2nthrt (problem 3.4.6)

Percentage Accurate: 53.1% → 92.1%

Time: 41.3s

Alternatives: 19

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: 53.1% 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: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 320:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 320.0)
   (/
    (log
     (/
      x
      (exp
       (+
        (log1p x)
        (/
         (fma
          0.5
          (- (pow (log1p x) 2.0) (pow (log x) 2.0))
          (/
           (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
           n))
         n)))))
    (- n))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 320.0) {
		tmp = log((x / exp((log1p(x) + (fma(0.5, (pow(log1p(x), 2.0) - pow(log(x), 2.0)), ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n)) / n))))) / -n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 320.0)
		tmp = Float64(log(Float64(x / exp(Float64(log1p(x) + Float64(fma(0.5, Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)), Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n)) / n))))) / Float64(-n));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 320.0], N[(N[Log[N[(x / N[Exp[N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 320

   1. Initial program 42.3%

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

   3. Taylor expanded in n around -inf 78.8%

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

   5. Simplified 78.8%

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

      1. add-log-exp 87.1%

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

      2. diff-log 87.1%

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

   7. Applied egg-rr 87.1%

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

   if 320 < x

   1. Initial program 73.2%

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

   3. Taylor expanded in x around inf 82.3%

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

   5. Simplified 82.3%

      \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{-\log x}{n}}}{n} + e^{-\frac{-\log x}{n}} \cdot \frac{\frac{0.5}{{n}^{2}} - \frac{0.5}{n}}{x}}{x}} \]

   6. Taylor expanded in x around inf 99.3%

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

      1. distribute-frac-neg2 99.3%

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

      2. log-rec 99.3%

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

      3. distribute-frac-neg 99.3%

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

      4. distribute-neg-frac2 99.3%

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

      5. neg-mul-1 99.3%

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

      6. distribute-lft-neg-in 99.3%

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

      7. metadata-eval 99.3%

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

      8. *-commutative 99.3%

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

      9. *-rgt-identity 99.3%

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

      10. associate-*r/ 99.3%

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

      11. *-rgt-identity 99.3%

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

      12. exp-to-pow 99.3%

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

   8. Simplified 99.3%

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

Alternative 2: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -t\_1, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (sqrt t_0)))
   (if (<= (/ 1.0 n) -8e-31)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-18)
       (/ (log (/ (+ x 1.0) x)) n)
       (fma t_1 (- t_1) (exp (/ (log1p x) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = sqrt(t_0);
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(t_1, -t_1, exp((log1p(x) / n)));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -8e-31)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = fma(t_1, Float64(-t_1), exp(Float64(log1p(x) / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -8e-31], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(t$95$1 * (-t$95$1) + N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -8.000000000000001e-31

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf 29.9%

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

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{-\log x}{n}}}{n} + e^{-\frac{-\log x}{n}} \cdot \frac{\frac{0.5}{{n}^{2}} - \frac{0.5}{n}}{x}}{x}} \]

   6. Taylor expanded in x around inf 95.6%

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

      1. distribute-frac-neg2 95.6%

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

      2. log-rec 95.6%

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

      3. distribute-frac-neg 95.6%

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

      4. distribute-neg-frac2 95.6%

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

      5. neg-mul-1 95.6%

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

      6. distribute-lft-neg-in 95.6%

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

      7. metadata-eval 95.6%

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

      8. *-commutative 95.6%

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

      9. *-rgt-identity 95.6%

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

      10. associate-*r/ 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. exp-to-pow 95.6%

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

   8. Simplified 95.6%

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

   if -8.000000000000001e-31 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 82.5%

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

      1. diff-log 82.7%

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

      2. +-commutative 82.7%

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

   5. Applied egg-rr 82.7%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.5%

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

      1. sub-neg 47.5%

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

      2. +-commutative 47.5%

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

      3. sqr-pow 47.5%

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

      4. distribute-rgt-neg-in 47.5%

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

      5. fma-define 47.6%

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

      6. sqrt-pow1 47.6%

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

      7. sqrt-pow1 47.6%

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

      8. pow-to-exp 47.6%

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

      9. un-div-inv 47.6%

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

      10. +-commutative 47.6%

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

      11. log1p-define 97.2%

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

   4. Applied egg-rr 97.2%

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

Alternative 3: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -8e-31)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-18)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (pow E (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = pow(((double) M_E), (log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.pow(Math.E, (Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -8e-31:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.pow(math.e, (math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -8e-31)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64((exp(1) ^ Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -8e-31], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Power[E, N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -8.000000000000001e-31

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf 29.9%

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

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{-\log x}{n}}}{n} + e^{-\frac{-\log x}{n}} \cdot \frac{\frac{0.5}{{n}^{2}} - \frac{0.5}{n}}{x}}{x}} \]

