2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 85.7%

Time: 23.0s

Alternatives: 16

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1500000 \lor \neg \left(n \leq 4.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + \left(\frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{2}{n}\right)}}, -e^{0.5 \cdot \frac{\log x}{n}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -1500000.0) (not (<= n 4.6e-5)))
   (+
    (/ (- (log1p x) (log x)) n)
    (+
     (/
      (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
      (pow n 3.0))
     (*
      0.5
      (-
       (/ (pow (log1p x) 2.0) (pow n 2.0))
       (/ (pow (log x) 2.0) (pow n 2.0))))))
   (fma
    (sqrt (pow (sqrt x) (/ 2.0 n)))
    (- (exp (* 0.5 (/ (log x) n))))
    (exp (/ (log1p x) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -1500000.0) || !(n <= 4.6e-5)) {
		tmp = ((log1p(x) - log(x)) / n) + (((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / pow(n, 3.0)) + (0.5 * ((pow(log1p(x), 2.0) / pow(n, 2.0)) - (pow(log(x), 2.0) / pow(n, 2.0)))));
	} else {
		tmp = fma(sqrt(pow(sqrt(x), (2.0 / n))), -exp((0.5 * (log(x) / n))), exp((log1p(x) / n)));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -1500000.0) || !(n <= 4.6e-5))
		tmp = Float64(Float64(Float64(log1p(x) - log(x)) / n) + Float64(Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / (n ^ 3.0)) + Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) / (n ^ 2.0)) - Float64((log(x) ^ 2.0) / (n ^ 2.0))))));
	else
		tmp = fma(sqrt((sqrt(x) ^ Float64(2.0 / n))), Float64(-exp(Float64(0.5 * Float64(log(x) / n)))), exp(Float64(log1p(x) / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -1500000.0], N[Not[LessEqual[n, 4.6e-5]], $MachinePrecision]], N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(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[Power[n, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[Power[N[Sqrt[x], $MachinePrecision], N[(2.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-N[Exp[N[(0.5 * N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) + N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.5e6 or 4.6e-5 < n

   1. Initial program 26.4%

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

   2. Taylor expanded in n around -inf 78.7%

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

      1. associate--l+ 78.7%

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

      2. associate-*r/ 78.7%

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

      3. mul-1-neg 78.7%

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

      4. distribute-lft-out-- 78.7%

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

      5. mul-1-neg 78.7%

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

      6. remove-double-neg 78.7%

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

      7. log1p-def 78.7%

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

      8. associate--l+ 78.7%

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

   4. Simplified 78.7%

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

   if -1.5e6 < n < 4.6e-5

   1. Initial program 85.7%

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

      1. sub-neg 85.7%

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

      2. +-commutative 85.7%

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

      3. add-sqr-sqrt 85.8%

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

      4. distribute-rgt-neg-in 85.8%

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

      5. fma-def 85.8%

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

      6. add-exp-log 85.8%

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

      7. log-pow 85.8%

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

      8. +-commutative 85.8%

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

      9. log1p-udef 99.2%

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

      10. *-commutative 99.2%

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

      11. un-div-inv 99.2%

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

   3. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}, -\sqrt{{x}^{\left(\frac{1}{n}\right)}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)} \]
   4. Step-by-step derivation

      1. pow1/2 99.2%

         \[\leadsto \mathsf{fma}\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}, -\color{blue}{{\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.5}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]

      2. pow-to-exp 99.2%

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

      3. pow-exp 99.2%

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

      4. div-inv 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. add-sqr-sqrt 99.2%

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

      2. unpow-prod-down 99.2%

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

   7. Applied egg-rr 99.2%

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

      1. pow-sqr 99.2%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

   9. Simplified 99.2%

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{\left(\frac{2}{n}\right)}}}, -e^{\frac{\log x}{n} \cdot 0.5}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1500000 \lor \neg \left(n \leq 4.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + \left(\frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{{\left(\sqrt{x}\right)}^{\left(\frac{2}{n}\right)}}, -e^{0.5 \cdot \frac{\log x}{n}}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \]

Alternative 2: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -25000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{elif}\;n \leq 44000000000:\\ \;\;\;\;{\left(\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -25000000.0)
   (+
    (/ (- (log1p x) (log x)) n)
    (*
     0.5
     (-
      (/ (pow (log1p x) 2.0) (pow n 2.0))
      (/ (pow (log x) 2.0) (pow n 2.0)))))
   (if (<= n 44000000000.0)
     (pow (cbrt (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))) 3.0)
     (/ (log (/ (+ x 1.0) x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -25000000.0) {
		tmp = ((log1p(x) - log(x)) / n) + (0.5 * ((pow(log1p(x), 2.0) / pow(n, 2.0)) - (pow(log(x), 2.0) / pow(n, 2.0))));
	} else if (n <= 44000000000.0) {
		tmp = pow(cbrt((exp((log1p(x) / n)) - pow(x, (1.0 / n)))), 3.0);
	} else {
		tmp = log(((x + 1.0) / x)) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -25000000.0) {
		tmp = ((Math.log1p(x) - Math.log(x)) / n) + (0.5 * ((Math.pow(Math.log1p(x), 2.0) / Math.pow(n, 2.0)) - (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))));
	} else if (n <= 44000000000.0) {
		tmp = Math.pow(Math.cbrt((Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n)))), 3.0);
	} else {
		tmp = Math.log(((x + 1.0) / x)) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -25000000.0)
		tmp = Float64(Float64(Float64(log1p(x) - log(x)) / n) + Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) / (n ^ 2.0)) - Float64((log(x) ^ 2.0) / (n ^ 2.0)))));
	elseif (n <= 44000000000.0)
		tmp = cbrt(Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)))) ^ 3.0;
	else
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -25000000.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 44000000000.0], N[Power[N[Power[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.5e7

