2nthrt (problem 3.4.6)

Percentage Accurate: 53.9% → 84.0%

Time: 17.4s

Alternatives: 18

Speedup: 2.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{\mathsf{log1p}\left(x\right) - \log x}{n} - \frac{-0.16666666666666666}{\frac{{n}^{3}}{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}}\right) + \mathsf{fma}\left(0.5, {n}^{-2} \cdot {\left(\mathsf{log1p}\left(x\right)\right)}^{2}, \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-122)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-9)
       (+
        (-
         (/ (- (log1p x) (log x)) n)
         (/
          -0.16666666666666666
          (/ (pow n 3.0) (- (pow (log1p x) 3.0) (pow (log x) 3.0)))))
        (fma
         0.5
         (* (pow n -2.0) (pow (log1p x) 2.0))
         (* (/ (pow (log x) 2.0) (* n n)) -0.5)))
       (- (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) <= -2e-122) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-9) {
		tmp = (((log1p(x) - log(x)) / n) - (-0.16666666666666666 / (pow(n, 3.0) / (pow(log1p(x), 3.0) - pow(log(x), 3.0))))) + fma(0.5, (pow(n, -2.0) * pow(log1p(x), 2.0)), ((pow(log(x), 2.0) / (n * n)) * -0.5));
	} else {
		tmp = exp((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) <= -2e-122)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-9)
		tmp = Float64(Float64(Float64(Float64(log1p(x) - log(x)) / n) - Float64(-0.16666666666666666 / Float64((n ^ 3.0) / Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))))) + fma(0.5, Float64((n ^ -2.0) * (log1p(x) ^ 2.0)), Float64(Float64((log(x) ^ 2.0) / Float64(n * n)) * -0.5)));
	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], -2e-122], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-9], N[(N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[(-0.16666666666666666 / N[(N[Power[n, 3.0], $MachinePrecision] / N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Power[n, -2.0], $MachinePrecision] * N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2.00000000000000012e-122

   1. Initial program 76.2%

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

   2. Taylor expanded in x around inf 89.4%

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

      1. log-rec 89.4%

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

      2. mul-1-neg 89.4%

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

      3. associate-*r/ 89.4%

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

      4. neg-mul-1 89.4%

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

      5. mul-1-neg 89.4%

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

      6. remove-double-neg 89.4%

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

      7. *-commutative 89.4%

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

   4. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. div-inv 89.4%

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

      3. pow-to-exp 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. *-lft-identity 89.4%

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

   8. Simplified 89.4%

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

   if -2.00000000000000012e-122 < (/.f64 1 n) < 5.0000000000000001e-9

   1. Initial program 33.4%

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

   2. Taylor expanded in n around -inf 82.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + -1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n}\right)\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}} \]

   4. Simplified 82.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{log1p}\left(x\right) - \log x}{n} - \frac{-0.16666666666666666}{\frac{{n}^{3}}{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}}\right) + \mathsf{fma}\left(0.5, {n}^{-2} \cdot {\left(\mathsf{log1p}\left(x\right)\right)}^{2}, \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\right)} \]

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

   1. Initial program 52.4%

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

   2. Taylor expanded in n around 0 52.4%

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

      1. log1p-def 97.3%

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

   4. Simplified 97.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{\mathsf{log1p}\left(x\right) - \log x}{n} - \frac{-0.16666666666666666}{\frac{{n}^{3}}{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}}\right) + \mathsf{fma}\left(0.5, {n}^{-2} \cdot {\left(\mathsf{log1p}\left(x\right)\right)}^{2}, \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-122)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-9)
       (fma
        0.5
        (/ (pow (log1p x) 2.0) (* n n))
        (- (/ (- (log1p x) (log x)) n) (* 0.5 (/ (pow (log x) 2.0) (* n 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) <= -2e-122) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-9) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), (((log1p(x) - log(x)) / n) - (0.5 * (pow(log(x), 2.0) / (n * n)))));
	} else {
		tmp = exp((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) <= -2e-122)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-9)
		tmp = fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), Float64(Float64(Float64(log1p(x) - log(x)) / n) - Float64(0.5 * Float64((log(x) ^ 2.0) / Float64(n * 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], -2e-122], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-9], N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2.00000000000000012e-122

   1. Initial program 76.2%

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

   2. Taylor expanded in x around inf 89.4%

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

      1. log-rec 89.4%

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

      2. mul-1-neg 89.4%

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

      3. associate-*r/ 89.4%

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

      4. neg-mul-1 89.4%

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

      5. mul-1-neg 89.4%

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

      6. remove-double-neg 89.4%

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

      7. *-commutative 89.4%

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

   4. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. div-inv 89.4%

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

      3. pow-to-exp 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. *-lft-identity 89.4%

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

   8. Simplified 89.4%

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

   if -2.00000000000000012e-122 < (/.f64 1 n) < 5.0000000000000001e-9

   1. Initial program 33.4%

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

   2. Taylor expanded in n around inf 82.4%

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

      1. associate--l+ 79.3%

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

      2. fma-def 79.3%

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

      3. log1p-def 79.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2}}{{n}^{2}}, \frac{\log \left(1 + x\right)}{n} - \left(\frac{\log x}{n} + 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)\right) \]

      4. unpow2 79.3%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{\color{blue}{n \cdot n}}, \frac{\log \left(1 + x\right)}{n} - \left(\frac{\log x}{n} + 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)\right) \]

      5. associate--r+ 82.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\left(\frac{\log \left(1 + x\right)}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}}\right) \]