   6. Taylor expanded in x around inf 95.6%

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

      1. distribute-frac-neg2 95.6%

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

      2. log-rec 95.6%

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

      3. distribute-frac-neg 95.6%

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

      4. distribute-neg-frac2 95.6%

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

      5. neg-mul-1 95.6%

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

      6. distribute-lft-neg-in 95.6%

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

      7. metadata-eval 95.6%

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

      8. *-commutative 95.6%

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

      9. *-rgt-identity 95.6%

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

      10. associate-*r/ 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. exp-to-pow 95.6%

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

   8. Simplified 95.6%

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

   if -8.000000000000001e-31 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 82.5%

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

      1. diff-log 82.7%

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

      2. +-commutative 82.7%

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

   5. Applied egg-rr 82.7%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.5%

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

   3. Taylor expanded in n around 0 47.5%

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

      1. log1p-define 97.1%

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

   5. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. exp-prod 97.2%

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

   7. Applied egg-rr 97.2%

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

      1. exp-1-e 97.2%

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

   9. Simplified 97.2%

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

Alternative 4: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -8e-31)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-18)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -8e-31:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -8e-31)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -8e-31], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -8.000000000000001e-31

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf 29.9%

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

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{-\log x}{n}}}{n} + e^{-\frac{-\log x}{n}} \cdot \frac{\frac{0.5}{{n}^{2}} - \frac{0.5}{n}}{x}}{x}} \]

   6. Taylor expanded in x around inf 95.6%

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

      1. distribute-frac-neg2 95.6%

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

      2. log-rec 95.6%

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

      3. distribute-frac-neg 95.6%

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

      4. distribute-neg-frac2 95.6%

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

      5. neg-mul-1 95.6%

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

      6. distribute-lft-neg-in 95.6%

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

      7. metadata-eval 95.6%

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

      8. *-commutative 95.6%

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

      9. *-rgt-identity 95.6%

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

      10. associate-*r/ 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. exp-to-pow 95.6%

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

   8. Simplified 95.6%

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

   if -8.000000000000001e-31 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 82.5%

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

      1. diff-log 82.7%

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

      2. +-commutative 82.7%

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

   5. Applied egg-rr 82.7%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.5%

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

   3. Taylor expanded in n around 0 47.5%

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

      1. log1p-define 97.1%

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

   5. Simplified 97.1%

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

Alternative 5: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -8e-31)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-18)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+157)
         (- (+ 1.0 (/ x n)) t_0)
         (* -0.16666666666666666 (pow (/ (log x) n) 3.0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = -0.16666666666666666 * pow((log(x) / n), 3.0);
	}
	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) <= (-8d-31)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2d-18) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+157) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (-0.16666666666666666d0) * ((log(x) / n) ** 3.0d0)
    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) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = -0.16666666666666666 * Math.pow((Math.log(x) / n), 3.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -8e-31:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+157:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = -0.16666666666666666 * math.pow((math.log(x) / n), 3.0)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -8e-31)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+157)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(-0.16666666666666666 * (Float64(log(x) / n) ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -8e-31)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e-18)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+157)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = -0.16666666666666666 * ((log(x) / n) ^ 3.0);
	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], -8e-31], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+157], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(-0.16666666666666666 * N[Power[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -8.000000000000001e-31

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf 29.9%

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

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{-\log x}{n}}}{n} + e^{-\frac{-\log x}{n}} \cdot \frac{\frac{0.5}{{n}^{2}} - \frac{0.5}{n}}{x}}{x}} \]

   6. Taylor expanded in x around inf 95.6%

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

      1. distribute-frac-neg2 95.6%

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

      2. log-rec 95.6%

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

      3. distribute-frac-neg 95.6%

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

      4. distribute-neg-frac2 95.6%

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

      5. neg-mul-1 95.6%

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

      6. distribute-lft-neg-in 95.6%

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

      7. metadata-eval 95.6%

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

      8. *-commutative 95.6%

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

      9. *-rgt-identity 95.6%

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

      10. associate-*r/ 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. exp-to-pow 95.6%

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

   8. Simplified 95.6%

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

   if -8.000000000000001e-31 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 82.5%

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

      1. diff-log 82.7%

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

      2. +-commutative 82.7%

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

   5. Applied egg-rr 82.7%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999976e157

   1. Initial program 73.9%

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

   3. Taylor expanded in x around 0 78.5%

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

   if 4.99999999999999976e157 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 28.0%