   1. Initial program 32.8%

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

   2. Taylor expanded in n around inf 81.2%

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

      1. associate--l+ 72.0%

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

      2. +-commutative 72.0%

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

      3. associate--r+ 81.2%

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

      4. div-sub 81.3%

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

      5. remove-double-neg 81.3%

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

      6. mul-1-neg 81.3%

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

      7. distribute-lft-out-- 81.3%

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

      8. distribute-neg-frac 81.3%

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

      9. mul-1-neg 81.3%

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

   4. Simplified 81.3%

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

   if -2.5e7 < n < 4.4e10

   1. Initial program 83.7%

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

      1. add-cube-cbrt 83.7%

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

      2. pow3 83.7%

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

      3. add-exp-log 83.7%

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

      4. log-pow 83.7%

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

      5. +-commutative 83.7%

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

      6. log1p-udef 96.7%

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

      7. *-commutative 96.7%

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

      8. un-div-inv 96.7%

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

   3. Applied egg-rr 96.7%

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

   if 4.4e10 < n

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 78.3%

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

      1. diff-log 78.7%

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

      2. +-commutative 78.7%

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

   4. Applied egg-rr 78.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -25000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{elif}\;n \leq 44000000000:\\ \;\;\;\;{\left(\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \]

Alternative 3: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -31500000000:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq 33000000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -31500000000.0)
   (- (/ (log (/ x (+ x 1.0))) n))
   (if (<= n 33000000000.0)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))
     (/ (log (/ (+ x 1.0) x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -31500000000.0) {
		tmp = -(log((x / (x + 1.0))) / n);
	} else if (n <= 33000000000.0) {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log(((x + 1.0) / x)) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -31500000000.0) {
		tmp = -(Math.log((x / (x + 1.0))) / n);
	} else if (n <= 33000000000.0) {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log(((x + 1.0) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -31500000000.0:
		tmp = -(math.log((x / (x + 1.0))) / n)
	elif n <= 33000000000.0:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log(((x + 1.0) / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -31500000000.0)
		tmp = Float64(-Float64(log(Float64(x / Float64(x + 1.0))) / n));
	elseif (n <= 33000000000.0)
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -31500000000.0], (-N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[n, 33000000000.0], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.15e10

   1. Initial program 31.1%

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

   2. Taylor expanded in n around inf 80.9%

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

      1. diff-log 81.1%

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

      2. +-commutative 81.1%

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

   4. Applied egg-rr 81.1%

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

      1. clear-num 81.1%

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

      2. log-rec 81.1%

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

   6. Applied egg-rr 81.1%

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

   if -3.15e10 < n < 3.3e10

   1. Initial program 83.4%

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

   2. Taylor expanded in n around 0 83.4%

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

      1. log1p-def 96.1%

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

   4. Simplified 96.1%

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

   if 3.3e10 < n

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 78.3%

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

      1. diff-log 78.7%

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

      2. +-commutative 78.7%

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

   4. Applied egg-rr 78.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -31500000000:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq 33000000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \]

Alternative 4: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-77}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+177}:\\ \;\;\;\;1 + t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-8)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-77)
       (- (/ (log (/ x (+ x 1.0))) n))
       (if (<= (/ 1.0 n) 2e-70)
         (* (/ 1.0 n) (/ 1.0 x))
         (if (<= (/ 1.0 n) 2e-7)
           (/ (log (/ (+ x 1.0) x)) n)
           (if (<= (/ 1.0 n) 1e+177)
             (+ 1.0 t_0)
             (sqrt (pow (* n x) -2.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-8) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-77) {
		tmp = -(log((x / (x + 1.0))) / n);
	} else if ((1.0 / n) <= 2e-70) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+177) {
		tmp = 1.0 + t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-8)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 5d-77) then
        tmp = -(log((x / (x + 1.0d0))) / n)
    else if ((1.0d0 / n) <= 2d-70) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 1d+177) then
        tmp = 1.0d0 + t_0
    else
        tmp = sqrt(((n * x) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-8) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-77) {
		tmp = -(Math.log((x / (x + 1.0))) / n);
	} else if ((1.0 / n) <= 2e-70) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+177) {
		tmp = 1.0 + t_0;
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-8:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-77:
		tmp = -(math.log((x / (x + 1.0))) / n)
	elif (1.0 / n) <= 2e-70:
		tmp = (1.0 / n) * (1.0 / x)
	elif (1.0 / n) <= 2e-7:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 1e+177:
		tmp = 1.0 + t_0
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-8)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-77)
		tmp = Float64(-Float64(log(Float64(x / Float64(x + 1.0))) / n));
	elseif (Float64(1.0 / n) <= 2e-70)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+177)
		tmp = Float64(1.0 + t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-8)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 5e-77)
		tmp = -(log((x / (x + 1.0))) / n);
	elseif ((1.0 / n) <= 2e-70)
		tmp = (1.0 / n) * (1.0 / x);
	elseif ((1.0 / n) <= 2e-7)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 1e+177)
		tmp = 1.0 + t_0;
	else
		tmp = sqrt(((n * x) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-8], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-77], (-N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-70], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+177], N[(1.0 + t$95$0), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -4.9999999999999998e-8

