2nthrt (problem 3.4.6)

Percentage Accurate: 54.4% → 86.9%

Time: 26.6s

Alternatives: 14

Speedup: 1.9×

• could not determine a ground truth

  1. x = -2.9606585483957903e+296
  2. n = -2.4056081506525054e+300

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: 54.4% 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: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1500:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1500.0)
   (/
    (+
     (/
      (fma
       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)
     (- (log1p x) (log x)))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1500.0) {
		tmp = ((fma(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) + (log1p(x) - log(x))) / n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1500.0)
		tmp = Float64(Float64(Float64(fma(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) + Float64(log1p(x) - log(x))) / n);
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1500.0], N[(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] + 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] + N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1500

   1. Initial program 41.6%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3} - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{0 - n}} \]

   if 1500 < x

   1. Initial program 69.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 69.0

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 97.4

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

   7. Simplified 97.4%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. pow-flip N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. pow-lowering-pow.f64 N/A

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

      9. /-lowering-/.f64 99.3

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1500:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 245:\\ \;\;\;\;\frac{1}{n} \cdot \left(\frac{\mathsf{fma}\left(0.5, \log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right), \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}\right)}{n} - \log \left(\frac{x}{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 245.0)
   (*
    (/ 1.0 n)
    (-
     (/
      (fma
       0.5
       (* (log (* x (+ x 1.0))) (log (/ (+ x 1.0) x)))
       (* (- (pow (log1p x) 3.0) (pow (log x) 3.0)) (/ 0.16666666666666666 n)))
      n)
     (log (/ x (+ x 1.0)))))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 245.0) {
		tmp = (1.0 / n) * ((fma(0.5, (log((x * (x + 1.0))) * log(((x + 1.0) / x))), ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) * (0.16666666666666666 / n))) / n) - log((x / (x + 1.0))));
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 245.0)
		tmp = Float64(Float64(1.0 / n) * Float64(Float64(fma(0.5, Float64(log(Float64(x * Float64(x + 1.0))) * log(Float64(Float64(x + 1.0) / x))), Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) * Float64(0.16666666666666666 / n))) / n) - log(Float64(x / Float64(x + 1.0)))));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 245.0], N[(N[(1.0 / n), $MachinePrecision] * N[(N[(N[(0.5 * N[(N[Log[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 245

   1. Initial program 41.6%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3} - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}{0 - n}} \]

   7. Applied egg-rr 78.1%

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

   if 245 < x

   1. Initial program 69.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 69.0

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 97.4

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

   7. Simplified 97.4%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. pow-flip N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. pow-lowering-pow.f64 N/A

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

      9. /-lowering-/.f64 99.3

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 245:\\ \;\;\;\;\frac{1}{n} \cdot \left(\frac{\mathsf{fma}\left(0.5, \log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right), \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}\right)}{n} - \log \left(\frac{x}{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.46:\\ \;\;\;\;\frac{\log x + \frac{\mathsf{fma}\left(0.5, {\log x}^{2}, {\log x}^{3} \cdot \frac{0.16666666666666666}{n}\right)}{n}}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.46)
   (/
    (+
     (log x)
     (/
      (fma
       0.5
       (pow (log x) 2.0)
       (* (pow (log x) 3.0) (/ 0.16666666666666666 n)))
      n))
    (- 0.0 n))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.46) {
		tmp = (log(x) + (fma(0.5, pow(log(x), 2.0), (pow(log(x), 3.0) * (0.16666666666666666 / n))) / n)) / (0.0 - n);
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.46)
		tmp = Float64(Float64(log(x) + Float64(fma(0.5, (log(x) ^ 2.0), Float64((log(x) ^ 3.0) * Float64(0.16666666666666666 / n))) / n)) / Float64(0.0 - n));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.46], N[(N[(N[Log[x], $MachinePrecision] + N[(N[(0.5 * N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] * N[(0.16666666666666666 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.46000000000000002

   1. Initial program 41.6%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 40.9

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

   5. Simplified 40.9%

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

   6. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{24} \cdot \frac{{\log x}^{4}}{n} - \frac{-1}{6} \cdot {\log x}^{3}}{n} - \frac{1}{2} \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n}} \]