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

   3. Taylor expanded in x around 0 19.6%

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

   4. Taylor expanded in n around -inf 74.7%

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

   5. Taylor expanded in n around 0 74.7%

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

      1. cube-div 74.7%

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

   7. Simplified 74.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\frac{x}{n}\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -8e-31)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-18)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (pow E (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = pow(((double) M_E), (x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.pow(Math.E, (x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -8e-31:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.pow(math.e, (x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -8e-31)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64((exp(1) ^ Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -8e-31)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e-18)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = (2.71828182845904523536 ^ (x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -8e-31], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Power[E, N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -8.000000000000001e-31

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf 29.9%

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

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{-\log x}{n}}}{n} + e^{-\frac{-\log x}{n}} \cdot \frac{\frac{0.5}{{n}^{2}} - \frac{0.5}{n}}{x}}{x}} \]

   6. Taylor expanded in x around inf 95.6%

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

      1. distribute-frac-neg2 95.6%

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

      2. log-rec 95.6%

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

      3. distribute-frac-neg 95.6%

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

      4. distribute-neg-frac2 95.6%

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

      5. neg-mul-1 95.6%

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

      6. distribute-lft-neg-in 95.6%

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

      7. metadata-eval 95.6%

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

      8. *-commutative 95.6%

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

      9. *-rgt-identity 95.6%

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

      10. associate-*r/ 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. exp-to-pow 95.6%

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

   8. Simplified 95.6%

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

   if -8.000000000000001e-31 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 82.5%

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

      1. diff-log 82.7%

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

      2. +-commutative 82.7%

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

   5. Applied egg-rr 82.7%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.5%

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

   3. Taylor expanded in n around 0 47.5%

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

      1. log1p-define 97.1%

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

   5. Simplified 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. exp-prod 97.2%

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

   7. Applied egg-rr 97.2%

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

      1. exp-1-e 97.2%

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

   9. Simplified 97.2%

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

   10. Taylor expanded in x around 0 97.1%

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

Alternative 7: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -8e-31)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-18)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-8d-31)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2d-18) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -8e-31:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -8e-31)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -8e-31)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e-18)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -8e-31], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -8.000000000000001e-31

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf 29.9%

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

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{-\log x}{n}}}{n} + e^{-\frac{-\log x}{n}} \cdot \frac{\frac{0.5}{{n}^{2}} - \frac{0.5}{n}}{x}}{x}} \]

   6. Taylor expanded in x around inf 95.6%

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

      1. distribute-frac-neg2 95.6%

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

      2. log-rec 95.6%

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

      3. distribute-frac-neg 95.6%

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

      4. distribute-neg-frac2 95.6%

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

      5. neg-mul-1 95.6%

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

      6. distribute-lft-neg-in 95.6%

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

      7. metadata-eval 95.6%

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

      8. *-commutative 95.6%

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

      9. *-rgt-identity 95.6%

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

      10. associate-*r/ 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. exp-to-pow 95.6%

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

   8. Simplified 95.6%

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

   if -8.000000000000001e-31 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 82.5%

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

      1. diff-log 82.7%

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

      2. +-commutative 82.7%

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

   5. Applied egg-rr 82.7%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.5%

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

   3. Taylor expanded in n around 0 47.5%

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

      1. log1p-define 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0 97.1%

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

Alternative 8: 82.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\frac{1 + \frac{-1 + 1.1666666666666667 \cdot \frac{1}{x}}{x}}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -8e-31)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-18)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+157)
         (- (+ 1.0 (/ x n)) t_0)
         (/
          (expm1 (/ (+ 1.0 (/ (+ -1.0 (* 1.1666666666666667 (/ 1.0 x))) x)) x))
          n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = expm1(((1.0 + ((-1.0 + (1.1666666666666667 * (1.0 / x))) / x)) / x)) / n;
	}
	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) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.expm1(((1.0 + ((-1.0 + (1.1666666666666667 * (1.0 / x))) / x)) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -8e-31:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+157:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.expm1(((1.0 + ((-1.0 + (1.1666666666666667 * (1.0 / x))) / x)) / x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -8e-31)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+157)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(expm1(Float64(Float64(1.0 + Float64(Float64(-1.0 + Float64(1.1666666666666667 * Float64(1.0 / x))) / x)) / x)) / n);
	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], -8e-31], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+157], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(Exp[N[(N[(1.0 + N[(N[(-1.0 + N[(1.1666666666666667 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]] - 1), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -8.000000000000001e-31

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf 29.9%

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

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{-\log x}{n}}}{n} + e^{-\frac{-\log x}{n}} \cdot \frac{\frac{0.5}{{n}^{2}} - \frac{0.5}{n}}{x}}{x}} \]