      6. +-rgt-identity 82.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 82.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 82.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      9. log1p-def 82.4%

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

      10. unpow2 82.4%

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

   4. Simplified 82.4%

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

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

   1. Initial program 52.4%

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

   2. Taylor expanded in n around 0 52.4%

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

      1. log1p-def 97.3%

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

   4. Simplified 97.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-122)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-15)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-122) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-122) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-122:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-15:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-122)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-122], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2.00000000000000012e-122

   1. Initial program 76.2%

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

   2. Taylor expanded in x around inf 89.4%

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

      1. log-rec 89.4%

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

      2. mul-1-neg 89.4%

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

      3. associate-*r/ 89.4%

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

      4. neg-mul-1 89.4%

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

      5. mul-1-neg 89.4%

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

      6. remove-double-neg 89.4%

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

      7. *-commutative 89.4%

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

   4. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. div-inv 89.4%

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

      3. pow-to-exp 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. *-lft-identity 89.4%

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

   8. Simplified 89.4%

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

   if -2.00000000000000012e-122 < (/.f64 1 n) < 1.0000000000000001e-15

   1. Initial program 33.0%

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

   2. Taylor expanded in n around inf 82.2%

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

      1. +-rgt-identity 82.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 82.2%

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

      3. log1p-def 82.2%

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

   4. Simplified 82.2%

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

   if 1.0000000000000001e-15 < (/.f64 1 n)

   1. Initial program 53.0%

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

   2. Taylor expanded in n around 0 53.0%

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

      1. log1p-def 96.7%

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

   4. Simplified 96.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-122)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-15)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-122) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-122) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-122:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-15:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-122)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-122], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2.00000000000000012e-122

   1. Initial program 76.2%

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

   2. Taylor expanded in x around inf 89.4%

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

      1. log-rec 89.4%

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

      2. mul-1-neg 89.4%

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

      3. associate-*r/ 89.4%

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

      4. neg-mul-1 89.4%

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

      5. mul-1-neg 89.4%

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

      6. remove-double-neg 89.4%

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

      7. *-commutative 89.4%

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

   4. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. div-inv 89.4%

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

      3. pow-to-exp 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. *-lft-identity 89.4%

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

   8. Simplified 89.4%

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

   if -2.00000000000000012e-122 < (/.f64 1 n) < 1.0000000000000001e-15

   1. Initial program 33.0%

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

   2. Taylor expanded in n around inf 82.2%

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

      1. +-rgt-identity 82.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 82.2%

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

      3. log1p-def 82.2%

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

   4. Simplified 82.2%

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

   if 1.0000000000000001e-15 < (/.f64 1 n)

   1. Initial program 53.0%

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

   2. Taylor expanded in n around 0 53.0%

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

      1. log1p-def 96.7%

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

   4. Simplified 96.7%

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

   5. Taylor expanded in x around 0 96.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \log x \cdot \frac{-1}{n}\\ \mathbf{if}\;x \leq 2 \cdot 10^{-218}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-154}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (* (log x) (/ -1.0 n))))
   (if (<= x 2e-218)
     (- (/ (log x) n))
     (if (<= x 1.4e-154)
       (- (+ 1.0 (/ x n)) t_0)
       (if (<= x 3.4e-118)
         t_1
         (if (<= x 6.5e-92)
           (log1p (expm1 (/ 1.0 (* n x))))
           (if (<= x 1.02e-25) t_1 (/ t_0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(x) * (-1.0 / n);
	double tmp;
	if (x <= 2e-218) {
		tmp = -(log(x) / n);
	} else if (x <= 1.4e-154) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 3.4e-118) {
		tmp = t_1;
	} else if (x <= 6.5e-92) {
		tmp = log1p(expm1((1.0 / (n * x))));
	} else if (x <= 1.02e-25) {
		tmp = t_1;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(x) * (-1.0 / n);
	double tmp;
	if (x <= 2e-218) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 1.4e-154) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 3.4e-118) {
		tmp = t_1;
	} else if (x <= 6.5e-92) {
		tmp = Math.log1p(Math.expm1((1.0 / (n * x))));
	} else if (x <= 1.02e-25) {
		tmp = t_1;
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(x) * (-1.0 / n)
	tmp = 0
	if x <= 2e-218:
		tmp = -(math.log(x) / n)
	elif x <= 1.4e-154:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 3.4e-118:
		tmp = t_1
	elif x <= 6.5e-92:
		tmp = math.log1p(math.expm1((1.0 / (n * x))))
	elif x <= 1.02e-25:
		tmp = t_1
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(x) * Float64(-1.0 / n))
	tmp = 0.0
	if (x <= 2e-218)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 1.4e-154)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 3.4e-118)
		tmp = t_1;
	elseif (x <= 6.5e-92)
		tmp = log1p(expm1(Float64(1.0 / Float64(n * x))));
	elseif (x <= 1.02e-25)
		tmp = t_1;
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e-218], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 1.4e-154], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 3.4e-118], t$95$1, If[LessEqual[x, 6.5e-92], N[Log[1 + N[(Exp[N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.02e-25], t$95$1, N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.0000000000000001e-218

   1. Initial program 45.0%

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

   2. Taylor expanded in x around 0 45.0%

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

   3. Taylor expanded in n around inf 57.0%

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

      1. associate-*r/ 57.0%

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

      2. neg-mul-1 57.0%

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

   5. Simplified 57.0%

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

   if 2.0000000000000001e-218 < x < 1.40000000000000006e-154

   1. Initial program 63.2%

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

   2. Taylor expanded in x around 0 63.2%

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

   if 1.40000000000000006e-154 < x < 3.39999999999999991e-118 or 6.50000000000000035e-92 < x < 1.01999999999999998e-25