   8. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(0.041666666666666664, \frac{{\log x}^{4}}{n}, {\log x}^{3} \cdot 0.16666666666666666\right)}{-n} + -0.5 \cdot {\log x}^{2}}{-n} + \log x}{-n}} \]

   9. Taylor expanded in n around inf

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{6} \cdot \frac{{\log x}^{3}}{n} + \frac{1}{2} \cdot {\log x}^{2}}{n}} + \log x}{\mathsf{neg}\left(n\right)} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{6} \cdot \frac{{\log x}^{3}}{n} + \frac{1}{2} \cdot {\log x}^{2}}{n}} + \log x}{\mathsf{neg}\left(n\right)} \]

       2. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} \cdot {\log x}^{2} + \frac{1}{6} \cdot \frac{{\log x}^{3}}{n}}}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, \frac{1}{6} \cdot \frac{{\log x}^{3}}{n}\right)}}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{{\log x}^{2}}, \frac{1}{6} \cdot \frac{{\log x}^{3}}{n}\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       5. log-lowering-log.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\color{blue}{\log x}}^{2}, \frac{1}{6} \cdot \frac{{\log x}^{3}}{n}\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       6. associate-*r/ N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, \color{blue}{\frac{\frac{1}{6} \cdot {\log x}^{3}}{n}}\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, \frac{\color{blue}{{\log x}^{3} \cdot \frac{1}{6}}}{n}\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       8. associate-/l* N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, \color{blue}{{\log x}^{3} \cdot \frac{\frac{1}{6}}{n}}\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       9. metadata-eval N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, {\log x}^{3} \cdot \frac{\color{blue}{\frac{1}{6} \cdot 1}}{n}\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       10. associate-*r/ N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, {\log x}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{n}\right)}\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, \color{blue}{{\log x}^{3} \cdot \left(\frac{1}{6} \cdot \frac{1}{n}\right)}\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       12. pow-lowering-pow.f64 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, \color{blue}{{\log x}^{3}} \cdot \left(\frac{1}{6} \cdot \frac{1}{n}\right)\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       13. log-lowering-log.f64 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, {\color{blue}{\log x}}^{3} \cdot \left(\frac{1}{6} \cdot \frac{1}{n}\right)\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       14. associate-*r/ N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, {\log x}^{3} \cdot \color{blue}{\frac{\frac{1}{6} \cdot 1}{n}}\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       15. metadata-eval N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{1}{2}, {\log x}^{2}, {\log x}^{3} \cdot \frac{\color{blue}{\frac{1}{6}}}{n}\right)}{n} + \log x}{\mathsf{neg}\left(n\right)} \]

       16. /-lowering-/.f64 77.8

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

   11. Simplified 77.8%

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

   if 0.46000000000000002 < x

   1. Initial program 69.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 69.0

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 97.4

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

   7. Simplified 97.4%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. pow-flip N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. pow-lowering-pow.f64 N/A

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

      9. /-lowering-/.f64 99.3

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.46:\\ \;\;\;\;\frac{\log x + \frac{\mathsf{fma}\left(0.5, {\log x}^{2}, {\log x}^{3} \cdot \frac{0.16666666666666666}{n}\right)}{n}}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.1 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\log x, \frac{-1}{x \cdot n}, \frac{1}{n}\right)\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 1.1e-83)
     (* x (fma (log x) (/ -1.0 (* x n)) (/ 1.0 n)))
     (if (<= x 0.88) (- (exp (/ (log1p x) n)) t_0) (/ (/ t_0 n) x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.1e-83) {
		tmp = x * fma(log(x), (-1.0 / (x * n)), (1.0 / n));
	} else if (x <= 0.88) {
		tmp = exp((log1p(x) / n)) - t_0;
	} 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 <= 1.1e-83)
		tmp = Float64(x * fma(log(x), Float64(-1.0 / Float64(x * n)), Float64(1.0 / n)));
	elseif (x <= 0.88)
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.1e-83], N[(x * N[(N[Log[x], $MachinePrecision] * N[(-1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.10000000000000004e-83

   1. Initial program 44.5%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 44.8

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

   5. Simplified 44.8%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 53.7

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

   8. Simplified 53.7%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. log-rec N/A

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

       3. distribute-frac-neg N/A

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

       4. unsub-neg N/A

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

       5. --lowering--.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. log-lowering-log.f64 N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 78.3