   6. Taylor expanded in x around inf 95.6%

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

      1. distribute-frac-neg2 95.6%

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

      2. log-rec 95.6%

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

      3. distribute-frac-neg 95.6%

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

      4. distribute-neg-frac2 95.6%

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

      5. neg-mul-1 95.6%

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

      6. distribute-lft-neg-in 95.6%

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

      7. metadata-eval 95.6%

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

      8. *-commutative 95.6%

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

      9. *-rgt-identity 95.6%

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

      10. associate-*r/ 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. exp-to-pow 95.6%

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

   8. Simplified 95.6%

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

   if -8.000000000000001e-31 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 82.5%

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

      1. diff-log 82.7%

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

      2. +-commutative 82.7%

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

   5. Applied egg-rr 82.7%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999976e157

   1. Initial program 73.9%

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

   3. Taylor expanded in x around 0 78.5%

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

   if 4.99999999999999976e157 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf 5.8%

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

      1. expm1-log1p-u 5.8%

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

      2. log1p-define 5.8%

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

   5. Applied egg-rr 5.8%

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

   6. Taylor expanded in x around -inf 74.7%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{1.1666666666666667 \cdot \frac{1}{x} - 1}{x} - 1}{x}}\right)}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\frac{1 + \frac{-1 + 1.1666666666666667 \cdot \frac{1}{x}}{x}}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-262}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -50000000.0)
   (/ 0.3333333333333333 (* n (pow x 3.0)))
   (if (<= (/ 1.0 n) 2e-262)
     (/ 1.0 (* x (+ n (* 0.5 (/ n x)))))
     (if (<= (/ 1.0 n) 2e-18)
       (* (log x) (/ -1.0 n))
       (if (<= (/ 1.0 n) 1e+144)
         (- 1.0 (pow x (/ 1.0 n)))
         (/ (log1p (+ x -1.0)) (- n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -50000000.0) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((1.0 / n) <= 2e-262) {
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(x) * (-1.0 / n);
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -50000000.0) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((1.0 / n) <= 2e-262) {
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	} else if ((1.0 / n) <= 2e-18) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -50000000.0:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (1.0 / n) <= 2e-262:
		tmp = 1.0 / (x * (n + (0.5 * (n / x))))
	elif (1.0 / n) <= 2e-18:
		tmp = math.log(x) * (-1.0 / n)
	elif (1.0 / n) <= 1e+144:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -50000000.0)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (Float64(1.0 / n) <= 2e-262)
		tmp = Float64(1.0 / Float64(x * Float64(n + Float64(0.5 * Float64(n / x)))));
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (Float64(1.0 / n) <= 1e+144)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -50000000.0], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-262], N[(1.0 / N[(x * N[(n + N[(0.5 * N[(n / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+144], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e7

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 52.6%

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

   4. Taylor expanded in x around -inf 45.4%

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

   5. Taylor expanded in x around 0 77.1%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]

   if -5e7 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-262

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. clear-num 79.9%

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

      2. inv-pow 79.9%

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

      3. log1p-define 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. unpow-1 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in x around inf 58.7%

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

   if 2.00000000000000002e-262 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf 78.8%

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

   4. Taylor expanded in x around 0 58.2%

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

      1. neg-mul-1 58.2%

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

   6. Simplified 58.2%

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

      1. div-inv 58.3%

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

   8. Applied egg-rr 58.3%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e144

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0 76.7%

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

   if 1.00000000000000002e144 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.9%

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

   3. Taylor expanded in n around inf 5.7%

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

   4. Taylor expanded in x around 0 5.7%

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

      1. neg-mul-1 5.7%

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

   6. Simplified 5.7%

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

      1. log1p-expm1-u 65.4%

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

      2. expm1-undefine 65.4%

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

      3. add-exp-log 65.4%

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

   8. Applied egg-rr 65.4%

      \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(x - 1\right)}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-262}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -8e-31)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-18)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+157)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (log1p (+ x -1.0)) (- n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	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) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -8e-31:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+157:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -8e-31)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+157)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	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], -8e-31], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+157], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -8.000000000000001e-31

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf 29.9%

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

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{-\log x}{n}}}{n} + e^{-\frac{-\log x}{n}} \cdot \frac{\frac{0.5}{{n}^{2}} - \frac{0.5}{n}}{x}}{x}} \]

   6. Taylor expanded in x around inf 95.6%

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

      1. distribute-frac-neg2 95.6%

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

      2. log-rec 95.6%

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

      3. distribute-frac-neg 95.6%

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

      4. distribute-neg-frac2 95.6%

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

      5. neg-mul-1 95.6%

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

      6. distribute-lft-neg-in 95.6%

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

      7. metadata-eval 95.6%

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

      8. *-commutative 95.6%

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

      9. *-rgt-identity 95.6%

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

      10. associate-*r/ 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. exp-to-pow 95.6%