   1. Initial program 30.8%

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

   2. Taylor expanded in x around 0 30.8%

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

   3. Taylor expanded in n around inf 58.9%

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

      1. associate-*r/ 58.9%

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

      2. neg-mul-1 58.9%

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

   5. Simplified 58.9%

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

      1. add-cube-cbrt 58.0%

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

      2. pow3 58.0%

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

   7. Applied egg-rr 58.0%

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

      1. rem-cube-cbrt 58.9%

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

      2. frac-2neg 58.9%

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

      3. div-inv 59.0%

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

      4. remove-double-neg 59.0%

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

      5. metadata-eval 59.0%

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

      6. distribute-neg-frac 59.0%

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

      7. metadata-eval 59.0%

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

      8. frac-2neg 59.0%

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

      9. distribute-neg-frac 59.0%

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

      10. metadata-eval 59.0%

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

   9. Applied egg-rr 59.0%

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

   if 3.39999999999999991e-118 < x < 6.50000000000000035e-92

   1. Initial program 38.3%

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

   2. Taylor expanded in x around inf 30.7%

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

      1. log-rec 30.7%

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

      2. mul-1-neg 30.7%

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

      3. associate-*r/ 30.7%

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

      4. neg-mul-1 30.7%

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

      5. mul-1-neg 30.7%

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

      6. remove-double-neg 30.7%

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

      7. *-commutative 30.7%

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

   4. Simplified 30.7%

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

   5. Taylor expanded in n around inf 26.7%

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

      1. *-commutative 26.7%

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

   7. Simplified 26.7%

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

      1. log1p-expm1-u 65.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]

   9. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]

   if 1.01999999999999998e-25 < x

   1. Initial program 68.9%

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

   2. Taylor expanded in x around inf 96.8%

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

      1. log-rec 96.8%

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

      2. mul-1-neg 96.8%

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

      3. associate-*r/ 96.8%

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

      4. neg-mul-1 96.8%

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

      5. mul-1-neg 96.8%

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

      6. remove-double-neg 96.8%

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

      7. *-commutative 96.8%

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

   4. Simplified 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. div-inv 96.8%

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

      3. pow-to-exp 96.8%

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

   6. Applied egg-rr 96.8%

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

      1. *-lft-identity 96.8%

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

   8. Simplified 96.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-218}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-154}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-118}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-25}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]

Alternative 6: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-122)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-15)
       (/ (- (log1p x) (log x)) n)
       (-
        (+ (* (- (/ 0.5 (* n n)) (/ 0.5 n)) (* x x)) (+ 1.0 (/ 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) <= -2e-122) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (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) <= -2e-122) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (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) <= -2e-122:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-15:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (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) <= -2e-122)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)) * Float64(x * x)) + Float64(1.0 + Float64(x / n))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-122], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2.00000000000000012e-122

   1. Initial program 76.2%

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

   2. Taylor expanded in x around inf 89.4%

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

      1. log-rec 89.4%

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

      2. mul-1-neg 89.4%

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

      3. associate-*r/ 89.4%

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

      4. neg-mul-1 89.4%

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

      5. mul-1-neg 89.4%

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

      6. remove-double-neg 89.4%

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

      7. *-commutative 89.4%

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

   4. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. div-inv 89.4%

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

      3. pow-to-exp 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. *-lft-identity 89.4%

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

   8. Simplified 89.4%

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

   if -2.00000000000000012e-122 < (/.f64 1 n) < 1.0000000000000001e-15

   1. Initial program 33.0%

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

   2. Taylor expanded in n around inf 82.2%

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

      1. +-rgt-identity 82.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 82.2%

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

      3. log1p-def 82.2%

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

   4. Simplified 82.2%

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

   if 1.0000000000000001e-15 < (/.f64 1 n)

   1. Initial program 53.0%

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

   2. Taylor expanded in x around 0 68.8%

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

      1. associate-+r+ 68.8%

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

      2. +-commutative 68.8%

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

      3. associate-*r/ 68.8%

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

      4. metadata-eval 68.8%

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

      5. unpow2 68.8%

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

      6. associate-*r/ 68.8%

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

      7. metadata-eval 68.8%

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

      8. unpow2 68.8%

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

   4. Simplified 68.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 7: 54.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 8.4 \cdot 10^{-218}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-24}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 4800000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 8.4e-218)
     (- (/ (log x) n))
     (if (<= x 1.4e-154)
       t_0
       (if (<= x 4.2e-24)
         (* (log x) (/ -1.0 n))
         (if (<= x 4800000000.0) t_0 (/ (/ 1.0 n) x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 8.4e-218) {
		tmp = -(log(x) / n);
	} else if (x <= 1.4e-154) {
		tmp = t_0;
	} else if (x <= 4.2e-24) {
		tmp = log(x) * (-1.0 / n);
	} else if (x <= 4800000000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 8.4d-218) then
        tmp = -(log(x) / n)
    else if (x <= 1.4d-154) then
        tmp = t_0
    else if (x <= 4.2d-24) then
        tmp = log(x) * ((-1.0d0) / n)
    else if (x <= 4800000000.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 8.4e-218) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 1.4e-154) {
		tmp = t_0;
	} else if (x <= 4.2e-24) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if (x <= 4800000000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 8.4e-218:
		tmp = -(math.log(x) / n)
	elif x <= 1.4e-154:
		tmp = t_0
	elif x <= 4.2e-24:
		tmp = math.log(x) * (-1.0 / n)
	elif x <= 4800000000.0:
		tmp = t_0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 8.4e-218)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 1.4e-154)
		tmp = t_0;
	elseif (x <= 4.2e-24)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (x <= 4800000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 8.4e-218)
		tmp = -(log(x) / n);
	elseif (x <= 1.4e-154)
		tmp = t_0;
	elseif (x <= 4.2e-24)
		tmp = log(x) * (-1.0 / n);
	elseif (x <= 4800000000.0)
		tmp = t_0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 8.4e-218], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 1.4e-154], t$95$0, If[LessEqual[x, 4.2e-24], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4800000000.0], t$95$0, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.39999999999999976e-218