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

   11. Simplified 78.3%

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

       1. sub-neg N/A

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

       2. +-commutative N/A

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

       3. div-inv N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\log x, \mathsf{neg}\left(\frac{1}{x \cdot n}\right), \frac{1}{n}\right)} \]

       6. log-lowering-log.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\log x}, \mathsf{neg}\left(\frac{1}{x \cdot n}\right), \frac{1}{n}\right) \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\log x, \color{blue}{\mathsf{neg}\left(\frac{1}{x \cdot n}\right)}, \frac{1}{n}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\log x, \mathsf{neg}\left(\color{blue}{\frac{1}{x \cdot n}}\right), \frac{1}{n}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\log x, \mathsf{neg}\left(\frac{1}{\color{blue}{x \cdot n}}\right), \frac{1}{n}\right) \]

       10. /-lowering-/.f64 78.3

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

   13. Applied egg-rr 78.3%

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

   if 1.10000000000000004e-83 < x < 0.880000000000000004

   1. Initial program 33.4%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 66.1

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

   4. Applied egg-rr 66.1%

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

   if 0.880000000000000004 < x

   1. Initial program 69.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 69.0

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 97.4

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

   7. Simplified 97.4%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. pow-flip N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. pow-lowering-pow.f64 N/A

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

      9. /-lowering-/.f64 99.3

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\log x, \frac{-1}{x \cdot n}, \frac{1}{n}\right)\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\log x, \frac{-1}{x \cdot n}, \frac{1}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.35)
   (* x (fma (log x) (/ -1.0 (* x n)) (/ 1.0 n)))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.35) {
		tmp = x * fma(log(x), (-1.0 / (x * n)), (1.0 / n));
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.35)
		tmp = Float64(x * fma(log(x), Float64(-1.0 / Float64(x * n)), Float64(1.0 / n)));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.35], N[(x * N[(N[Log[x], $MachinePrecision] * N[(-1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.34999999999999998

   1. Initial program 41.6%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 41.3

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

   5. Simplified 41.3%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 49.9

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

   8. Simplified 49.9%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. log-rec N/A

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

       3. distribute-frac-neg N/A

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

       4. unsub-neg N/A

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

       5. --lowering--.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. log-lowering-log.f64 N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 70.7

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

   11. Simplified 70.7%

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

       1. sub-neg N/A

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

       2. +-commutative N/A

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

       3. div-inv N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\log x, \mathsf{neg}\left(\frac{1}{x \cdot n}\right), \frac{1}{n}\right)} \]

       6. log-lowering-log.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\log x}, \mathsf{neg}\left(\frac{1}{x \cdot n}\right), \frac{1}{n}\right) \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\log x, \color{blue}{\mathsf{neg}\left(\frac{1}{x \cdot n}\right)}, \frac{1}{n}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\log x, \mathsf{neg}\left(\color{blue}{\frac{1}{x \cdot n}}\right), \frac{1}{n}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\log x, \mathsf{neg}\left(\frac{1}{\color{blue}{x \cdot n}}\right), \frac{1}{n}\right) \]

       10. /-lowering-/.f64 70.7

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

   13. Applied egg-rr 70.7%

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

   if 0.34999999999999998 < x

   1. Initial program 69.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 69.0

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 97.4

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

   7. Simplified 97.4%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. pow-flip N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. pow-lowering-pow.f64 N/A

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

      9. /-lowering-/.f64 99.3

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\log x, \frac{-1}{x \cdot n}, \frac{1}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.33:\\ \;\;\;\;x \cdot \left(\frac{1}{n} - \frac{\log x}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.33)
   (* x (- (/ 1.0 n) (/ (log x) (* x n))))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.33) {
		tmp = x * ((1.0 / n) - (log(x) / (x * n)));
	} else {
		tmp = (pow(x, (1.0 / n)) / 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 <= 0.33d0) then
        tmp = x * ((1.0d0 / n) - (log(x) / (x * n)))
    else
        tmp = ((x ** (1.0d0 / n)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.33) {
		tmp = x * ((1.0 / n) - (Math.log(x) / (x * n)));
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.33:
		tmp = x * ((1.0 / n) - (math.log(x) / (x * n)))
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.33)
		tmp = Float64(x * Float64(Float64(1.0 / n) - Float64(log(x) / Float64(x * n))));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.33)
		tmp = x * ((1.0 / n) - (log(x) / (x * n)));
	else
		tmp = ((x ^ (1.0 / n)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.33], N[(x * N[(N[(1.0 / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.330000000000000016