   1. Initial program 97.6%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

      1. expm1-log1p-u 55.4%

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

      2. expm1-udef 53.1%

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

      3. div-inv 53.1%

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

      4. exp-to-pow 53.1%

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

   6. Applied egg-rr 53.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 55.4%

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

      2. expm1-log1p 98.9%

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

   8. Simplified 98.9%

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

   if -4.9999999999999998e-8 < (/.f64 1 n) < 4.99999999999999963e-77

   1. Initial program 29.5%

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

   2. Taylor expanded in n around inf 82.9%

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

      1. diff-log 83.2%

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

      2. +-commutative 83.2%

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

   4. Applied egg-rr 83.2%

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

      1. clear-num 83.2%

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

      2. log-rec 83.3%

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

   6. Applied egg-rr 83.3%

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

   if 4.99999999999999963e-77 < (/.f64 1 n) < 1.99999999999999999e-70

   1. Initial program 6.2%

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

   2. Taylor expanded in n around inf 6.2%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. associate-/r* 99.1%

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

      2. div-inv 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 1.99999999999999999e-70 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 10.0%

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

   2. Taylor expanded in n around inf 64.3%

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

      1. diff-log 64.3%

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

      2. +-commutative 64.3%

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

   4. Applied egg-rr 64.3%

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

   if 1.9999999999999999e-7 < (/.f64 1 n) < 1e177

   1. Initial program 83.7%

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

   2. Taylor expanded in x around 0 75.7%

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

      1. sub-neg 75.7%

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

      2. +-commutative 75.7%

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

      3. add-sqr-sqrt 75.7%

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

      4. sqrt-unprod 79.8%

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

      5. sqr-neg 79.8%

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

      6. sqrt-unprod 79.8%

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

      7. add-sqr-sqrt 79.8%

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

   4. Applied egg-rr 79.8%

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

   if 1e177 < (/.f64 1 n)

   1. Initial program 24.7%

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

   2. Taylor expanded in n around inf 7.0%

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

   3. Taylor expanded in x around inf 68.6%

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

      1. *-commutative 68.6%

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

   5. Simplified 68.6%

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

      1. add-sqr-sqrt 68.6%

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

      2. sqrt-unprod 83.9%

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

      3. inv-pow 83.9%

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

      4. inv-pow 83.9%

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

      5. pow-prod-up 83.9%

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

      6. metadata-eval 83.9%

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

   7. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-77}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+177}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]

Alternative 5: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-77}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+177}:\\ \;\;\;\;1 + t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-8)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-77)
       (- (/ (log (/ x (+ x 1.0))) n))
       (if (<= (/ 1.0 n) 2e-70)
         (* (/ 1.0 n) (/ 1.0 x))
         (if (<= (/ 1.0 n) 2e-7)
           (/ (- (log1p x) (log x)) n)
           (if (<= (/ 1.0 n) 1e+177)
             (+ 1.0 t_0)
             (sqrt (pow (* n x) -2.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-8) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-77) {
		tmp = -(log((x / (x + 1.0))) / n);
	} else if ((1.0 / n) <= 2e-70) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1e+177) {
		tmp = 1.0 + t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-8) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-77) {
		tmp = -(Math.log((x / (x + 1.0))) / n);
	} else if ((1.0 / n) <= 2e-70) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e+177) {
		tmp = 1.0 + t_0;
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-8:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-77:
		tmp = -(math.log((x / (x + 1.0))) / n)
	elif (1.0 / n) <= 2e-70:
		tmp = (1.0 / n) * (1.0 / x)
	elif (1.0 / n) <= 2e-7:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1e+177:
		tmp = 1.0 + t_0
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-8)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-77)
		tmp = Float64(-Float64(log(Float64(x / Float64(x + 1.0))) / n));
	elseif (Float64(1.0 / n) <= 2e-70)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e+177)
		tmp = Float64(1.0 + t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-8], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-77], (-N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-70], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+177], N[(1.0 + t$95$0), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -4.9999999999999998e-8

   1. Initial program 97.6%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

      1. expm1-log1p-u 55.4%

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

      2. expm1-udef 53.1%

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

      3. div-inv 53.1%

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

      4. exp-to-pow 53.1%

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

   6. Applied egg-rr 53.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 55.4%

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

      2. expm1-log1p 98.9%

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

   8. Simplified 98.9%

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

   if -4.9999999999999998e-8 < (/.f64 1 n) < 4.99999999999999963e-77

   1. Initial program 29.5%

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

   2. Taylor expanded in n around inf 82.9%

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

      1. diff-log 83.2%

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

      2. +-commutative 83.2%

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

   4. Applied egg-rr 83.2%

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

      1. clear-num 83.2%

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

      2. log-rec 83.3%

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

   6. Applied egg-rr 83.3%

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

   if 4.99999999999999963e-77 < (/.f64 1 n) < 1.99999999999999999e-70

   1. Initial program 6.2%

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

   2. Taylor expanded in n around inf 6.2%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. associate-/r* 99.1%

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

      2. div-inv 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 1.99999999999999999e-70 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 10.0%