   1. Initial program 45.0%

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

   2. Taylor expanded in x around 0 45.0%

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

   3. Taylor expanded in n around inf 57.0%

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

      1. associate-*r/ 57.0%

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

      2. neg-mul-1 57.0%

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

   5. Simplified 57.0%

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

   if 8.39999999999999976e-218 < x < 1.40000000000000006e-154 or 4.1999999999999999e-24 < x < 4.8e9

   1. Initial program 70.1%

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

   2. Taylor expanded in x around 0 60.1%

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

   if 1.40000000000000006e-154 < x < 4.1999999999999999e-24

   1. Initial program 32.2%

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

   2. Taylor expanded in x around 0 32.2%

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

   3. Taylor expanded in n around inf 54.0%

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

      1. associate-*r/ 54.0%

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

      2. neg-mul-1 54.0%

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

   5. Simplified 54.0%

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

      1. add-cube-cbrt 53.2%

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

      2. pow3 53.2%

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

   7. Applied egg-rr 53.2%

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

      1. rem-cube-cbrt 54.0%

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

      2. frac-2neg 54.0%

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

      3. div-inv 54.1%

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

      4. remove-double-neg 54.1%

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

      5. metadata-eval 54.1%

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

      6. distribute-neg-frac 54.1%

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

      7. metadata-eval 54.1%

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

      8. frac-2neg 54.1%

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

      9. distribute-neg-frac 54.1%

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

      10. metadata-eval 54.1%

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

   9. Applied egg-rr 54.1%

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

   if 4.8e9 < x

   1. Initial program 67.1%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. log-rec 98.9%

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

      2. mul-1-neg 98.9%

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

      3. associate-*r/ 98.9%

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

      4. neg-mul-1 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. div-inv 98.9%

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

      3. pow-to-exp 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-lft-identity 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in n around inf 67.6%

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

       1. associate-/r* 68.6%

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

   11. Simplified 68.6%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.4 \cdot 10^{-218}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-154}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-24}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 4800000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 8: 69.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 6.2 \cdot 10^{-218}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-154}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-24}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 6.2e-218)
     (- (/ (log x) n))
     (if (<= x 3.2e-154)
       (- 1.0 t_0)
       (if (<= x 4.2e-24) (* (log x) (/ -1.0 n)) (/ t_0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 6.2e-218) {
		tmp = -(log(x) / n);
	} else if (x <= 3.2e-154) {
		tmp = 1.0 - t_0;
	} else if (x <= 4.2e-24) {
		tmp = log(x) * (-1.0 / n);
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 6.2d-218) then
        tmp = -(log(x) / n)
    else if (x <= 3.2d-154) then
        tmp = 1.0d0 - t_0
    else if (x <= 4.2d-24) then
        tmp = log(x) * ((-1.0d0) / n)
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 6.2e-218) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 3.2e-154) {
		tmp = 1.0 - t_0;
	} else if (x <= 4.2e-24) {
		tmp = Math.log(x) * (-1.0 / n);
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 6.2e-218:
		tmp = -(math.log(x) / n)
	elif x <= 3.2e-154:
		tmp = 1.0 - t_0
	elif x <= 4.2e-24:
		tmp = math.log(x) * (-1.0 / n)
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 6.2e-218)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 3.2e-154)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 4.2e-24)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 6.2e-218)
		tmp = -(log(x) / n);
	elseif (x <= 3.2e-154)
		tmp = 1.0 - t_0;
	elseif (x <= 4.2e-24)
		tmp = log(x) * (-1.0 / n);
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 6.2e-218], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 3.2e-154], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 4.2e-24], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.19999999999999994e-218