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

   8. Simplified 95.6%

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

   if -8.000000000000001e-31 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 82.5%

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

      1. diff-log 82.7%

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

      2. +-commutative 82.7%

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

   5. Applied egg-rr 82.7%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999976e157

   1. Initial program 73.9%

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

   3. Taylor expanded in x around 0 78.5%

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

   if 4.99999999999999976e157 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 28.0%

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

   3. Taylor expanded in n around inf 5.8%

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

   4. Taylor expanded in x around 0 5.8%

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

      1. neg-mul-1 5.8%

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

   6. Simplified 5.8%

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

      1. log1p-expm1-u 66.6%

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

      2. expm1-undefine 66.6%

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

      3. add-exp-log 66.6%

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

   8. Applied egg-rr 66.6%

      \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(x - 1\right)}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -8e-31)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-18)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 1e+144) (- 1.0 t_0) (/ (log1p (+ x -1.0)) (- n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - t_0;
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	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) <= -8e-31) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -8e-31:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 1e+144:
		tmp = 1.0 - t_0
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -8e-31)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+144)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	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], -8e-31], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+144], N[(1.0 - t$95$0), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -8.000000000000001e-31

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf 29.9%

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

   5. Simplified 29.9%

      \[\leadsto \color{blue}{\frac{\frac{e^{-\frac{-\log x}{n}}}{n} + e^{-\frac{-\log x}{n}} \cdot \frac{\frac{0.5}{{n}^{2}} - \frac{0.5}{n}}{x}}{x}} \]

   6. Taylor expanded in x around inf 95.6%

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

      1. distribute-frac-neg2 95.6%

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

      2. log-rec 95.6%

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

      3. distribute-frac-neg 95.6%

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

      4. distribute-neg-frac2 95.6%

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

      5. neg-mul-1 95.6%

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

      6. distribute-lft-neg-in 95.6%

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

      7. metadata-eval 95.6%

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

      8. *-commutative 95.6%

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

      9. *-rgt-identity 95.6%

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

      10. associate-*r/ 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. exp-to-pow 95.6%

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

   8. Simplified 95.6%

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

   if -8.000000000000001e-31 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 34.8%

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

   3. Taylor expanded in n around inf 82.5%

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

      1. diff-log 82.7%

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

      2. +-commutative 82.7%

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

   5. Applied egg-rr 82.7%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e144

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0 76.7%

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

   if 1.00000000000000002e144 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.9%

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

   3. Taylor expanded in n around inf 5.7%

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

   4. Taylor expanded in x around 0 5.7%

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

      1. neg-mul-1 5.7%

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

   6. Simplified 5.7%

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

      1. log1p-expm1-u 65.4%

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

      2. expm1-undefine 65.4%

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

      3. add-exp-log 65.4%

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

   8. Applied egg-rr 65.4%

      \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(x - 1\right)}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-5)
   (/ 0.3333333333333333 (* n (pow x 3.0)))
   (if (<= (/ 1.0 n) 2e-18)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= (/ 1.0 n) 1e+144)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (log1p (+ x -1.0)) (- n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-5) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((1.0 / n) <= 2e-18) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-5) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((1.0 / n) <= 2e-18) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-5:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (1.0 / n) <= 2e-18:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 1e+144:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-5)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (Float64(1.0 / n) <= 2e-18)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+144)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-5], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-18], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+144], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000024e-5

   1. Initial program 98.7%

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

   3. Taylor expanded in n around inf 52.0%

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

   4. Taylor expanded in x around -inf 45.2%

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

   5. Taylor expanded in x around 0 76.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]

   if -5.00000000000000024e-5 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-18

   1. Initial program 34.0%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. diff-log 80.3%

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

      2. +-commutative 80.3%

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

   5. Applied egg-rr 80.3%

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

   if 2.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e144

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0 76.7%

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

   if 1.00000000000000002e144 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.9%

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

   3. Taylor expanded in n around inf 5.7%

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

   4. Taylor expanded in x around 0 5.7%

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

      1. neg-mul-1 5.7%

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

   6. Simplified 5.7%

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

      1. log1p-expm1-u 65.4%

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

      2. expm1-undefine 65.4%

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

      3. add-exp-log 65.4%

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

   8. Applied egg-rr 65.4%

      \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(x - 1\right)}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot -0.3333333333333333 + n \cdot 0.25\\ \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+118}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{\frac{\left(-0.5 \cdot \frac{t\_0}{x} + \left(\frac{n}{x} \cdot -0.25 + 0.16666666666666666 \cdot \frac{n}{x}\right)\right) - t\_0}{x} + n \cdot -0.5}{x} - n\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ (* n -0.3333333333333333) (* n 0.25))))
   (if (<= x 0.35)
     (* (log x) (/ -1.0 n))
     (if (<= x 1.9e+118)
       (/
        -1.0
        (*
         x
         (-
          (/
           (+
            (/
             (-
              (+
               (* -0.5 (/ t_0 x))
               (+ (* (/ n x) -0.25) (* 0.16666666666666666 (/ n x))))
              t_0)
             x)
            (* n -0.5))
           x)
          n)))
       0.0))))