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

   2. Taylor expanded in n around inf 64.3%

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

      1. +-rgt-identity 64.3%

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

      2. +-rgt-identity 64.3%

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

      3. log1p-def 64.3%

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

   4. Simplified 64.3%

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

   if 1.9999999999999999e-7 < (/.f64 1 n) < 1e177

   1. Initial program 83.7%

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

   2. Taylor expanded in x around 0 75.7%

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

      1. sub-neg 75.7%

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

      2. +-commutative 75.7%

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

      3. add-sqr-sqrt 75.7%

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

      4. sqrt-unprod 79.8%

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

      5. sqr-neg 79.8%

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

      6. sqrt-unprod 79.8%

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

      7. add-sqr-sqrt 79.8%

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

   4. Applied egg-rr 79.8%

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

   if 1e177 < (/.f64 1 n)

   1. Initial program 24.7%

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

   2. Taylor expanded in n around inf 7.0%

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

   3. Taylor expanded in x around inf 68.6%

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

      1. *-commutative 68.6%

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

   5. Simplified 68.6%

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

      1. add-sqr-sqrt 68.6%

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

      2. sqrt-unprod 83.9%

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

      3. inv-pow 83.9%

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

      4. inv-pow 83.9%

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

      5. pow-prod-up 83.9%

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

      6. metadata-eval 83.9%

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

   7. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-77}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+177}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]

Alternative 6: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -15200000000:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq 7200000000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -15200000000.0)
   (- (/ (log (/ x (+ x 1.0))) n))
   (if (<= n 7200000000.0)
     (- (exp (/ x n)) (pow x (/ 1.0 n)))
     (/ (log (/ (+ x 1.0) x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -15200000000.0) {
		tmp = -(log((x / (x + 1.0))) / n);
	} else if (n <= 7200000000.0) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log(((x + 1.0) / x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-15200000000.0d0)) then
        tmp = -(log((x / (x + 1.0d0))) / n)
    else if (n <= 7200000000.0d0) then
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    else
        tmp = log(((x + 1.0d0) / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -15200000000.0) {
		tmp = -(Math.log((x / (x + 1.0))) / n);
	} else if (n <= 7200000000.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log(((x + 1.0) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -15200000000.0:
		tmp = -(math.log((x / (x + 1.0))) / n)
	elif n <= 7200000000.0:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log(((x + 1.0) / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -15200000000.0)
		tmp = Float64(-Float64(log(Float64(x / Float64(x + 1.0))) / n));
	elseif (n <= 7200000000.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -15200000000.0)
		tmp = -(log((x / (x + 1.0))) / n);
	elseif (n <= 7200000000.0)
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	else
		tmp = log(((x + 1.0) / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -15200000000.0], (-N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[n, 7200000000.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.52e10

   1. Initial program 31.1%

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

   2. Taylor expanded in n around inf 80.9%

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

      1. diff-log 81.1%

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

      2. +-commutative 81.1%

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

   4. Applied egg-rr 81.1%

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

      1. clear-num 81.1%

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

      2. log-rec 81.1%

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

   6. Applied egg-rr 81.1%

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

   if -1.52e10 < n < 7.2e9

   1. Initial program 83.4%

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

   2. Taylor expanded in n around 0 83.4%

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

      1. log1p-def 96.1%

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

   4. Simplified 96.1%

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

   5. Taylor expanded in x around 0 96.1%

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

   if 7.2e9 < n

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 78.3%

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

      1. diff-log 78.7%

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

      2. +-commutative 78.7%

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

   4. Applied egg-rr 78.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -15200000000:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq 7200000000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \]

Alternative 7: 80.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-77}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+234}:\\ \;\;\;\;1 + t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-8)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-77)
       (- (/ (log (/ x (+ x 1.0))) n))
       (if (<= (/ 1.0 n) 2e-70)
         (* (/ 1.0 n) (/ 1.0 x))
         (if (<= (/ 1.0 n) 2e-7)
           (/ (log (/ (+ x 1.0) x)) n)
           (if (<= (/ 1.0 n) 2e+234) (+ 1.0 t_0) (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-8) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-77) {
		tmp = -(log((x / (x + 1.0))) / n);
	} else if ((1.0 / n) <= 2e-70) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+234) {
		tmp = 1.0 + t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-8)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 5d-77) then
        tmp = -(log((x / (x + 1.0d0))) / n)
    else if ((1.0d0 / n) <= 2d-70) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+234) then
        tmp = 1.0d0 + t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-8) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-77) {
		tmp = -(Math.log((x / (x + 1.0))) / n);
	} else if ((1.0 / n) <= 2e-70) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+234) {
		tmp = 1.0 + t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-8:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-77:
		tmp = -(math.log((x / (x + 1.0))) / n)
	elif (1.0 / n) <= 2e-70:
		tmp = (1.0 / n) * (1.0 / x)
	elif (1.0 / n) <= 2e-7:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+234:
		tmp = 1.0 + t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-8)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-77)
		tmp = Float64(-Float64(log(Float64(x / Float64(x + 1.0))) / n));
	elseif (Float64(1.0 / n) <= 2e-70)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+234)
		tmp = Float64(1.0 + t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-8)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 5e-77)
		tmp = -(log((x / (x + 1.0))) / n);
	elseif ((1.0 / n) <= 2e-70)
		tmp = (1.0 / n) * (1.0 / x);
	elseif ((1.0 / n) <= 2e-7)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+234)
		tmp = 1.0 + t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-8], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-77], (-N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-70], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+234], N[(1.0 + t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -4.9999999999999998e-8

