2nthrt (problem 3.4.6)

Percentage Accurate: 53.0% → 86.1%

Time: 45.5s

Alternatives: 15

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 520000:\\ \;\;\;\;\left(\log \left(\frac{x}{x + 1}\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}\right) \cdot \frac{-1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 520000.0)
   (*
    (-
     (log (/ x (+ x 1.0)))
     (/
      (fma
       0.16666666666666666
       (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)
       (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))))
      n))
    (/ -1.0 n))
   (/ (cbrt (pow x (/ 3.0 n))) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 520000.0) {
		tmp = (log((x / (x + 1.0))) - (fma(0.16666666666666666, ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n), (0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0)))) / n)) * (-1.0 / n);
	} else {
		tmp = cbrt(pow(x, (3.0 / n))) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 520000.0)
		tmp = Float64(Float64(log(Float64(x / Float64(x + 1.0))) - Float64(fma(0.16666666666666666, Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n), Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)))) / n)) * Float64(-1.0 / n));
	else
		tmp = Float64(cbrt((x ^ Float64(3.0 / n))) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 520000.0], N[(N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2e5

   1. Initial program 35.7%

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

   3. Taylor expanded in n around -inf 82.1%

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

   5. Simplified 82.1%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. div-inv 82.0%

         \[\leadsto \color{blue}{\left(\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)\right) \cdot \frac{1}{-n}} \]

      2. associate--r+ 82.0%

         \[\leadsto \color{blue}{\left(\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right)} \cdot \frac{1}{-n} \]

      3. +-commutative 82.0%

         \[\leadsto \left(\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{\color{blue}{\frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n} + 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}}{n}\right) \cdot \frac{1}{-n} \]

      4. associate-/l* 82.0%

         \[\leadsto \left(\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{\color{blue}{0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}} + 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right) \cdot \frac{1}{-n} \]

      5. fma-define 82.0%

         \[\leadsto \left(\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{\color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}}{n}\right) \cdot \frac{1}{-n} \]

   7. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{\left(\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}\right) \cdot \frac{1}{-n}} \]
   8. Step-by-step derivation

      1. log1p-undefine 82.0%

         \[\leadsto \left(\left(\log x - \color{blue}{\log \left(1 + x\right)}\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}\right) \cdot \frac{1}{-n} \]

      2. diff-log 82.2%

         \[\leadsto \left(\color{blue}{\log \left(\frac{x}{1 + x}\right)} - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}\right) \cdot \frac{1}{-n} \]

   9. Applied egg-rr 82.2%

      \[\leadsto \left(\color{blue}{\log \left(\frac{x}{1 + x}\right)} - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}\right) \cdot \frac{1}{-n} \]
   10. Step-by-step derivation

       1. +-commutative 82.2%

          \[\leadsto \left(\log \left(\frac{x}{\color{blue}{x + 1}}\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}\right) \cdot \frac{1}{-n} \]

   11. Simplified 82.2%

       \[\leadsto \left(\color{blue}{\log \left(\frac{x}{x + 1}\right)} - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}\right) \cdot \frac{1}{-n} \]

   if 5.2e5 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. log-rec 98.8%

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

      3. mul-1-neg 98.8%

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

      4. distribute-neg-frac 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-commutative 98.8%

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

   5. Simplified 98.8%

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

      1. div-inv 98.7%

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

      2. pow-to-exp 98.7%

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

      3. add-cbrt-cube 98.7%

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

      4. pow1/3 98.7%

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

      5. pow3 98.7%

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

      6. pow-pow 98.8%

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

   7. Applied egg-rr 98.8%

      \[\leadsto \frac{\color{blue}{{\left({x}^{\left(\frac{1}{n} \cdot 3\right)}\right)}^{0.3333333333333333}}}{x \cdot n} \]
   8. Step-by-step derivation