   1. Initial program 41.6%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 41.3

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

   5. Simplified 41.3%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 49.9

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

   8. Simplified 49.9%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. log-rec N/A

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

       3. distribute-frac-neg N/A

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

       4. unsub-neg N/A

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

       5. --lowering--.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. log-lowering-log.f64 N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 70.7

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

   11. Simplified 70.7%

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

   if 0.330000000000000016 < x

   1. Initial program 69.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 69.0

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 97.4

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

   7. Simplified 97.4%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. pow-flip N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. pow-lowering-pow.f64 N/A

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

      9. /-lowering-/.f64 99.3

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

   9. Applied egg-rr 99.3%

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

Alternative 7: 77.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.33:\\ \;\;\;\;x \cdot \frac{\log x}{0 - x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.33)
   (* x (/ (log x) (- 0.0 (* x n))))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.33) {
		tmp = x * (log(x) / (0.0 - (x * n)));
	} else {
		tmp = (pow(x, (1.0 / n)) / 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 <= 0.33d0) then
        tmp = x * (log(x) / (0.0d0 - (x * n)))
    else
        tmp = ((x ** (1.0d0 / n)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.33) {
		tmp = x * (Math.log(x) / (0.0 - (x * n)));
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.33:
		tmp = x * (math.log(x) / (0.0 - (x * n)))
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.33)
		tmp = Float64(x * Float64(log(x) / Float64(0.0 - Float64(x * n))));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.33)
		tmp = x * (log(x) / (0.0 - (x * n)));
	else
		tmp = ((x ^ (1.0 / n)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.33], N[(x * N[(N[Log[x], $MachinePrecision] / N[(0.0 - N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.330000000000000016

   1. Initial program 41.6%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 41.3

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

   5. Simplified 41.3%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 49.9

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

   8. Simplified 49.9%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. log-rec N/A

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

       3. distribute-frac-neg N/A

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

       4. unsub-neg N/A

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

       5. --lowering--.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. log-lowering-log.f64 N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 70.7

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

   11. Simplified 70.7%

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

   12. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. log-lowering-log.f64 N/A

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

       5. neg-sub0 N/A

          \[\leadsto x \cdot \frac{\log x}{\color{blue}{0 - n \cdot x}} \]

       6. --lowering--.f64 N/A

          \[\leadsto x \cdot \frac{\log x}{\color{blue}{0 - n \cdot x}} \]

       7. *-lowering-*.f64 66.7

          \[\leadsto x \cdot \frac{\log x}{0 - \color{blue}{n \cdot x}} \]

   14. Simplified 66.7%

       \[\leadsto x \cdot \color{blue}{\frac{\log x}{0 - n \cdot x}} \]

   if 0.330000000000000016 < x

   1. Initial program 69.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 69.0

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 97.4

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

   7. Simplified 97.4%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. pow-flip N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. pow-lowering-pow.f64 N/A

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

      9. /-lowering-/.f64 99.3

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.33:\\ \;\;\;\;x \cdot \frac{\log x}{0 - x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.33:\\ \;\;\;\;x \cdot \frac{\log x}{0 - x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x \cdot {x}^{\left(\frac{-1}{n}\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.33)
   (* x (/ (log x) (- 0.0 (* x n))))
   (/ 1.0 (* n (* x (pow x (/ -1.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.33) {
		tmp = x * (log(x) / (0.0 - (x * n)));
	} else {
		tmp = 1.0 / (n * (x * 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 <= 0.33d0) then
        tmp = x * (log(x) / (0.0d0 - (x * n)))
    else
        tmp = 1.0d0 / (n * (x * (x ** ((-1.0d0) / n))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.33) {
		tmp = x * (Math.log(x) / (0.0 - (x * n)));
	} else {
		tmp = 1.0 / (n * (x * Math.pow(x, (-1.0 / n))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.33:
		tmp = x * (math.log(x) / (0.0 - (x * n)))
	else:
		tmp = 1.0 / (n * (x * math.pow(x, (-1.0 / n))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.33)
		tmp = Float64(x * Float64(log(x) / Float64(0.0 - Float64(x * n))));
	else
		tmp = Float64(1.0 / Float64(n * Float64(x * (x ^ Float64(-1.0 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.33)
		tmp = x * (log(x) / (0.0 - (x * n)));
	else
		tmp = 1.0 / (n * (x * (x ^ (-1.0 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.33], N[(x * N[(N[Log[x], $MachinePrecision] / N[(0.0 - N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * N[(x * N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.330000000000000016