C

double code(double x, double n) {
	double t_0 = (n * -0.3333333333333333) + (n * 0.25);
	double tmp;
	if (x <= 0.35) {
		tmp = log(x) * (-1.0 / n);
	} else if (x <= 1.9e+118) {
		tmp = -1.0 / (x * (((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + (0.16666666666666666 * (n / x)))) - t_0) / x) + (n * -0.5)) / 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) :: t_0
    real(8) :: tmp
    t_0 = (n * (-0.3333333333333333d0)) + (n * 0.25d0)
    if (x <= 0.35d0) then
        tmp = log(x) * ((-1.0d0) / n)
    else if (x <= 1.9d+118) then
        tmp = (-1.0d0) / (x * ((((((((-0.5d0) * (t_0 / x)) + (((n / x) * (-0.25d0)) + (0.16666666666666666d0 * (n / x)))) - t_0) / x) + (n * (-0.5d0))) / x) - n))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (n * -0.3333333333333333) + (n * 0.25);
	double tmp;
	if (x <= 0.35) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if (x <= 1.9e+118) {
		tmp = -1.0 / (x * (((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + (0.16666666666666666 * (n / x)))) - t_0) / x) + (n * -0.5)) / x) - n));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (n * -0.3333333333333333) + (n * 0.25)
	tmp = 0
	if x <= 0.35:
		tmp = math.log(x) * (-1.0 / n)
	elif x <= 1.9e+118:
		tmp = -1.0 / (x * (((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + (0.16666666666666666 * (n / x)))) - t_0) / x) + (n * -0.5)) / x) - n))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(n * -0.3333333333333333) + Float64(n * 0.25))
	tmp = 0.0
	if (x <= 0.35)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (x <= 1.9e+118)
		tmp = Float64(-1.0 / Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(t_0 / x)) + Float64(Float64(Float64(n / x) * -0.25) + Float64(0.16666666666666666 * Float64(n / x)))) - t_0) / x) + Float64(n * -0.5)) / x) - n)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (n * -0.3333333333333333) + (n * 0.25);
	tmp = 0.0;
	if (x <= 0.35)
		tmp = log(x) * (-1.0 / n);
	elseif (x <= 1.9e+118)
		tmp = -1.0 / (x * (((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + (0.16666666666666666 * (n / x)))) - t_0) / x) + (n * -0.5)) / x) - n));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(n * -0.3333333333333333), $MachinePrecision] + N[(n * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.35], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e+118], N[(-1.0 / N[(x * N[(N[(N[(N[(N[(N[(N[(-0.5 * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n / x), $MachinePrecision] * -0.25), $MachinePrecision] + N[(0.16666666666666666 * N[(n / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / x), $MachinePrecision] + N[(n * -0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.34999999999999998

   1. Initial program 42.6%

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

   3. Taylor expanded in n around inf 50.2%

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

   4. Taylor expanded in x around 0 50.2%

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

      1. neg-mul-1 50.2%

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

   6. Simplified 50.2%

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

      1. div-inv 50.2%

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

   8. Applied egg-rr 50.2%

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

   if 0.34999999999999998 < x < 1.90000000000000008e118

   1. Initial program 46.5%

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

   3. Taylor expanded in n around inf 47.8%

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

      1. clear-num 47.8%

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

      2. inv-pow 47.8%

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

      3. log1p-define 47.8%

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

   5. Applied egg-rr 47.8%

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

      1. unpow-1 47.8%

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

   7. Simplified 47.8%

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

   8. Taylor expanded in x around -inf 61.8%

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot n + -1 \cdot \frac{-1 \cdot \frac{\left(-0.5 \cdot \frac{-0.3333333333333333 \cdot n + 0.25 \cdot n}{x} + \left(-0.25 \cdot \frac{n}{x} + 0.16666666666666666 \cdot \frac{n}{x}\right)\right) - \left(-0.3333333333333333 \cdot n + 0.25 \cdot n\right)}{x} - -0.5 \cdot n}{x}\right)\right)}} \]

   if 1.90000000000000008e118 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0 55.5%