      1. unpow1/3 98.8%

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

      2. associate-*l/ 98.8%

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

      3. metadata-eval 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 520000:\\ \;\;\;\;\left(\log \left(\frac{x}{x + 1}\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}\right) \cdot \frac{-1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 520000:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 520000.0)
   (/
    (-
     (+
      (log1p x)
      (/
       (+
        (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
        (/
         (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
         n))
       n))
     (log x))
    n)
   (/ (cbrt (pow x (/ 3.0 n))) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 520000.0) {
		tmp = ((log1p(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) + ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n)) / n)) - log(x)) / n;
	} else {
		tmp = cbrt(pow(x, (3.0 / n))) / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 520000.0) {
		tmp = ((Math.log1p(x) + (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) + ((0.16666666666666666 * (Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0))) / n)) / n)) - Math.log(x)) / n;
	} else {
		tmp = Math.cbrt(Math.pow(x, (3.0 / n))) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 520000.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) + Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n)) / n)) - log(x)) / n);
	else
		tmp = Float64(cbrt((x ^ Float64(3.0 / n))) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 520000.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2e5

   1. Initial program 35.7%

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

   3. Taylor expanded in n around -inf 82.1%

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

   5. Simplified 82.1%

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

   if 5.2e5 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. log-rec 98.8%

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

      3. mul-1-neg 98.8%

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

      4. distribute-neg-frac 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-commutative 98.8%

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

   5. Simplified 98.8%

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

      1. div-inv 98.7%

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

      2. pow-to-exp 98.7%

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

      3. add-cbrt-cube 98.7%

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

      4. pow1/3 98.7%

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

      5. pow3 98.7%

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

      6. pow-pow 98.8%

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

   7. Applied egg-rr 98.8%

      \[\leadsto \frac{\color{blue}{{\left({x}^{\left(\frac{1}{n} \cdot 3\right)}\right)}^{0.3333333333333333}}}{x \cdot n} \]
   8. Step-by-step derivation

      1. unpow1/3 98.8%

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

      2. associate-*l/ 98.8%

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

      3. metadata-eval 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 520000:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{elif}\;x \leq 1850000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - 2 \cdot \log \left(\sqrt{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.3e-14)
   (/
    (-
     (/
      (+
       (* -0.16666666666666666 (/ (pow (log x) 3.0) n))
       (* (pow (log x) 2.0) -0.5))
      n)
     (log x))
    n)
   (if (<= x 1850000.0)
     (/ (- (log1p x) (* 2.0 (log (sqrt x)))) n)
     (/ (cbrt (pow x (/ 3.0 n))) (* x n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.3e-14) {
		tmp = ((((-0.16666666666666666 * (pow(log(x), 3.0) / n)) + (pow(log(x), 2.0) * -0.5)) / n) - log(x)) / n;
	} else if (x <= 1850000.0) {
		tmp = (log1p(x) - (2.0 * log(sqrt(x)))) / n;
	} else {
		tmp = cbrt(pow(x, (3.0 / n))) / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.3e-14) {
		tmp = ((((-0.16666666666666666 * (Math.pow(Math.log(x), 3.0) / n)) + (Math.pow(Math.log(x), 2.0) * -0.5)) / n) - Math.log(x)) / n;
	} else if (x <= 1850000.0) {
		tmp = (Math.log1p(x) - (2.0 * Math.log(Math.sqrt(x)))) / n;
	} else {
		tmp = Math.cbrt(Math.pow(x, (3.0 / n))) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.3e-14)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) / n)) + Float64((log(x) ^ 2.0) * -0.5)) / n) - log(x)) / n);
	elseif (x <= 1850000.0)
		tmp = Float64(Float64(log1p(x) - Float64(2.0 * log(sqrt(x)))) / n);
	else
		tmp = Float64(cbrt((x ^ Float64(3.0 / n))) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.3e-14], N[(N[(N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1850000.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[(2.0 * N[Log[N[Sqrt[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.29999999999999998e-14

   1. Initial program 38.8%

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

   3. Taylor expanded in x around 0 38.8%

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

      1. *-rgt-identity 38.8%

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

      2. associate-/l* 38.8%

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

      3. exp-to-pow 38.8%

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

   5. Simplified 38.8%

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

   6. Taylor expanded in n around -inf 82.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} - 0.5 \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n}} \]
   7. Step-by-step derivation

      1. mul-1-neg 82.4%

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

   8. Simplified 82.4%

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

   if 2.29999999999999998e-14 < x < 1.85e6

   1. Initial program 8.8%

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

   3. Taylor expanded in n around inf 79.6%

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

      1. log1p-define 79.6%

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

   5. Simplified 79.6%

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

      1. add-sqr-sqrt 79.3%

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

      2. log-prod 79.8%

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

   7. Applied egg-rr 79.8%

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

      1. count-2 79.8%

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

   9. Simplified 79.8%

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

   if 1.85e6 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. log-rec 98.8%

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

      3. mul-1-neg 98.8%

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

      4. distribute-neg-frac 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. *-commutative 98.8%