   1. Initial program 41.6%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 41.3

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

   5. Simplified 41.3%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 49.9

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

   8. Simplified 49.9%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. log-rec N/A

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

       3. distribute-frac-neg N/A

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

       4. unsub-neg N/A

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

       5. --lowering--.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. log-lowering-log.f64 N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 70.7

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

   11. Simplified 70.7%

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

   12. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. log-lowering-log.f64 N/A

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

       5. neg-sub0 N/A

          \[\leadsto x \cdot \frac{\log x}{\color{blue}{0 - n \cdot x}} \]

       6. --lowering--.f64 N/A

          \[\leadsto x \cdot \frac{\log x}{\color{blue}{0 - n \cdot x}} \]

       7. *-lowering-*.f64 66.7

          \[\leadsto x \cdot \frac{\log x}{0 - \color{blue}{n \cdot x}} \]

   14. Simplified 66.7%

       \[\leadsto x \cdot \color{blue}{\frac{\log x}{0 - n \cdot x}} \]

   if 0.330000000000000016 < x

   1. Initial program 69.6%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 69.0

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 97.4

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

   7. Simplified 97.4%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. /-lowering-/.f64 97.4

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

   10. Simplified 97.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.33:\\ \;\;\;\;x \cdot \frac{\log x}{0 - x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x \cdot {x}^{\left(\frac{-1}{n}\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.33:\\ \;\;\;\;x \cdot \frac{\log x}{0 - x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.33)
   (* x (/ (log x) (- 0.0 (* x n))))
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.33) {
		tmp = x * (log(x) / (0.0 - (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 <= 0.33d0) then
        tmp = x * (log(x) / (0.0d0 - (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 <= 0.33) {
		tmp = x * (Math.log(x) / (0.0 - (x * n)));
	} else {
		tmp = Math.pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.33:
		tmp = x * (math.log(x) / (0.0 - (x * n)))
	else:
		tmp = math.pow(x, (1.0 / n)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.33)
		tmp = Float64(x * Float64(log(x) / Float64(0.0 - Float64(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 <= 0.33)
		tmp = x * (log(x) / (0.0 - (x * n)));
	else
		tmp = (x ^ (1.0 / n)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.33], N[(x * N[(N[Log[x], $MachinePrecision] / N[(0.0 - N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 0.330000000000000016

   1. Initial program 41.6%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 41.3

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

   5. Simplified 41.3%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 49.9

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

   8. Simplified 49.9%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. log-rec N/A

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

       3. distribute-frac-neg N/A

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

       4. unsub-neg N/A

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

       5. --lowering--.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. log-lowering-log.f64 N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 70.7

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

   11. Simplified 70.7%

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

   12. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. log-lowering-log.f64 N/A

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

       5. neg-sub0 N/A

          \[\leadsto x \cdot \frac{\log x}{\color{blue}{0 - n \cdot x}} \]

       6. --lowering--.f64 N/A

          \[\leadsto x \cdot \frac{\log x}{\color{blue}{0 - n \cdot x}} \]

       7. *-lowering-*.f64 66.7

          \[\leadsto x \cdot \frac{\log x}{0 - \color{blue}{n \cdot x}} \]