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

   4. Taylor expanded in n around inf 88.5%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 88.5%

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

   6. Applied egg-rr 88.5%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+118}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{\frac{\left(-0.5 \cdot \frac{n \cdot -0.3333333333333333 + n \cdot 0.25}{x} + \left(\frac{n}{x} \cdot -0.25 + 0.16666666666666666 \cdot \frac{n}{x}\right)\right) - \left(n \cdot -0.3333333333333333 + n \cdot 0.25\right)}{x} + n \cdot -0.5}{x} - n\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot -0.3333333333333333 + n \cdot 0.25\\ \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+118}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{\frac{\left(-0.5 \cdot \frac{t\_0}{x} + \left(\frac{n}{x} \cdot -0.25 + 0.16666666666666666 \cdot \frac{n}{x}\right)\right) - t\_0}{x} + n \cdot -0.5}{x} - n\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ (* n -0.3333333333333333) (* n 0.25))))
   (if (<= x 0.35)
     (/ (log x) (- n))
     (if (<= x 2.35e+118)
       (/
        -1.0
        (*
         x
         (-
          (/
           (+
            (/
             (-
              (+
               (* -0.5 (/ t_0 x))
               (+ (* (/ n x) -0.25) (* 0.16666666666666666 (/ n x))))
              t_0)
             x)
            (* n -0.5))
           x)
          n)))
       0.0))))

C

double code(double x, double n) {
	double t_0 = (n * -0.3333333333333333) + (n * 0.25);
	double tmp;
	if (x <= 0.35) {
		tmp = log(x) / -n;
	} else if (x <= 2.35e+118) {
		tmp = -1.0 / (x * (((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + (0.16666666666666666 * (n / x)))) - t_0) / x) + (n * -0.5)) / 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) :: t_0
    real(8) :: tmp
    t_0 = (n * (-0.3333333333333333d0)) + (n * 0.25d0)
    if (x <= 0.35d0) then
        tmp = log(x) / -n
    else if (x <= 2.35d+118) then
        tmp = (-1.0d0) / (x * ((((((((-0.5d0) * (t_0 / x)) + (((n / x) * (-0.25d0)) + (0.16666666666666666d0 * (n / x)))) - t_0) / x) + (n * (-0.5d0))) / x) - n))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (n * -0.3333333333333333) + (n * 0.25);
	double tmp;
	if (x <= 0.35) {
		tmp = Math.log(x) / -n;
	} else if (x <= 2.35e+118) {
		tmp = -1.0 / (x * (((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + (0.16666666666666666 * (n / x)))) - t_0) / x) + (n * -0.5)) / x) - n));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (n * -0.3333333333333333) + (n * 0.25)
	tmp = 0
	if x <= 0.35:
		tmp = math.log(x) / -n
	elif x <= 2.35e+118:
		tmp = -1.0 / (x * (((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + (0.16666666666666666 * (n / x)))) - t_0) / x) + (n * -0.5)) / x) - n))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(n * -0.3333333333333333) + Float64(n * 0.25))
	tmp = 0.0
	if (x <= 0.35)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 2.35e+118)
		tmp = Float64(-1.0 / Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(t_0 / x)) + Float64(Float64(Float64(n / x) * -0.25) + Float64(0.16666666666666666 * Float64(n / x)))) - t_0) / x) + Float64(n * -0.5)) / x) - n)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (n * -0.3333333333333333) + (n * 0.25);
	tmp = 0.0;
	if (x <= 0.35)
		tmp = log(x) / -n;
	elseif (x <= 2.35e+118)
		tmp = -1.0 / (x * (((((((-0.5 * (t_0 / x)) + (((n / x) * -0.25) + (0.16666666666666666 * (n / x)))) - t_0) / x) + (n * -0.5)) / x) - n));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(n * -0.3333333333333333), $MachinePrecision] + N[(n * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.35], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 2.35e+118], N[(-1.0 / N[(x * N[(N[(N[(N[(N[(N[(N[(-0.5 * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n / x), $MachinePrecision] * -0.25), $MachinePrecision] + N[(0.16666666666666666 * N[(n / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / x), $MachinePrecision] + N[(n * -0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.34999999999999998

   1. Initial program 42.6%

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

   3. Taylor expanded in n around inf 50.2%

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

   4. Taylor expanded in x around 0 50.2%

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

      1. neg-mul-1 50.2%

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

   6. Simplified 50.2%

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

   if 0.34999999999999998 < x < 2.3499999999999999e118

   1. Initial program 46.5%

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

   3. Taylor expanded in n around inf 47.8%

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

      1. clear-num 47.8%

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

      2. inv-pow 47.8%

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

      3. log1p-define 47.8%

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

   5. Applied egg-rr 47.8%

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

      1. unpow-1 47.8%

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

   7. Simplified 47.8%

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

   8. Taylor expanded in x around -inf 61.8%

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot n + -1 \cdot \frac{-1 \cdot \frac{\left(-0.5 \cdot \frac{-0.3333333333333333 \cdot n + 0.25 \cdot n}{x} + \left(-0.25 \cdot \frac{n}{x} + 0.16666666666666666 \cdot \frac{n}{x}\right)\right) - \left(-0.3333333333333333 \cdot n + 0.25 \cdot n\right)}{x} - -0.5 \cdot n}{x}\right)\right)}} \]

   if 2.3499999999999999e118 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0 55.5%