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

   5. Simplified 98.8%

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

      1. div-inv 98.7%

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

      2. pow-to-exp 98.7%

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

      3. add-cbrt-cube 98.7%

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

      4. pow1/3 98.7%

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

      5. pow3 98.7%

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

      6. pow-pow 98.8%

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

   7. Applied egg-rr 98.8%

      \[\leadsto \frac{\color{blue}{{\left({x}^{\left(\frac{1}{n} \cdot 3\right)}\right)}^{0.3333333333333333}}}{x \cdot n} \]
   8. Step-by-step derivation

      1. unpow1/3 98.8%

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

      2. associate-*l/ 98.8%

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

      3. metadata-eval 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{elif}\;x \leq 1850000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - 2 \cdot \log \left(\sqrt{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.72)
   (* x (+ (/ 1.0 n) (/ (log (/ 1.0 x)) (* x n))))
   (/ (cbrt (pow x (/ 3.0 n))) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = x * ((1.0 / n) + (log((1.0 / x)) / (x * n)));
	} else {
		tmp = cbrt(pow(x, (3.0 / n))) / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = x * ((1.0 / n) + (Math.log((1.0 / x)) / (x * n)));
	} else {
		tmp = Math.cbrt(Math.pow(x, (3.0 / n))) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.72)
		tmp = Float64(x * Float64(Float64(1.0 / n) + Float64(log(Float64(1.0 / x)) / Float64(x * n))));
	else
		tmp = Float64(cbrt((x ^ Float64(3.0 / n))) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.72], N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. log1p-define 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around 0 59.5%

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

      1. neg-mul-1 59.5%

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

      2. distribute-neg-frac 59.5%

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

      3. log-rec 59.5%

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

      4. +-commutative 59.5%

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

      5. log-rec 59.5%

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

      6. distribute-neg-frac 59.5%

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

      7. unsub-neg 59.5%

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

   8. Simplified 59.5%

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

   9. Taylor expanded in x around inf 75.9%

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

   if 0.71999999999999997 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. log-rec 95.6%

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

      3. mul-1-neg 95.6%

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

      4. distribute-neg-frac 95.6%

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

      5. mul-1-neg 95.6%

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

      6. remove-double-neg 95.6%

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

      7. *-commutative 95.6%

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

   5. Simplified 95.6%

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

      1. div-inv 95.6%

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

      2. pow-to-exp 95.6%

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

      3. add-cbrt-cube 95.6%

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

      4. pow1/3 95.6%

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

      5. pow3 95.6%

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

      6. pow-pow 95.6%

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

   7. Applied egg-rr 95.6%

      \[\leadsto \frac{\color{blue}{{\left({x}^{\left(\frac{1}{n} \cdot 3\right)}\right)}^{0.3333333333333333}}}{x \cdot n} \]
   8. Step-by-step derivation

      1. unpow1/3 95.6%

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

      2. associate-*l/ 95.7%

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

      3. metadata-eval 95.7%

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

   9. Simplified 95.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left({x}^{\left(\frac{1}{n}\right)} + -1\right)}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.72)
   (* x (+ (/ 1.0 n) (/ (log (/ 1.0 x)) (* x n))))
   (/ (+ 1.0 (+ (pow x (/ 1.0 n)) -1.0)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = x * ((1.0 / n) + (log((1.0 / x)) / (x * n)));
	} else {
		tmp = (1.0 + (pow(x, (1.0 / n)) + -1.0)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = x * ((1.0d0 / n) + (log((1.0d0 / x)) / (x * n)))
    else
        tmp = (1.0d0 + ((x ** (1.0d0 / n)) + (-1.0d0))) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = x * ((1.0 / n) + (Math.log((1.0 / x)) / (x * n)));
	} else {
		tmp = (1.0 + (Math.pow(x, (1.0 / n)) + -1.0)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.72:
		tmp = x * ((1.0 / n) + (math.log((1.0 / x)) / (x * n)))
	else:
		tmp = (1.0 + (math.pow(x, (1.0 / n)) + -1.0)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.72)
		tmp = Float64(x * Float64(Float64(1.0 / n) + Float64(log(Float64(1.0 / x)) / Float64(x * n))));
	else
		tmp = Float64(Float64(1.0 + Float64((x ^ Float64(1.0 / n)) + -1.0)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = x * ((1.0 / n) + (log((1.0 / x)) / (x * n)));
	else
		tmp = (1.0 + ((x ^ (1.0 / n)) + -1.0)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.72], N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. log1p-define 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around 0 59.5%