   14. Simplified 66.7%

       \[\leadsto x \cdot \color{blue}{\frac{\log x}{0 - n \cdot x}} \]

   if 0.330000000000000016 < x

   1. Initial program 69.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 97.4

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

   5. Simplified 97.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.33:\\ \;\;\;\;x \cdot \frac{\log x}{0 - x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.32:\\ \;\;\;\;x \cdot \frac{\log x}{0 - x \cdot n}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.32)
   (* x (/ (log x) (- 0.0 (* x n))))
   (if (<= x 2e+143) (/ 1.0 (* x n)) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.32) {
		tmp = x * (log(x) / (0.0 - (x * n)));
	} else if (x <= 2e+143) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.32d0) then
        tmp = x * (log(x) / (0.0d0 - (x * n)))
    else if (x <= 2d+143) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.32) {
		tmp = x * (Math.log(x) / (0.0 - (x * n)));
	} else if (x <= 2e+143) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.32:
		tmp = x * (math.log(x) / (0.0 - (x * n)))
	elif x <= 2e+143:
		tmp = 1.0 / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.32)
		tmp = Float64(x * Float64(log(x) / Float64(0.0 - Float64(x * n))));
	elseif (x <= 2e+143)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.32)
		tmp = x * (log(x) / (0.0 - (x * n)));
	elseif (x <= 2e+143)
		tmp = 1.0 / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.32], N[(x * N[(N[Log[x], $MachinePrecision] / N[(0.0 - N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+143], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.320000000000000007

   1. Initial program 41.1%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 41.5

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

   5. Simplified 41.5%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 50.2

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

   8. Simplified 50.2%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. log-rec N/A

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

       3. distribute-frac-neg N/A

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

       4. unsub-neg N/A

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

       5. --lowering--.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. log-lowering-log.f64 N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 71.2

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

   11. Simplified 71.2%

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

   12. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. log-lowering-log.f64 N/A

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

       5. neg-sub0 N/A

          \[\leadsto x \cdot \frac{\log x}{\color{blue}{0 - n \cdot x}} \]

       6. --lowering--.f64 N/A

          \[\leadsto x \cdot \frac{\log x}{\color{blue}{0 - n \cdot x}} \]

       7. *-lowering-*.f64 67.2

          \[\leadsto x \cdot \frac{\log x}{0 - \color{blue}{n \cdot x}} \]

   14. Simplified 67.2%

       \[\leadsto x \cdot \color{blue}{\frac{\log x}{0 - n \cdot x}} \]

   if 0.320000000000000007 < x < 2e143

   1. Initial program 52.2%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 52.1

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

   4. Applied egg-rr 52.1%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 95.2

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

   7. Simplified 95.2%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 59.8%

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

   if 2e143 < x

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 54.8

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

   5. Simplified 54.8%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 83.0%

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

      1. metadata-eval 83.0

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

   10. Applied egg-rr 83.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.32:\\ \;\;\;\;x \cdot \frac{\log x}{0 - x \cdot n}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-190}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+145}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.4e-190)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.32)
     (/ (- x (log x)) n)
     (if (<= x 1.15e+145) (/ 1.0 (* x n)) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.4e-190) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.32) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.15e+145) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.4d-190) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.32d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.15d+145) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.4e-190) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.32) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.15e+145) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.4e-190:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.32:
		tmp = (x - math.log(x)) / n
	elif x <= 1.15e+145:
		tmp = 1.0 / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.4e-190)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.32)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.15e+145)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.4e-190)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.32)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.15e+145)
		tmp = 1.0 / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.4e-190], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.32], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.15e+145], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.39999999999999981e-190

   1. Initial program 60.2%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 60.2

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

   5. Simplified 60.2%

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

   if 3.39999999999999981e-190 < x < 0.320000000000000007

   1. Initial program 29.7%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 30.1

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

   5. Simplified 30.1%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 52.9

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

   8. Simplified 52.9%

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

   if 0.320000000000000007 < x < 1.15e145

   1. Initial program 52.2%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 52.1

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

   4. Applied egg-rr 52.1%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 95.2

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

   7. Simplified 95.2%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 59.8%

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

   if 1.15e145 < x

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 54.8

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

   5. Simplified 54.8%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 83.0%

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

      1. metadata-eval 83.0

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

   10. Applied egg-rr 83.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.32:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.32) (/ (- x (log x)) n) (if (<= x 2.8e+143) (/ 1.0 (* x n)) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.32) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.8e+143) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.32d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.8d+143) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.32) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.8e+143) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.32:
		tmp = (x - math.log(x)) / n
	elif x <= 2.8e+143:
		tmp = 1.0 / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.32)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.8e+143)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.32)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.8e+143)
		tmp = 1.0 / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.32], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.8e+143], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.320000000000000007