   1. Initial program 45.0%

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

   2. Taylor expanded in x around 0 45.0%

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

   3. Taylor expanded in n around inf 57.0%

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

      1. associate-*r/ 57.0%

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

      2. neg-mul-1 57.0%

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

   5. Simplified 57.0%

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

   if 6.19999999999999994e-218 < x < 3.20000000000000005e-154

   1. Initial program 63.2%

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

   2. Taylor expanded in x around 0 63.2%

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

   if 3.20000000000000005e-154 < x < 4.1999999999999999e-24

   1. Initial program 32.2%

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

   2. Taylor expanded in x around 0 32.2%

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

   3. Taylor expanded in n around inf 54.0%

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

      1. associate-*r/ 54.0%

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

      2. neg-mul-1 54.0%

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

   5. Simplified 54.0%

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

      1. add-cube-cbrt 53.2%

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

      2. pow3 53.2%

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

   7. Applied egg-rr 53.2%

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

      1. rem-cube-cbrt 54.0%

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

      2. frac-2neg 54.0%

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

      3. div-inv 54.1%

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

      4. remove-double-neg 54.1%

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

      5. metadata-eval 54.1%

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

      6. distribute-neg-frac 54.1%

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

      7. metadata-eval 54.1%

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

      8. frac-2neg 54.1%

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

      9. distribute-neg-frac 54.1%

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

      10. metadata-eval 54.1%

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

   9. Applied egg-rr 54.1%

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

   if 4.1999999999999999e-24 < x

   1. Initial program 68.9%

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

   2. Taylor expanded in x around inf 96.8%

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

      1. log-rec 96.8%

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

      2. mul-1-neg 96.8%

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

      3. associate-*r/ 96.8%

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

      4. neg-mul-1 96.8%

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

      5. mul-1-neg 96.8%

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

      6. remove-double-neg 96.8%

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

      7. *-commutative 96.8%

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

   4. Simplified 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. div-inv 96.8%

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

      3. pow-to-exp 96.8%

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

   6. Applied egg-rr 96.8%

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

      1. *-lft-identity 96.8%

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

   8. Simplified 96.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-218}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-154}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-24}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]

Alternative 9: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.9 \cdot 10^{-218}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-154}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-24}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 1.9e-218)
     (- (/ (log x) n))
     (if (<= x 2.85e-154)
       (- (+ 1.0 (/ x n)) t_0)
       (if (<= x 3.8e-24) (* (log x) (/ -1.0 n)) (/ t_0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.9e-218) {
		tmp = -(log(x) / n);
	} else if (x <= 2.85e-154) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 3.8e-24) {
		tmp = log(x) * (-1.0 / n);
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 1.9d-218) then
        tmp = -(log(x) / n)
    else if (x <= 2.85d-154) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 3.8d-24) then
        tmp = log(x) * ((-1.0d0) / n)
    else
        tmp = t_0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.9e-218) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 2.85e-154) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 3.8e-24) {
		tmp = Math.log(x) * (-1.0 / n);
	} else {
		tmp = t_0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.9e-218:
		tmp = -(math.log(x) / n)
	elif x <= 2.85e-154:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 3.8e-24:
		tmp = math.log(x) * (-1.0 / n)
	else:
		tmp = t_0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.9e-218)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 2.85e-154)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 3.8e-24)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	else
		tmp = Float64(t_0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 1.9e-218)
		tmp = -(log(x) / n);
	elseif (x <= 2.85e-154)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 3.8e-24)
		tmp = log(x) * (-1.0 / n);
	else
		tmp = t_0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.9e-218], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 2.85e-154], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 3.8e-24], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.8999999999999999e-218

   1. Initial program 45.0%

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

   2. Taylor expanded in x around 0 45.0%

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

   3. Taylor expanded in n around inf 57.0%

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

      1. associate-*r/ 57.0%

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

      2. neg-mul-1 57.0%

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

   5. Simplified 57.0%

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

   if 1.8999999999999999e-218 < x < 2.8499999999999999e-154

   1. Initial program 63.2%

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

   2. Taylor expanded in x around 0 63.2%

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

   if 2.8499999999999999e-154 < x < 3.80000000000000026e-24

   1. Initial program 32.2%

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

   2. Taylor expanded in x around 0 32.2%

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

   3. Taylor expanded in n around inf 54.0%

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

      1. associate-*r/ 54.0%

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

      2. neg-mul-1 54.0%

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

   5. Simplified 54.0%

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

      1. add-cube-cbrt 53.2%

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

      2. pow3 53.2%

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

   7. Applied egg-rr 53.2%

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

      1. rem-cube-cbrt 54.0%

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

      2. frac-2neg 54.0%

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

      3. div-inv 54.1%

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

      4. remove-double-neg 54.1%

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

      5. metadata-eval 54.1%

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

      6. distribute-neg-frac 54.1%

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

      7. metadata-eval 54.1%

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

      8. frac-2neg 54.1%

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

      9. distribute-neg-frac 54.1%

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

      10. metadata-eval 54.1%

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

   9. Applied egg-rr 54.1%

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

   if 3.80000000000000026e-24 < x

   1. Initial program 68.9%

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

   2. Taylor expanded in x around inf 96.8%

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

      1. log-rec 96.8%

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

      2. mul-1-neg 96.8%

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

      3. associate-*r/ 96.8%

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

      4. neg-mul-1 96.8%

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

      5. mul-1-neg 96.8%

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

      6. remove-double-neg 96.8%

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

      7. *-commutative 96.8%

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

   4. Simplified 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. div-inv 96.8%

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

      3. pow-to-exp 96.8%

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

   6. Applied egg-rr 96.8%

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

      1. *-lft-identity 96.8%

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

   8. Simplified 96.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-218}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-154}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-24}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \end{array} \]

Alternative 10: 56.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0175:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0175) (* (log x) (/ -1.0 n)) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0175) {
		tmp = log(x) * (-1.0 / n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.0175d0) then
        tmp = log(x) * ((-1.0d0) / n)
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.0175) {
		tmp = Math.log(x) * (-1.0 / n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0175:
		tmp = math.log(x) * (-1.0 / n)
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0175)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0175)
		tmp = log(x) * (-1.0 / n);
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0175], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.017500000000000002