   1. Initial program 97.6%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   4. Simplified 98.9%

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

      1. expm1-log1p-u 55.4%

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

      2. expm1-udef 53.1%

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

      3. div-inv 53.1%

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

      4. exp-to-pow 53.1%

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

   6. Applied egg-rr 53.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 55.4%

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

      2. expm1-log1p 98.9%

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

   8. Simplified 98.9%

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

   if -4.9999999999999998e-8 < (/.f64 1 n) < 4.99999999999999963e-77

   1. Initial program 29.5%

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

   2. Taylor expanded in n around inf 82.9%

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

      1. diff-log 83.2%

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

      2. +-commutative 83.2%

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

   4. Applied egg-rr 83.2%

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

      1. clear-num 83.2%

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

      2. log-rec 83.3%

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

   6. Applied egg-rr 83.3%

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

   if 4.99999999999999963e-77 < (/.f64 1 n) < 1.99999999999999999e-70

   1. Initial program 6.2%

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

   2. Taylor expanded in n around inf 6.2%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. associate-/r* 99.1%

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

      2. div-inv 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 1.99999999999999999e-70 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 10.0%

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

   2. Taylor expanded in n around inf 64.3%

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

      1. diff-log 64.3%

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

      2. +-commutative 64.3%

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

   4. Applied egg-rr 64.3%

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

   if 1.9999999999999999e-7 < (/.f64 1 n) < 2.00000000000000004e234

   1. Initial program 78.0%

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

   2. Taylor expanded in x around 0 68.6%

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

      1. sub-neg 68.6%

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

      2. +-commutative 68.6%

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

      3. add-sqr-sqrt 68.6%

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

      4. sqrt-unprod 71.9%

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

      5. sqr-neg 71.9%

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

      6. sqrt-unprod 71.9%

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

      7. add-sqr-sqrt 71.9%

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

   4. Applied egg-rr 71.9%

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

   if 2.00000000000000004e234 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 8.4%

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

   3. Taylor expanded in x around inf 92.1%

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

      1. *-commutative 92.1%

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

   5. Simplified 92.1%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-77}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+234}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 8: 66.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq 5 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+234}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) 5e-77)
     t_0
     (if (<= (/ 1.0 n) 2e-70)
       (* (/ 1.0 n) (/ 1.0 x))
       (if (<= (/ 1.0 n) 2e-7)
         t_0
         (if (<= (/ 1.0 n) 2e+234)
           (+ 1.0 (pow x (/ 1.0 n)))
           (/ 1.0 (* n x))))))))

C

double code(double x, double n) {
	double t_0 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= 5e-77) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-70) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+234) {
		tmp = 1.0 + pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= 5d-77) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-70) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d+234) then
        tmp = 1.0d0 + (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= 5e-77) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-70) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+234) {
		tmp = 1.0 + Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= 5e-77:
		tmp = t_0
	elif (1.0 / n) <= 2e-70:
		tmp = (1.0 / n) * (1.0 / x)
	elif (1.0 / n) <= 2e-7:
		tmp = t_0
	elif (1.0 / n) <= 2e+234:
		tmp = 1.0 + math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 5e-77)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-70)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e+234)
		tmp = Float64(1.0 + (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= 5e-77)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-70)
		tmp = (1.0 / n) * (1.0 / x);
	elseif ((1.0 / n) <= 2e-7)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e+234)
		tmp = 1.0 + (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-77], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-70], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+234], N[(1.0 + N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < 4.99999999999999963e-77 or 1.99999999999999999e-70 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 55.4%

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

   2. Taylor expanded in n around inf 70.4%

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

      1. diff-log 70.6%

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

      2. +-commutative 70.6%

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

   4. Applied egg-rr 70.6%

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

   if 4.99999999999999963e-77 < (/.f64 1 n) < 1.99999999999999999e-70

   1. Initial program 6.2%

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

   2. Taylor expanded in n around inf 6.2%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. associate-/r* 99.1%

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

      2. div-inv 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 1.9999999999999999e-7 < (/.f64 1 n) < 2.00000000000000004e234

   1. Initial program 78.0%

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

   2. Taylor expanded in x around 0 68.6%

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

      1. sub-neg 68.6%

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

      2. +-commutative 68.6%

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

      3. add-sqr-sqrt 68.6%

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

      4. sqrt-unprod 71.9%

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

      5. sqr-neg 71.9%

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

      6. sqrt-unprod 71.9%

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

      7. add-sqr-sqrt 71.9%

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

   4. Applied egg-rr 71.9%

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

   if 2.00000000000000004e234 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 8.4%