   1. Initial program 41.1%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 41.5

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

   5. Simplified 41.5%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 50.2

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

   8. Simplified 50.2%

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

   if 0.320000000000000007 < x < 2.79999999999999998e143

   1. Initial program 52.2%

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

      1. pow-to-exp N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 52.1

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

   4. Applied egg-rr 52.1%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. exp-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. exp-to-pow N/A

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

      12. pow-lowering-pow.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 95.2

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

   7. Simplified 95.2%

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

   8. Taylor expanded in n around inf

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

   10. Simplified 59.8%

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

   if 2.79999999999999998e143 < x

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 54.8

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

   5. Simplified 54.8%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 83.0%

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

      1. metadata-eval 83.0

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

   10. Applied egg-rr 83.0%

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

Alternative 13: 47.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= (/ 1.0 n) -1.0) 0.0 (/ 1.0 (* x n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = 0.0;
	} else {
		tmp = 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 ((1.0d0 / n) <= (-1.0d0)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

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

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1.0:
		tmp = 0.0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. --lowering--.f64 N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

      16. pow-lowering-pow.f64 N/A

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

      17. /-lowering-/.f64 48.1

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

   5. Simplified 48.1%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 54.3%

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

      1. metadata-eval 54.3

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

   10. Applied egg-rr 54.3%

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

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

   1. Initial program 37.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

      3. un-div-inv N/A

         \[\leadsto e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

      5. +-commutative N/A

         \[\leadsto e^{\frac{\log \color{blue}{\left(1 + x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

      6. accelerator-lowering-log1p.f64 48.4

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied egg-rr 48.4%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}}{n \cdot x} \]

      3. log-rec N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)}}{n \cdot x} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\color{blue}{-1 \cdot \log x}}{n}\right)}}{n \cdot x} \]

      5. exp-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}}}{n \cdot x} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}}}{n \cdot x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{\log x \cdot -1}}{n}}}}{n \cdot x} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{e^{\color{blue}{\log x \cdot \frac{-1}{n}}}}}{n \cdot x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{e^{\log x \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{n}}}}{n \cdot x} \]

      10. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{1}{e^{\log x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{n}\right)\right)}}}}{n \cdot x} \]

      11. exp-to-pow N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{n}\right)\right)}}}}{n \cdot x} \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{{x}^{\left(\mathsf{neg}\left(\frac{1}{n}\right)\right)}}}}{n \cdot x} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{1}{{x}^{\color{blue}{\left(\frac{\mathsf{neg}\left(1\right)}{n}\right)}}}}{n \cdot x} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{{x}^{\left(\frac{\color{blue}{-1}}{n}\right)}}}{n \cdot x} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{1}{{x}^{\color{blue}{\left(\frac{-1}{n}\right)}}}}{n \cdot x} \]

      16. *-commutative N/A

          \[\leadsto \frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{\color{blue}{x \cdot n}} \]

      17. *-lowering-*.f64 44.3

          \[\leadsto \frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{\color{blue}{x \cdot n}} \]

   7. Simplified 44.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]

   8. Taylor expanded in n around inf

      \[\leadsto \frac{\color{blue}{1}}{x \cdot n} \]
   9. Step-by-step derivation

   10. Simplified 48.6%

       \[\leadsto \frac{\color{blue}{1}}{x \cdot n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 31.9% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 54.6%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
4. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

   2. mul-1-neg N/A

      \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

   3. distribute-neg-frac N/A

      \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

   5. log-rec N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

   6. mul-1-neg N/A

      \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   7. --lowering--.f64 N/A

      \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   8. log-rec N/A

      \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

   9. mul-1-neg N/A

      \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

   10. associate-*r/ N/A

       \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

   11. associate-*r* N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

   12. metadata-eval N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

   13. *-commutative N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

   14. associate-/l* N/A

       \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

   15. exp-to-pow N/A

       \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

   16. pow-lowering-pow.f64 N/A

       \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

   17. /-lowering-/.f64 40.2

       \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

5. Simplified 40.2%

   \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

6. Taylor expanded in n around inf

   \[\leadsto 1 - \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 34.3%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 34.3

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 34.3%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024197 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))