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

      1. neg-mul-1 59.5%

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

      2. distribute-neg-frac 59.5%

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

      3. log-rec 59.5%

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

      4. +-commutative 59.5%

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

      5. log-rec 59.5%

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

      6. distribute-neg-frac 59.5%

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

      7. unsub-neg 59.5%

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

   8. Simplified 59.5%

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

   9. Taylor expanded in x around inf 75.9%

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

   if 0.71999999999999997 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. log-rec 95.6%

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

      3. mul-1-neg 95.6%

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

      4. distribute-neg-frac 95.6%

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

      5. mul-1-neg 95.6%

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

      6. remove-double-neg 95.6%

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

      7. *-commutative 95.6%

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

   5. Simplified 95.6%

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

      1. div-inv 95.6%

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

      2. pow-to-exp 95.6%

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

      3. expm1-log1p-u 95.7%

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

      4. expm1-undefine 95.6%

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

      5. log1p-undefine 95.6%

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

      6. add-exp-log 95.6%

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

   7. Applied egg-rr 95.6%

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

      1. associate--l+ 95.6%

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

   9. Simplified 95.6%

      \[\leadsto \frac{\color{blue}{1 + \left({x}^{\left(\frac{1}{n}\right)} - 1\right)}}{x \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left({x}^{\left(\frac{1}{n}\right)} + -1\right)}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot -0.5\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left({x}^{\left(\frac{1}{n}\right)} + -1\right)}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.72)
   (/ (- (* x (+ 1.0 (* x -0.5))) (log x)) n)
   (/ (+ 1.0 (+ (pow x (/ 1.0 n)) -1.0)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = ((x * (1.0 + (x * -0.5))) - log(x)) / n;
	} else {
		tmp = (1.0 + (pow(x, (1.0 / n)) + -1.0)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = ((x * (1.0d0 + (x * (-0.5d0)))) - log(x)) / n
    else
        tmp = (1.0d0 + ((x ** (1.0d0 / n)) + (-1.0d0))) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = ((x * (1.0 + (x * -0.5))) - Math.log(x)) / n;
	} else {
		tmp = (1.0 + (Math.pow(x, (1.0 / n)) + -1.0)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.72:
		tmp = ((x * (1.0 + (x * -0.5))) - math.log(x)) / n
	else:
		tmp = (1.0 + (math.pow(x, (1.0 / n)) + -1.0)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.72)
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64((x ^ Float64(1.0 / n)) + -1.0)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = ((x * (1.0 + (x * -0.5))) - log(x)) / n;
	else
		tmp = (1.0 + ((x ^ (1.0 / n)) + -1.0)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.72], N[(N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. log1p-define 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around 0 59.7%

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

      1. *-commutative 59.7%

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

   8. Simplified 59.7%

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

   if 0.71999999999999997 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. log-rec 95.6%