   1. Initial program 43.3%

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

   2. Taylor expanded in x around 0 43.3%

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

   3. Taylor expanded in n around inf 50.6%

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

      1. associate-*r/ 50.6%

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

      2. neg-mul-1 50.6%

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

   5. Simplified 50.6%

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

      1. add-cube-cbrt 49.9%

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

      2. pow3 49.9%

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

   7. Applied egg-rr 49.9%

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

      1. rem-cube-cbrt 50.6%

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

      2. frac-2neg 50.6%

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

      3. div-inv 50.6%

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

      4. remove-double-neg 50.6%

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

      5. metadata-eval 50.6%

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

      6. distribute-neg-frac 50.6%

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

      7. metadata-eval 50.6%

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

      8. frac-2neg 50.6%

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

      9. distribute-neg-frac 50.6%

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

      10. metadata-eval 50.6%

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

   9. Applied egg-rr 50.6%

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

   if 0.017500000000000002 < x

   1. Initial program 68.6%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. log-rec 98.9%

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

      2. mul-1-neg 98.9%

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

      3. associate-*r/ 98.9%

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

      4. neg-mul-1 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. div-inv 98.9%

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

      3. pow-to-exp 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-lft-identity 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in n around inf 64.6%

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

       1. associate-/r* 65.6%

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

   11. Simplified 65.6%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0175:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 11: 56.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0175:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0175) (/ (- x (log x)) n) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0175) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.0175d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.0175) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0175:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0175)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0175)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0175], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.017500000000000002

   1. Initial program 43.3%

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

   2. Taylor expanded in x around 0 33.1%

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

      1. associate-+r+ 33.1%

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

      2. +-commutative 33.1%

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

      3. associate-*r/ 33.1%

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

      4. metadata-eval 33.1%

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

      5. unpow2 33.1%

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

      6. associate-*r/ 33.1%

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

      7. metadata-eval 33.1%

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

      8. unpow2 33.1%

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

   4. Simplified 33.1%

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

   5. Taylor expanded in n around inf 51.0%

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

      1. *-commutative 51.0%

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

      2. fma-def 51.0%

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

      3. unpow2 51.0%

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

   7. Simplified 51.0%

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

   8. Taylor expanded in x around 0 50.8%

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

   if 0.017500000000000002 < x

   1. Initial program 68.6%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. log-rec 98.9%

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

      2. mul-1-neg 98.9%

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

      3. associate-*r/ 98.9%

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

      4. neg-mul-1 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. div-inv 98.9%

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

      3. pow-to-exp 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-lft-identity 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in n around inf 64.6%

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

       1. associate-/r* 65.6%

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

   11. Simplified 65.6%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0175:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 12: 56.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0175:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0175) (- (/ (log x) n)) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0175) {
		tmp = -(log(x) / n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.0175d0) then
        tmp = -(log(x) / n)
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.0175) {
		tmp = -(Math.log(x) / n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0175:
		tmp = -(math.log(x) / n)
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0175)
		tmp = Float64(-Float64(log(x) / n));
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0175)
		tmp = -(log(x) / n);
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0175], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.017500000000000002

   1. Initial program 43.3%

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

   2. Taylor expanded in x around 0 43.3%

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

   3. Taylor expanded in n around inf 50.6%

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

      1. associate-*r/ 50.6%

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

      2. neg-mul-1 50.6%

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

   5. Simplified 50.6%

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

   if 0.017500000000000002 < x

   1. Initial program 68.6%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. log-rec 98.9%

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

      2. mul-1-neg 98.9%

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

      3. associate-*r/ 98.9%

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

      4. neg-mul-1 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. div-inv 98.9%

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

      3. pow-to-exp 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-lft-identity 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in n around inf 64.6%

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

       1. associate-/r* 65.6%

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

   11. Simplified 65.6%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0175:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 13: 42.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{\left(n \cdot n\right) \cdot -0.5 + n \cdot 0.5}{n \cdot \left(n \cdot n\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 1e+45)
   (/ (/ 1.0 n) x)
   (* x (* x (/ (+ (* (* n n) -0.5) (* n 0.5)) (* n (* n n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 1e+45) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = x * (x * ((((n * n) * -0.5) + (n * 0.5)) / (n * (n * n))));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 1d+45) then
        tmp = (1.0d0 / n) / x
    else
        tmp = x * (x * ((((n * n) * (-0.5d0)) + (n * 0.5d0)) / (n * (n * n))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 1e+45) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = x * (x * ((((n * n) * -0.5) + (n * 0.5)) / (n * (n * n))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 1e+45:
		tmp = (1.0 / n) / x
	else:
		tmp = x * (x * ((((n * n) * -0.5) + (n * 0.5)) / (n * (n * n))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 1e+45)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(x * Float64(x * Float64(Float64(Float64(Float64(n * n) * -0.5) + Float64(n * 0.5)) / Float64(n * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 1e+45)
		tmp = (1.0 / n) / x;
	else
		tmp = x * (x * ((((n * n) * -0.5) + (n * 0.5)) / (n * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+45], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(x * N[(N[(N[(N[(n * n), $MachinePrecision] * -0.5), $MachinePrecision] + N[(n * 0.5), $MachinePrecision]), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < 9.9999999999999993e44

   1. Initial program 55.8%

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

   2. Taylor expanded in x around inf 66.8%

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

      1. log-rec 66.8%

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

      2. mul-1-neg 66.8%

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

      3. associate-*r/ 66.8%

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

      4. neg-mul-1 66.8%

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

      5. mul-1-neg 66.8%

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

      6. remove-double-neg 66.8%

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

      7. *-commutative 66.8%

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

   4. Simplified 66.8%

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

      1. *-un-lft-identity 66.8%

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

      2. div-inv 66.8%

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

      3. pow-to-exp 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. *-lft-identity 66.8%