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

   4. Taylor expanded in n around inf 88.5%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 88.5%

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

   6. Applied egg-rr 88.5%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+118}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{\frac{\left(-0.5 \cdot \frac{n \cdot -0.3333333333333333 + n \cdot 0.25}{x} + \left(\frac{n}{x} \cdot -0.25 + 0.16666666666666666 \cdot \frac{n}{x}\right)\right) - \left(n \cdot -0.3333333333333333 + n \cdot 0.25\right)}{x} + n \cdot -0.5}{x} - n\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.95e+118)
   (/ (+ (/ 1.0 n) (/ (+ -0.5 (/ 0.3333333333333333 x)) (* x n))) x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.95e+118) {
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * 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 (x <= 1.95d+118) then
        tmp = ((1.0d0 / n) + (((-0.5d0) + (0.3333333333333333d0 / x)) / (x * n))) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.95e+118) {
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.95e+118:
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.95e+118)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / Float64(x * n))) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.95e+118)
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.95e+118], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.95e118

   1. Initial program 43.5%

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

   3. Taylor expanded in n around inf 49.7%

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

   4. Taylor expanded in x around inf 25.6%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]

   5. Taylor expanded in x around -inf 41.3%

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

   7. Simplified 41.3%

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

   if 1.95e118 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0 55.5%

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

   4. Taylor expanded in n around inf 88.5%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 88.5%

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

   6. Applied egg-rr 88.5%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.4% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+118}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.1e+118)
   (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.1e+118) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (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 <= 2.1d+118) then
        tmp = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (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 <= 2.1e+118) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.1e+118:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.1e+118)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.1e+118)
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.1e+118], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.1e118

   1. Initial program 43.5%

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

   3. Taylor expanded in n around inf 49.7%

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

   4. Taylor expanded in x around -inf 41.3%

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

   5. Taylor expanded in n around -inf 41.3%

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

      1. mul-1-neg 41.3%

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

      2. unsub-neg 41.3%

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

      3. sub-neg 41.3%

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

      4. associate-*r/ 41.3%

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

      5. metadata-eval 41.3%

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

      6. distribute-neg-frac 41.3%

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

      7. metadata-eval 41.3%

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

      8. *-commutative 41.3%

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

   7. Simplified 41.3%

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

   if 2.1e118 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0 55.5%

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

   4. Taylor expanded in n around inf 88.5%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 88.5%

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

   6. Applied egg-rr 88.5%

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

Alternative 17: 46.4% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -50000000.0) 0.0 (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -50000000.0) {
		tmp = 0.0;
	} 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) :: tmp
    if ((1.0d0 / n) <= (-50000000.0d0)) then
        tmp = 0.0d0
    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) <= -50000000.0) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -50000000.0:
		tmp = 0.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -50000000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -50000000.0)
		tmp = 0.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -50000000.0], 0.0, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 51.6%

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

   4. Taylor expanded in n around inf 50.8%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 50.8%

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

   6. Applied egg-rr 50.8%

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

   if -5e7 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.8%

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

   3. Taylor expanded in n around inf 63.3%

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

   4. Taylor expanded in x around inf 43.9%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]

   5. Taylor expanded in x around inf 46.4%

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

Alternative 18: 43.5% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 1.8e+118) (/ 1.0 (* x n)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.8e+118) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.8d+118) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.8e+118) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.8e+118:
		tmp = 1.0 / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.8e+118)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.8e+118)
		tmp = 1.0 / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.8e+118], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.8e118

   1. Initial program 43.5%

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

   3. Taylor expanded in n around inf 49.7%

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

   4. Taylor expanded in x around inf 32.3%

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

      1. *-commutative 32.3%

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

   6. Simplified 32.3%

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

   if 1.8e118 < x

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0 55.5%

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

   4. Taylor expanded in n around inf 88.5%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 88.5%

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

   6. Applied egg-rr 88.5%

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

Alternative 19: 30.2% 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 55.6%

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

3. Taylor expanded in x around 0 40.3%

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

4. Taylor expanded in n around inf 33.5%

   \[\leadsto 1 - \color{blue}{1} \]
5. Step-by-step derivation

   1. metadata-eval 33.5%

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

6. Applied egg-rr 33.5%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))