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

   3. Taylor expanded in x around inf 92.1%

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

      1. *-commutative 92.1%

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

   5. Simplified 92.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+234}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 9: 66.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 5 \cdot 10^{-77}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+234}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 5e-77)
   (- (/ (log (/ x (+ x 1.0))) n))
   (if (<= (/ 1.0 n) 2e-70)
     (* (/ 1.0 n) (/ 1.0 x))
     (if (<= (/ 1.0 n) 2e-7)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e+234) (+ 1.0 (pow x (/ 1.0 n))) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 5e-77) {
		tmp = -(log((x / (x + 1.0))) / n);
	} else if ((1.0 / n) <= 2e-70) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+234) {
		tmp = 1.0 + pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 5d-77) then
        tmp = -(log((x / (x + 1.0d0))) / n)
    else if ((1.0d0 / n) <= 2d-70) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if ((1.0d0 / n) <= 2d-7) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+234) then
        tmp = 1.0d0 + (x ** (1.0d0 / n))
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 5e-77) {
		tmp = -(Math.log((x / (x + 1.0))) / n);
	} else if ((1.0 / n) <= 2e-70) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 2e-7) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+234) {
		tmp = 1.0 + Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 5e-77:
		tmp = -(math.log((x / (x + 1.0))) / n)
	elif (1.0 / n) <= 2e-70:
		tmp = (1.0 / n) * (1.0 / x)
	elif (1.0 / n) <= 2e-7:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+234:
		tmp = 1.0 + math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 5e-77)
		tmp = Float64(-Float64(log(Float64(x / Float64(x + 1.0))) / n));
	elseif (Float64(1.0 / n) <= 2e-70)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (Float64(1.0 / n) <= 2e-7)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+234)
		tmp = Float64(1.0 + (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 5e-77)
		tmp = -(log((x / (x + 1.0))) / n);
	elseif ((1.0 / n) <= 2e-70)
		tmp = (1.0 / n) * (1.0 / x);
	elseif ((1.0 / n) <= 2e-7)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+234)
		tmp = 1.0 + (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-77], (-N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-70], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-7], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+234], N[(1.0 + N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < 4.99999999999999963e-77

   1. Initial program 58.2%

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

   2. Taylor expanded in n around inf 70.8%

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

      1. diff-log 71.0%

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

      2. +-commutative 71.0%

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

   4. Applied egg-rr 71.0%

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

      1. clear-num 71.0%

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

      2. log-rec 71.0%

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

   6. Applied egg-rr 71.0%

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

   if 4.99999999999999963e-77 < (/.f64 1 n) < 1.99999999999999999e-70

   1. Initial program 6.2%

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

   2. Taylor expanded in n around inf 6.2%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. associate-/r* 99.1%

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

      2. div-inv 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 1.99999999999999999e-70 < (/.f64 1 n) < 1.9999999999999999e-7

   1. Initial program 10.0%

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

   2. Taylor expanded in n around inf 64.3%

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

      1. diff-log 64.3%

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

      2. +-commutative 64.3%

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

   4. Applied egg-rr 64.3%

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

   if 1.9999999999999999e-7 < (/.f64 1 n) < 2.00000000000000004e234

   1. Initial program 78.0%

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

   2. Taylor expanded in x around 0 68.6%

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

      1. sub-neg 68.6%

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

      2. +-commutative 68.6%

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

      3. add-sqr-sqrt 68.6%

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

      4. sqrt-unprod 71.9%

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

      5. sqr-neg 71.9%

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

      6. sqrt-unprod 71.9%

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

      7. add-sqr-sqrt 71.9%

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

   4. Applied egg-rr 71.9%

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

   if 2.00000000000000004e234 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 8.4%

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

   3. Taylor expanded in x around inf 92.1%

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

      1. *-commutative 92.1%

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

   5. Simplified 92.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 5 \cdot 10^{-77}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+234}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 10: 57.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.65 \cdot 10^{-243}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5.65e-243)
   (/ (- (log x)) n)
   (if (<= x 1.7e-224)
     (/ 1.0 (* n x))
     (if (<= x 1.45) (/ (- x (log x)) n) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5.65e-243) {
		tmp = -log(x) / n;
	} else if (x <= 1.7e-224) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.45) {
		tmp = (x - log(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 <= 5.65d-243) then
        tmp = -log(x) / n
    else if (x <= 1.7d-224) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 1.45d0) then
        tmp = (x - log(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 <= 5.65e-243) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.7e-224) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.45) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5.65e-243:
		tmp = -math.log(x) / n
	elif x <= 1.7e-224:
		tmp = 1.0 / (n * x)
	elif x <= 1.45:
		tmp = (x - math.log(x)) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5.65e-243)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.7e-224)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 1.45)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5.65e-243)
		tmp = -log(x) / n;
	elseif (x <= 1.7e-224)
		tmp = 1.0 / (n * x);
	elseif (x <= 1.45)
		tmp = (x - log(x)) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5.65e-243], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.7e-224], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.64999999999999979e-243

   1. Initial program 42.0%

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

   2. Taylor expanded in x around 0 42.0%

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

   3. Taylor expanded in n around inf 59.0%

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

      1. neg-mul-1 59.0%

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

      2. distribute-neg-frac 59.0%

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

   5. Simplified 59.0%

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

   if 5.64999999999999979e-243 < x < 1.69999999999999996e-224

   1. Initial program 82.4%

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

   2. Taylor expanded in n around inf 14.2%

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

   3. Taylor expanded in x around inf 56.1%

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

      1. *-commutative 56.1%

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

   5. Simplified 56.1%

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

   if 1.69999999999999996e-224 < x < 1.44999999999999996

   1. Initial program 41.6%

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

   2. Taylor expanded in n around inf 52.4%

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

   3. Taylor expanded in x around 0 51.3%

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

      1. neg-mul-1 51.3%

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

      2. unsub-neg 51.3%

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

   5. Simplified 51.3%

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

   if 1.44999999999999996 < x

   1. Initial program 70.5%

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

      1. sub-neg 70.5%

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

      2. +-commutative 70.5%

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

      3. add-log-exp 70.5%

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

      4. add-log-exp 70.5%

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

      5. sum-log 70.5%

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

      6. add-exp-log 70.5%

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

      7. log-pow 70.5%

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

      8. +-commutative 70.5%

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

      9. log1p-udef 70.5%

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

      10. *-commutative 70.5%

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

      11. un-div-inv 70.5%

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

   3. Applied egg-rr 70.5%

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

   4. Taylor expanded in x around inf 70.5%

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

      1. exp-neg 70.5%

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

      2. rgt-mult-inverse 70.5%

         \[\leadsto \log \color{blue}{1} \]