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

   8. Simplified 66.8%

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

   9. Taylor expanded in n around inf 42.1%

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

       1. associate-/r* 42.6%

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

   11. Simplified 42.6%

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

   if 9.9999999999999993e44 < (/.f64 1 n)

   1. Initial program 43.2%

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

   2. Taylor expanded in x around 0 64.0%

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

      1. associate-+r+ 64.0%

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

      2. +-commutative 64.0%

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

      3. associate-*r/ 64.0%

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

      4. metadata-eval 64.0%

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

      5. unpow2 64.0%

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

      6. associate-*r/ 64.0%

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

      7. metadata-eval 64.0%

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

      8. unpow2 64.0%

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

   4. Simplified 64.0%

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

   5. Taylor expanded in x around inf 35.4%

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

      1. *-commutative 35.4%

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

      2. unpow2 35.4%

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

      3. associate-*r/ 35.4%

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

      4. metadata-eval 35.4%

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

      5. unpow2 35.4%

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

      6. associate-*r/ 35.4%

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

      7. metadata-eval 35.4%

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

      8. associate-*l* 42.6%

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

      9. sub-neg 42.6%

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

      10. distribute-neg-frac 42.6%

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

      11. metadata-eval 42.6%

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

   7. Simplified 42.6%

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

      1. +-commutative 42.6%

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

      2. frac-add 48.7%

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

   9. Applied egg-rr 48.7%

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{-0.5 \cdot \left(n \cdot n\right) + n \cdot 0.5}{n \cdot \left(n \cdot n\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{\left(n \cdot n\right) \cdot -0.5 + n \cdot 0.5}{n \cdot \left(n \cdot n\right)}\right)\\ \end{array} \]

Alternative 14: 41.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 1e+45) (/ (/ 1.0 n) x) (* x (* x (/ 0.5 (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 1e+45) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = x * (x * (0.5 / (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 1d+45) then
        tmp = (1.0d0 / n) / x
    else
        tmp = x * (x * (0.5d0 / (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 1e+45) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = x * (x * (0.5 / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 1e+45:
		tmp = (1.0 / n) / x
	else:
		tmp = x * (x * (0.5 / (n * n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 1e+45)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 1e+45)
		tmp = (1.0 / n) / x;
	else
		tmp = x * (x * (0.5 / (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+45], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(x * N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < 9.9999999999999993e44

   1. Initial program 55.8%

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

   2. Taylor expanded in x around inf 66.8%

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

      1. log-rec 66.8%

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

      2. mul-1-neg 66.8%

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

      3. associate-*r/ 66.8%

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

      4. neg-mul-1 66.8%

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

      5. mul-1-neg 66.8%

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

      6. remove-double-neg 66.8%

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

      7. *-commutative 66.8%

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

   4. Simplified 66.8%

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

      1. *-un-lft-identity 66.8%

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

      2. div-inv 66.8%

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

      3. pow-to-exp 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. *-lft-identity 66.8%

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

   8. Simplified 66.8%

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

   9. Taylor expanded in n around inf 42.1%

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

       1. associate-/r* 42.6%

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

   11. Simplified 42.6%

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

   if 9.9999999999999993e44 < (/.f64 1 n)

   1. Initial program 43.2%

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

   2. Taylor expanded in x around 0 64.0%

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

      1. associate-+r+ 64.0%

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

      2. +-commutative 64.0%

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

      3. associate-*r/ 64.0%

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

      4. metadata-eval 64.0%

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

      5. unpow2 64.0%

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

      6. associate-*r/ 64.0%

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

      7. metadata-eval 64.0%

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

      8. unpow2 64.0%

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

   4. Simplified 64.0%

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

   5. Taylor expanded in x around inf 35.4%

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

      1. *-commutative 35.4%

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

      2. unpow2 35.4%

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

      3. associate-*r/ 35.4%

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

      4. metadata-eval 35.4%

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

      5. unpow2 35.4%

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

      6. associate-*r/ 35.4%

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

      7. metadata-eval 35.4%

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

      8. associate-*l* 42.6%

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

      9. sub-neg 42.6%

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

      10. distribute-neg-frac 42.6%

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

      11. metadata-eval 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in n around 0 42.6%

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

      1. unpow2 42.6%

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

   10. Simplified 42.6%

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{0.5}{n \cdot n}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{n \cdot n}\right)\\ \end{array} \]

Alternative 15: 39.5% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

2. Taylor expanded in x around inf 59.1%

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

   1. log-rec 59.1%

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

   2. mul-1-neg 59.1%

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

   3. associate-*r/ 59.1%

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

   4. neg-mul-1 59.1%

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

   5. mul-1-neg 59.1%

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

   6. remove-double-neg 59.1%

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

   7. *-commutative 59.1%

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

4. Simplified 59.1%

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

5. Taylor expanded in n around inf 40.4%

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

   1. *-commutative 40.4%

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

7. Simplified 40.4%

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

8. Final simplification 40.4%

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

Alternative 16: 40.1% 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 54.3%