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

      3. mul-1-neg 95.6%

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

      4. distribute-neg-frac 95.6%

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

      5. mul-1-neg 95.6%

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

      6. remove-double-neg 95.6%

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

      7. *-commutative 95.6%

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

   5. Simplified 95.6%

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

      1. div-inv 95.6%

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

      2. pow-to-exp 95.6%

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

      3. expm1-log1p-u 95.7%

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

      4. expm1-undefine 95.6%

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

      5. log1p-undefine 95.6%

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

      6. add-exp-log 95.6%

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

   7. Applied egg-rr 95.6%

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

      1. associate--l+ 95.6%

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

   9. Simplified 95.6%

      \[\leadsto \frac{\color{blue}{1 + \left({x}^{\left(\frac{1}{n}\right)} - 1\right)}}{x \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot -0.5\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left({x}^{\left(\frac{1}{n}\right)} + -1\right)}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\frac{x \cdot \left(1 + x \cdot -0.5\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{n}{{x}^{\left(\frac{1}{n}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.72)
   (/ (- (* x (+ 1.0 (* x -0.5))) (log x)) n)
   (/ 1.0 (* x (/ n (pow x (/ 1.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = ((x * (1.0 + (x * -0.5))) - log(x)) / n;
	} else {
		tmp = 1.0 / (x * (n / pow(x, (1.0 / n))));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = ((x * (1.0d0 + (x * (-0.5d0)))) - log(x)) / n
    else
        tmp = 1.0d0 / (x * (n / (x ** (1.0d0 / n))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = ((x * (1.0 + (x * -0.5))) - Math.log(x)) / n;
	} else {
		tmp = 1.0 / (x * (n / Math.pow(x, (1.0 / n))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.72:
		tmp = ((x * (1.0 + (x * -0.5))) - math.log(x)) / n
	else:
		tmp = 1.0 / (x * (n / math.pow(x, (1.0 / n))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.72)
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) - log(x)) / n);
	else
		tmp = Float64(1.0 / Float64(x * Float64(n / (x ^ Float64(1.0 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = ((x * (1.0 + (x * -0.5))) - log(x)) / n;
	else
		tmp = 1.0 / (x * (n / (x ^ (1.0 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.72], N[(N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(1.0 / N[(x * N[(n / N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. log1p-define 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around 0 59.7%

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

      1. *-commutative 59.7%

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

   8. Simplified 59.7%

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

   if 0.71999999999999997 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. log-rec 95.6%

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

      3. mul-1-neg 95.6%

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

      4. distribute-neg-frac 95.6%

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

      5. mul-1-neg 95.6%

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

      6. remove-double-neg 95.6%

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

      7. *-commutative 95.6%

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

   5. Simplified 95.6%

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

      1. clear-num 95.6%

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

      2. inv-pow 95.6%

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

      3. div-inv 95.6%

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

      4. pow-to-exp 95.6%

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

   7. Applied egg-rr 95.6%

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

      1. unpow-1 95.6%

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

      2. associate-/l* 95.6%

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

   9. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \frac{n}{{x}^{\left(\frac{1}{n}\right)}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{n}{{x}^{\left(\frac{1}{n}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.72) (/ (- x (log x)) n) (/ 1.0 (* x (/ n (pow x (/ 1.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = 1.0 / (x * (n / pow(x, (1.0 / n))));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = (x - log(x)) / n
    else
        tmp = 1.0d0 / (x * (n / (x ** (1.0d0 / n))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = 1.0 / (x * (n / Math.pow(x, (1.0 / n))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.72:
		tmp = (x - math.log(x)) / n
	else:
		tmp = 1.0 / (x * (n / math.pow(x, (1.0 / n))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.72)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(1.0 / Float64(x * Float64(n / (x ^ Float64(1.0 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = (x - log(x)) / n;
	else
		tmp = 1.0 / (x * (n / (x ^ (1.0 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.72], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(1.0 / N[(x * N[(n / N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. log1p-define 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around 0 59.5%

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

   if 0.71999999999999997 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. log-rec 95.6%

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

      3. mul-1-neg 95.6%

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

      4. distribute-neg-frac 95.6%

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

      5. mul-1-neg 95.6%

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

      6. remove-double-neg 95.6%

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

      7. *-commutative 95.6%

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

   5. Simplified 95.6%

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

      1. clear-num 95.6%

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

      2. inv-pow 95.6%

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

      3. div-inv 95.6%

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

      4. pow-to-exp 95.6%

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

   7. Applied egg-rr 95.6%

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

      1. unpow-1 95.6%

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

      2. associate-/l* 95.6%

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

   9. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \frac{n}{{x}^{\left(\frac{1}{n}\right)}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.72) (/ (- x (log x)) n) (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.72d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (x ** (1.0d0 / n)) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.72) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = Math.pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.72:
		tmp = (x - math.log(x)) / n
	else:
		tmp = math.pow(x, (1.0 / n)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.72)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.72)
		tmp = (x - log(x)) / n;
	else
		tmp = (x ^ (1.0 / n)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.72], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. log1p-define 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around 0 59.5%