      3. metadata-eval 70.5%

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

   6. Simplified 70.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.65 \cdot 10^{-243}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 57.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-185}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.35e-261)
   (/ (- (log x)) n)
   (if (<= x 5.5e-185)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 1.45) (/ (- x (log x)) n) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.35e-261) {
		tmp = -log(x) / n;
	} else if (x <= 5.5e-185) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.45) {
		tmp = (x - log(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 <= 1.35d-261) then
        tmp = -log(x) / n
    else if (x <= 5.5d-185) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.45d0) then
        tmp = (x - log(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 <= 1.35e-261) {
		tmp = -Math.log(x) / n;
	} else if (x <= 5.5e-185) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.45) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.35e-261:
		tmp = -math.log(x) / n
	elif x <= 5.5e-185:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.45:
		tmp = (x - math.log(x)) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.35e-261)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 5.5e-185)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.45)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.35e-261)
		tmp = -log(x) / n;
	elseif (x <= 5.5e-185)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.45)
		tmp = (x - log(x)) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.35e-261], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 5.5e-185], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.3499999999999999e-261

   1. Initial program 32.0%

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

   2. Taylor expanded in x around 0 32.0%

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

   3. Taylor expanded in n around inf 67.3%

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

      1. neg-mul-1 67.3%

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

      2. distribute-neg-frac 67.3%

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

   5. Simplified 67.3%

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

   if 1.3499999999999999e-261 < x < 5.4999999999999998e-185

   1. Initial program 67.5%

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

   2. Taylor expanded in x around 0 67.5%

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

   if 5.4999999999999998e-185 < x < 1.44999999999999996

   1. Initial program 37.6%

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

   2. Taylor expanded in n around inf 55.1%

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

   3. Taylor expanded in x around 0 53.7%

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

      1. neg-mul-1 53.7%

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

      2. unsub-neg 53.7%

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

   5. Simplified 53.7%

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

   if 1.44999999999999996 < x

   1. Initial program 70.5%

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

      1. sub-neg 70.5%

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

      2. +-commutative 70.5%

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

      3. add-log-exp 70.5%

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

      4. add-log-exp 70.5%

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

      5. sum-log 70.5%

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

      6. add-exp-log 70.5%

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

      7. log-pow 70.5%

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

      8. +-commutative 70.5%

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

      9. log1p-udef 70.5%

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

      10. *-commutative 70.5%

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

      11. un-div-inv 70.5%

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

   3. Applied egg-rr 70.5%

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

   4. Taylor expanded in x around inf 70.5%

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

      1. exp-neg 70.5%

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

      2. rgt-mult-inverse 70.5%

         \[\leadsto \log \color{blue}{1} \]

      3. metadata-eval 70.5%

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

   6. Simplified 70.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-185}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 5.65 \cdot 10^{-243}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 5.65e-243)
     t_0
     (if (<= x 1.7e-224) (/ 1.0 (* n x)) (if (<= x 1.0) t_0 0.0)))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 5.65e-243) {
		tmp = t_0;
	} else if (x <= 1.7e-224) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 5.65d-243) then
        tmp = t_0
    else if (x <= 1.7d-224) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 1.0d0) then
        tmp = t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 5.65e-243) {
		tmp = t_0;
	} else if (x <= 1.7e-224) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 5.65e-243:
		tmp = t_0
	elif x <= 1.7e-224:
		tmp = 1.0 / (n * x)
	elif x <= 1.0:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 5.65e-243)
		tmp = t_0;
	elseif (x <= 1.7e-224)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 5.65e-243)
		tmp = t_0;
	elseif (x <= 1.7e-224)
		tmp = 1.0 / (n * x);
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 5.65e-243], t$95$0, If[LessEqual[x, 1.7e-224], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], t$95$0, 0.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.64999999999999979e-243 or 1.69999999999999996e-224 < x < 1

   1. Initial program 42.0%

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

   2. Taylor expanded in x around 0 39.9%

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

   3. Taylor expanded in n around inf 53.2%

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

      1. neg-mul-1 53.2%

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

      2. distribute-neg-frac 53.2%

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

   5. Simplified 53.2%

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

   if 5.64999999999999979e-243 < x < 1.69999999999999996e-224

   1. Initial program 82.4%

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

   2. Taylor expanded in n around inf 14.2%

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

   3. Taylor expanded in x around inf 56.1%

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

      1. *-commutative 56.1%

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

   5. Simplified 56.1%

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

   if 1 < x

   1. Initial program 69.9%

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

      1. sub-neg 69.9%

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

      2. +-commutative 69.9%

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

      3. add-log-exp 69.9%

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

      4. add-log-exp 69.9%

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

      5. sum-log 69.9%

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

      6. add-exp-log 69.9%

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

      7. log-pow 69.9%

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

      8. +-commutative 69.9%

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

      9. log1p-udef 69.9%

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

      10. *-commutative 69.9%

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

      11. un-div-inv 69.9%

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

   3. Applied egg-rr 69.9%

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

   4. Taylor expanded in x around inf 69.9%

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

      1. exp-neg 69.8%

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

      2. rgt-mult-inverse 69.9%

         \[\leadsto \log \color{blue}{1} \]