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

2. Taylor expanded in x around inf 59.1%

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

   1. log-rec 59.1%

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

   2. mul-1-neg 59.1%

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

   3. associate-*r/ 59.1%

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

   4. neg-mul-1 59.1%

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

   5. mul-1-neg 59.1%

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

   6. remove-double-neg 59.1%

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

   7. *-commutative 59.1%

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

4. Simplified 59.1%

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

   1. *-un-lft-identity 59.1%

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

   2. div-inv 59.1%

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

   3. pow-to-exp 59.1%

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

6. Applied egg-rr 59.1%

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

   1. *-lft-identity 59.1%

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

8. Simplified 59.1%

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

9. Taylor expanded in n around inf 40.4%

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

    1. associate-/r* 40.8%

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

11. Simplified 40.8%

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

12. Final simplification 40.8%

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

Alternative 17: 3.2% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

2. Taylor expanded in x around inf 59.1%

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

   1. log-rec 59.1%

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

   2. mul-1-neg 59.1%

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

   3. associate-*r/ 59.1%

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

   4. neg-mul-1 59.1%

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

   5. mul-1-neg 59.1%

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

   6. remove-double-neg 59.1%

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

   7. *-commutative 59.1%

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

4. Simplified 59.1%

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

   1. *-un-lft-identity 59.1%

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

   2. div-inv 59.1%

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

   3. pow-to-exp 59.1%

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

6. Applied egg-rr 59.1%

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

   1. *-lft-identity 59.1%

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

8. Simplified 59.1%

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

9. Taylor expanded in n around inf 40.4%

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

    1. associate-/r* 40.8%

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

11. Simplified 40.8%

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

    1. associate-/l/ 40.4%

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

    2. add-exp-log 15.9%

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

    3. exp-neg 15.9%

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

    4. neg-sub0 15.9%

       \[\leadsto e^{\color{blue}{0 - \log \left(x \cdot n\right)}} \]

    5. metadata-eval 15.9%

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

    6. log-prod 16.2%

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

    7. associate--r+ 16.2%

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

    8. metadata-eval 16.2%

       \[\leadsto e^{\left(\color{blue}{0} - \log x\right) - \log n} \]

    9. neg-sub0 16.2%

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

    10. add-sqr-sqrt 4.1%

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

    11. sqrt-unprod 4.6%

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

    12. sqr-neg 4.6%

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

    13. sqrt-unprod 0.5%

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

    14. add-sqr-sqrt 2.1%

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

    15. log-div 2.5%

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

    16. add-exp-log 4.4%

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

    17. add-log-exp 23.3%

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

    18. *-un-lft-identity 23.3%

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

    19. log-prod 23.3%

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

    20. metadata-eval 23.3%

        \[\leadsto \color{blue}{0} + \log \left(e^{\frac{x}{n}}\right) \]

    21. add-log-exp 4.4%

        \[\leadsto 0 + \color{blue}{\frac{x}{n}} \]

    22. add-exp-log 2.5%

        \[\leadsto 0 + \color{blue}{e^{\log \left(\frac{x}{n}\right)}} \]

    23. log-div 2.1%

        \[\leadsto 0 + e^{\color{blue}{\log x - \log n}} \]

13. Applied egg-rr 3.2%

    \[\leadsto \color{blue}{0 + x \cdot n} \]
14. Step-by-step derivation

    1. +-lft-identity 3.2%

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

15. Simplified 3.2%

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

16. Final simplification 3.2%

    \[\leadsto n \cdot x \]

Alternative 18: 10.4% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

2. Taylor expanded in x around inf 59.1%

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

   1. log-rec 59.1%

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

   2. mul-1-neg 59.1%

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

   3. associate-*r/ 59.1%

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

   4. neg-mul-1 59.1%

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

   5. mul-1-neg 59.1%

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

   6. remove-double-neg 59.1%

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

   7. *-commutative 59.1%

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

4. Simplified 59.1%

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

   1. *-un-lft-identity 59.1%

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

   2. div-inv 59.1%

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

   3. pow-to-exp 59.1%

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

6. Applied egg-rr 59.1%

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

   1. *-lft-identity 59.1%

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

8. Simplified 59.1%

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

9. Taylor expanded in n around inf 40.4%

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

    1. associate-/r* 40.8%

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

11. Simplified 40.8%

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

    1. associate-/l/ 40.4%

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

    2. add-exp-log 15.9%

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

    3. exp-neg 15.9%

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

    4. add-sqr-sqrt 3.6%

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

    5. sqrt-unprod 4.1%

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

    6. sqr-neg 4.1%

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

    7. sqrt-unprod 0.5%

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

    8. add-sqr-sqrt 1.5%

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

    9. add-exp-log 3.2%

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

    10. *-commutative 3.2%

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

    11. add-exp-log 3.2%

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

    12. add-sqr-sqrt 1.1%

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

    13. sqrt-unprod 3.0%

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

    14. sqr-neg 3.0%

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

    15. sqrt-unprod 1.9%

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

    16. add-sqr-sqrt 10.7%

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

    17. rec-exp 10.7%

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

    18. add-exp-log 10.7%

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

    19. div-inv 10.7%

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

    20. add-cube-cbrt 10.7%

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

    21. *-un-lft-identity 10.7%

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

    22. times-frac 10.7%

        \[\leadsto \color{blue}{\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{1} \cdot \frac{\sqrt[3]{n}}{x}} \]

13. Applied egg-rr 10.7%

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{n}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{n}}{x}} \]
14. Step-by-step derivation

    1. /-rgt-identity 10.7%

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

    2. associate-*r/ 10.7%

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

    3. unpow2 10.7%

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

    4. rem-3cbrt-lft 10.7%

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

15. Simplified 10.7%

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

16. Final simplification 10.7%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))