Simplified 79.8%

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

   6. Taylor expanded in x around -inf 65.0%

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

   7. Taylor expanded in x around 0 79.8%

      \[\leadsto \color{blue}{\frac{-0.25}{n \cdot {x}^{4}}} \]
   8. Step-by-step derivation

      1. associate-/r* 79.8%

         \[\leadsto \color{blue}{\frac{\frac{-0.25}{n}}{{x}^{4}}} \]

   9. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{\frac{-0.25}{n}}{{x}^{4}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{n}}{{x}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.9)
   (/ (- x (log x)) n)
   (/ (/ (+ (/ (- (/ (+ 0.3333333333333333 (/ -0.25 x)) x) 0.5) x) 1.0) x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((((((0.3333333333333333d0 + ((-0.25d0) / x)) / x) - 0.5d0) / x) + 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.9:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.9)
		tmp = (x - log(x)) / n;
	else
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.3333333333333333 + N[(-0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   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 36.5%

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

      1. sub-neg 36.5%

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

      2. +-commutative 36.5%

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

      3. associate-+l+ 37.1%

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

      4. sub-neg 37.1%

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

      5. *-rgt-identity 37.1%

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

      6. associate-/l* 37.0%

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

      7. exp-to-pow 37.1%

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

   5. Simplified 37.1%

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

   6. Taylor expanded in n around inf 58.6%

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

   if 0.900000000000000022 < x

   1. Initial program 67.0%

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

   3. Taylor expanded in n around inf 68.6%

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

      1. log1p-define 68.6%

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

   5. Simplified 68.6%

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

   6. Taylor expanded in x around -inf 64.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. associate-*r/ 64.4%

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

      2. fmm-def 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x}, -1\right)}}{x}}{n} \]

      3. fmm-def 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\color{blue}{\mathsf{fma}\left(-1, \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x}, -0.5\right)}}{x}, -1\right)}{x}}{n} \]

      4. sub-neg 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\color{blue}{0.25 \cdot \frac{1}{x} + \left(-0.3333333333333333\right)}}{x}, -0.5\right)}{x}, -1\right)}{x}}{n} \]

      5. un-div-inv 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\color{blue}{\frac{0.25}{x}} + \left(-0.3333333333333333\right)}{x}, -0.5\right)}{x}, -1\right)}{x}}{n} \]

      6. metadata-eval 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\frac{0.25}{x} + \color{blue}{-0.3333333333333333}}{x}, -0.5\right)}{x}, -1\right)}{x}}{n} \]

      7. metadata-eval 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\frac{0.25}{x} + -0.3333333333333333}{x}, \color{blue}{-0.5}\right)}{x}, -1\right)}{x}}{n} \]

      8. metadata-eval 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\frac{0.25}{x} + -0.3333333333333333}{x}, -0.5\right)}{x}, \color{blue}{-1}\right)}{x}}{n} \]

   8. Applied egg-rr 64.4%

      \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\frac{0.25}{x} + -0.3333333333333333}{x}, -0.5\right)}{x}, -1\right)}{x}}}{n} \]

   10. Simplified 64.4%

       \[\leadsto \frac{\color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{x}}{x} - 0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = log(x) / -n;
	} else {
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	}
	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
        tmp = ((((((0.3333333333333333d0 + ((-0.25d0) / x)) / x) - 0.5d0) / x) + 1.0d0) / x) / n
    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 {
		tmp = ((((((0.3333333333333333 + (-0.25 / x)) / x) - 0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 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

   1. Initial program 67.0%

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

   3. Taylor expanded in n around inf 68.6%

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

      1. log1p-define 68.6%

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

   5. Simplified 68.6%

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

   6. Taylor expanded in x around -inf 64.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. associate-*r/ 64.4%

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

      2. fmm-def 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x}, -1\right)}}{x}}{n} \]

      3. fmm-def 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\color{blue}{\mathsf{fma}\left(-1, \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x}, -0.5\right)}}{x}, -1\right)}{x}}{n} \]

      4. sub-neg 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\color{blue}{0.25 \cdot \frac{1}{x} + \left(-0.3333333333333333\right)}}{x}, -0.5\right)}{x}, -1\right)}{x}}{n} \]

      5. un-div-inv 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\color{blue}{\frac{0.25}{x}} + \left(-0.3333333333333333\right)}{x}, -0.5\right)}{x}, -1\right)}{x}}{n} \]

      6. metadata-eval 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\frac{0.25}{x} + \color{blue}{-0.3333333333333333}}{x}, -0.5\right)}{x}, -1\right)}{x}}{n} \]