      3. metadata-eval 69.9%

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

   6. Simplified 69.9%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.65 \cdot 10^{-243}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 46.8% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -100000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -100000.0) 0.0 (* (/ 1.0 n) (/ 1.0 x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100000.0) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	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) <= (-100000.0d0)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / n) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -100000.0) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -100000.0:
		tmp = 0.0
	else:
		tmp = (1.0 / n) * (1.0 / x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -100000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -100000.0)
		tmp = 0.0;
	else
		tmp = (1.0 / n) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -100000.0], 0.0, N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -1e5

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. add-log-exp 100.0%

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

      4. add-log-exp 100.0%

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

      5. sum-log 100.0%

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

      6. add-exp-log 100.0%

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

      7. log-pow 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-udef 100.0%

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

      10. *-commutative 100.0%

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

      11. un-div-inv 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 55.7%

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

      1. exp-neg 55.7%

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

      2. rgt-mult-inverse 56.4%

         \[\leadsto \log \color{blue}{1} \]

      3. metadata-eval 56.4%

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

   6. Simplified 56.4%

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

   if -1e5 < (/.f64 1 n)

   1. Initial program 34.8%

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

   2. Taylor expanded in n around inf 59.5%

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

   3. Taylor expanded in x around inf 39.9%

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

      1. *-commutative 39.9%

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

   5. Simplified 39.9%

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

      1. associate-/r* 40.3%

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

      2. div-inv 40.3%

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

   7. Applied egg-rr 40.3%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -100000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 14: 46.3% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-5} \lor \neg \left(n \leq -1.95 \cdot 10^{-305}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -2.8e-5) (not (<= n -1.95e-305))) (/ 1.0 (* n x)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -2.8e-5) || !(n <= -1.95e-305)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.8d-5)) .or. (.not. (n <= (-1.95d-305)))) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -2.8e-5) || !(n <= -1.95e-305)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -2.8e-5) or not (n <= -1.95e-305):
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -2.8e-5) || !(n <= -1.95e-305))
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -2.8e-5) || ~((n <= -1.95e-305)))
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -2.8e-5], N[Not[LessEqual[n, -1.95e-305]], $MachinePrecision]], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -2.79999999999999996e-5 or -1.95000000000000013e-305 < n

   1. Initial program 35.2%

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

   2. Taylor expanded in n around inf 59.7%

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

   3. Taylor expanded in x around inf 40.2%

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

      1. *-commutative 40.2%

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

   5. Simplified 40.2%

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

   if -2.79999999999999996e-5 < n < -1.95000000000000013e-305

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. add-log-exp 100.0%

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

      4. add-log-exp 100.0%

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

      5. sum-log 100.0%

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

      6. add-exp-log 100.0%

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

      7. log-pow 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-udef 100.0%

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

      10. *-commutative 100.0%

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

      11. un-div-inv 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 56.4%

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

      1. exp-neg 56.4%

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

      2. rgt-mult-inverse 57.1%

         \[\leadsto \log \color{blue}{1} \]

      3. metadata-eval 57.1%

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

   6. Simplified 57.1%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-5} \lor \neg \left(n \leq -1.95 \cdot 10^{-305}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 46.9% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-5} \lor \neg \left(n \leq -1.95 \cdot 10^{-305}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -2.8e-5) (not (<= n -1.95e-305))) (/ (/ 1.0 x) n) 0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -2.8e-5) || !(n <= -1.95e-305)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.8d-5)) .or. (.not. (n <= (-1.95d-305)))) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -2.8e-5) || !(n <= -1.95e-305)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -2.8e-5) or not (n <= -1.95e-305):
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -2.8e-5) || !(n <= -1.95e-305))
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -2.8e-5) || ~((n <= -1.95e-305)))
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -2.8e-5], N[Not[LessEqual[n, -1.95e-305]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -2.79999999999999996e-5 or -1.95000000000000013e-305 < n

   1. Initial program 35.2%

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

   2. Taylor expanded in n around inf 59.7%

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

   3. Taylor expanded in x around inf 40.6%

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

   if -2.79999999999999996e-5 < n < -1.95000000000000013e-305

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. add-log-exp 100.0%

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

      4. add-log-exp 100.0%

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

      5. sum-log 100.0%

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

      6. add-exp-log 100.0%

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

      7. log-pow 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-udef 100.0%

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

      10. *-commutative 100.0%

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

      11. un-div-inv 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 56.4%

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

      1. exp-neg 56.4%

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

      2. rgt-mult-inverse 57.1%

         \[\leadsto \log \color{blue}{1} \]

      3. metadata-eval 57.1%

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

   6. Simplified 57.1%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-5} \lor \neg \left(n \leq -1.95 \cdot 10^{-305}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16: 30.8% accurate, 211.0× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 54.9%

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

   1. sub-neg 54.9%

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

   2. +-commutative 54.9%

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

   3. add-log-exp 54.6%

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

   4. add-log-exp 54.6%

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

   5. sum-log 54.6%

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

   6. add-exp-log 54.6%

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

   7. log-pow 54.6%

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

   8. +-commutative 54.6%

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

   9. log1p-udef 61.0%

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

   10. *-commutative 61.0%

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

   11. un-div-inv 61.0%

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

3. Applied egg-rr 61.0%

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

4. Taylor expanded in x around inf 30.1%

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

   1. exp-neg 30.1%

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

   2. rgt-mult-inverse 30.4%

      \[\leadsto \log \color{blue}{1} \]

   3. metadata-eval 30.4%

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

6. Simplified 30.4%

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

7. Final simplification 30.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))