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

   if 0.71999999999999997 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. log-rec 95.6%

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

      3. distribute-frac-neg 95.6%

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

      4. remove-double-neg 95.6%

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

      5. *-rgt-identity 95.6%

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

      6. associate-/l* 95.6%

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

      7. exp-to-pow 95.6%

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

      8. *-commutative 95.6%

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

   5. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.900000000000000022

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. log1p-define 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around 0 59.5%

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

   if 0.900000000000000022 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in n around inf 67.1%

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

      1. log1p-define 67.1%

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

   5. Simplified 67.1%

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

   6. Taylor expanded in x around -inf 74.9%

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

4. Final simplification 66.6%

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

Alternative 11: 56.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 37.6%

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

   3. Taylor expanded in x around 0 37.4%

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

      1. *-rgt-identity 37.4%

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

      2. associate-/l* 37.4%

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

      3. exp-to-pow 37.4%

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

   5. Simplified 37.4%

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

   6. Taylor expanded in n around inf 58.7%

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

      1. associate-*r/ 58.7%

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

      2. neg-mul-1 58.7%

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

   8. Simplified 58.7%

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

   if 0.69999999999999996 < x

   1. Initial program 62.0%

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

   3. Taylor expanded in n around inf 67.4%

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

      1. log1p-define 67.4%

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

   5. Simplified 67.4%

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

   6. Taylor expanded in x around -inf 74.4%

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

4. Final simplification 66.0%

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

Alternative 12: 45.6% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in n around inf 63.5%

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

   1. log1p-define 63.5%

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

5. Simplified 63.5%

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

6. Taylor expanded in x around -inf 50.1%

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

   1. mul-1-neg 50.1%

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

   2. mul-1-neg 50.1%

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

   3. associate-*r/ 50.1%

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

   4. metadata-eval 50.1%

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

   5. *-commutative 50.1%

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

   6. associate-*r/ 50.1%

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

   7. metadata-eval 50.1%

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

8. Simplified 50.1%

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

9. Final simplification 50.1%

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

Alternative 13: 39.9% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in x around inf 58.2%

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

   1. mul-1-neg 58.2%

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

   2. log-rec 58.2%

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

   3. mul-1-neg 58.2%

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

   4. distribute-neg-frac 58.2%

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

   5. mul-1-neg 58.2%

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

   6. remove-double-neg 58.2%

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

   7. *-commutative 58.2%

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

5. Simplified 58.2%

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

6. Taylor expanded in n around inf 45.9%

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

   1. *-commutative 45.9%

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

   2. associate-/r* 46.1%

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

8. Simplified 46.1%

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

Alternative 14: 39.3% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in x around inf 58.2%

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

   1. mul-1-neg 58.2%

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

   2. log-rec 58.2%

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

   3. mul-1-neg 58.2%

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

   4. distribute-neg-frac 58.2%

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

   5. mul-1-neg 58.2%

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

   6. remove-double-neg 58.2%

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

   7. *-commutative 58.2%

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

5. Simplified 58.2%

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

6. Taylor expanded in n around inf 45.9%

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

Alternative 15: 4.6% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in x around inf 58.2%

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

   1. mul-1-neg 58.2%

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

   2. log-rec 58.2%

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

   3. mul-1-neg 58.2%

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

   4. distribute-neg-frac 58.2%

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

   5. mul-1-neg 58.2%

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

   6. remove-double-neg 58.2%

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

   7. *-commutative 58.2%

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

5. Simplified 58.2%

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

6. Taylor expanded in n around inf 45.9%

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

   1. associate-/r* 46.1%

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

   2. div-inv 46.1%

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

   3. add-exp-log 45.3%

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

   4. add-sqr-sqrt 13.1%

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

   5. sqrt-unprod 14.9%

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

   6. log-rec 14.9%

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

   7. log-rec 14.9%

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

   8. sqr-neg 14.9%

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

   9. sqrt-unprod 1.9%

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

   10. add-sqr-sqrt 5.1%

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

   11. add-exp-log 5.1%

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

8. Applied egg-rr 5.1%

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

   1. associate-*r/ 5.1%

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

   2. *-rgt-identity 5.1%

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

10. Simplified 5.1%

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

Reproduce

herbie shell --seed 2024132 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))