      7. metadata-eval 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\frac{0.25}{x} + -0.3333333333333333}{x}, \color{blue}{-0.5}\right)}{x}, -1\right)}{x}}{n} \]

      8. metadata-eval 64.4%

         \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\frac{0.25}{x} + -0.3333333333333333}{x}, -0.5\right)}{x}, \color{blue}{-1}\right)}{x}}{n} \]

   8. Applied egg-rr 64.4%

      \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\frac{0.25}{x} + -0.3333333333333333}{x}, -0.5\right)}{x}, -1\right)}{x}}}{n} \]

   10. Simplified 64.4%

       \[\leadsto \frac{\color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

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

Alternative 11: 45.8% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{n \cdot x}}{x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (- (/ 1.0 n) (/ (- (/ 0.5 n) (/ 0.3333333333333333 (* n x))) x)) x))

C

double code(double x, double n) {
	return ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) - (((0.5d0 / n) - (0.3333333333333333d0 / (n * x))) / x)) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x;
}

Python

def code(x, n):
	return ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(0.5 / n) - Float64(0.3333333333333333 / Float64(n * x))) / x)) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (n * x))) / x)) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(0.5 / n), $MachinePrecision] - N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

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 n around inf 62.8%

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

   1. log1p-define 62.8%

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

5. Simplified 62.8%

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

6. Taylor expanded in x around -inf 27.2%

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

7. Taylor expanded in x around -inf 41.1%

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

   1. mul-1-neg 41.1%

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

   2. mul-1-neg 41.1%

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

   3. associate-*r/ 41.1%

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

   4. metadata-eval 41.1%

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

   5. *-commutative 41.1%

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

   6. associate-*r/ 41.1%

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

   7. metadata-eval 41.1%

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

9. Simplified 41.1%

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

10. Final simplification 41.1%

    \[\leadsto \frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{n \cdot x}}{x}}{x} \]
11. Add Preprocessing

Alternative 12: 45.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) x) n))

C

double code(double x, double n) {
	return ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / x) / n
end function

Java

public static double code(double x, double n) {
	return ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
}

Python

def code(x, n):
	return ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]

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 n around inf 62.8%

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

   1. log1p-define 62.8%

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

5. Simplified 62.8%

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

6. Taylor expanded in x around -inf 27.2%

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

7. Taylor expanded in x around -inf 41.1%

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

   1. mul-1-neg 41.1%

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

   2. distribute-neg-frac2 41.1%

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

   3. sub-neg 41.1%

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

   4. associate-*r/ 41.1%

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

   5. sub-neg 41.1%

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

   6. metadata-eval 41.1%

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

   7. distribute-lft-in 41.1%

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

   8. neg-mul-1 41.1%

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

   9. associate-*r/ 41.1%

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

   10. metadata-eval 41.1%

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

   11. distribute-neg-frac 41.1%

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

   12. metadata-eval 41.1%

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

   13. metadata-eval 41.1%

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

   14. metadata-eval 41.1%

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

9. Simplified 41.1%

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

10. Final simplification 41.1%

    \[\leadsto \frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n} \]
11. Add Preprocessing

Alternative 13: 40.3% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 n around inf 62.8%

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

   1. log1p-define 62.8%

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

5. Simplified 62.8%

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

6. Taylor expanded in x around -inf 27.2%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
7. Step-by-step derivation

   1. clear-num 26.7%

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

   2. inv-pow 26.7%

      \[\leadsto \color{blue}{{\left(\frac{n}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}\right)}^{-1}} \]

8. Applied egg-rr 26.7%

   \[\leadsto \color{blue}{{\left(\frac{n}{-\frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{\frac{0.25}{x} + -0.3333333333333333}{x}, -0.5\right)}{x}, -1\right)}{x}}\right)}^{-1}} \]

10. Simplified 26.7%

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

11. Taylor expanded in x around inf 37.6%

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

    1. associate-/r* 38.0%

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

13. Simplified 38.0%

    \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]
14. Add Preprocessing

Alternative 14: 39.8% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in n around inf 62.8%

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

   1. log1p-define 62.8%

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

5. Simplified 62.8%

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

6. Taylor expanded in x around inf 37.6%

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

   1. *-commutative 37.6%

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

8. Simplified 37.6%

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

9. Final simplification 37.6%

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

Alternative 15: 4.6% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 27.0%

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

   1. sub-neg 27.0%

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

   2. +-commutative 27.0%

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

   3. associate-+l+ 23.1%

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

   4. sub-neg 23.1%

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

   5. *-rgt-identity 23.1%

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

   6. associate-/l* 23.0%

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

   7. exp-to-pow 23.0%

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

5. Simplified 23.0%

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

6. Taylor expanded in x around inf 4.6%

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