2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 85.9%

Time: 24.3s

Alternatives: 12

Speedup: 1.9×

• could not determine a ground truth

  1. x = -8.657829216509739e-125
  2. n = 3.394922811342115e-93

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: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\log \left(\frac{1 + x}{x}\right), n, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -5e-31)
   (/ (/ (exp (/ (log x) n)) n) x)
   (if (<= (pow n -1.0) 2e-20)
     (/
      (/
       (fma
        (log (/ (+ 1.0 x) x))
        n
        (* (- (pow (log1p x) 2.0) (pow (log x) 2.0)) 0.5))
       n)
      n)
     (- (exp (/ (log1p x) n)) (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -5e-31) {
		tmp = (exp((log(x) / n)) / n) / x;
	} else if (pow(n, -1.0) <= 2e-20) {
		tmp = (fma(log(((1.0 + x) / x)), n, ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) * 0.5)) / n) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -5e-31)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	elseif ((n ^ -1.0) <= 2e-20)
		tmp = Float64(Float64(fma(log(Float64(Float64(1.0 + x) / x)), n, Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) * 0.5)) / n) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-31], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e-20], N[(N[(N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * n + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e-31

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 95.1%

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

   if -5e-31 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999989e-20

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 81.9%

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

   10. Applied rewrites 82.1%

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

   if 1.99999999999999989e-20 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.8%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 94.4

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

   4. Applied rewrites 94.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\log \left(\frac{1 + x}{x}\right), n, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;1 - e^{\frac{\log x}{n}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{0.5}{n} - 0.5, 1\right)}{n}, x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (pow (+ x 1.0) (pow n -1.0)) t_0)))
   (if (<= t_1 -2e-8)
     (- 1.0 (exp (/ (log x) n)))
     (if (<= t_1 0.0)
       (/ (- (log1p x) (log x)) n)
       (- (fma (/ (fma x (- (/ 0.5 n) 0.5) 1.0) n) x 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = pow((x + 1.0), pow(n, -1.0)) - t_0;
	double tmp;
	if (t_1 <= -2e-8) {
		tmp = 1.0 - exp((log(x) / n));
	} else if (t_1 <= 0.0) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma((fma(x, ((0.5 / n) - 0.5), 1.0) / n), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0)
	tmp = 0.0
	if (t_1 <= -2e-8)
		tmp = Float64(1.0 - exp(Float64(log(x) / n)));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(Float64(fma(x, Float64(Float64(0.5 / n) - 0.5), 1.0) / n), x, 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-8], N[(1.0 - N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(x * N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -2e-8

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. log-rec N/A

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

      12. mul-1-neg N/A

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

      13. associate-*r/ N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-exp.f64 N/A

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

   5. Applied rewrites 98.9%

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

   if -2e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 37.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 57.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(\frac{1 + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(-0.5, x, 0.5\right)}{n} + \left(0.3333333333333333 \cdot x - 0.5\right), 1\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}, 1\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 78.9%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{0.5}{n} - 0.5, 1\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -2 \cdot 10^{-8}:\\ \;\;\;\;1 - e^{\frac{\log x}{n}}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{0.5}{n} - 0.5, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -5e-31)
   (/ (/ (exp (/ (log x) n)) n) x)
   (if (<= (pow n -1.0) 2e-20)
     (/ (- (log1p x) (log x)) n)
     (- (exp (/ (log1p x) n)) (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -5e-31) {
		tmp = (exp((log(x) / n)) / n) / x;
	} else if (pow(n, -1.0) <= 2e-20) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -5e-31) {
		tmp = (Math.exp((Math.log(x) / n)) / n) / x;
	} else if (Math.pow(n, -1.0) <= 2e-20) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, Math.pow(n, -1.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -5e-31:
		tmp = (math.exp((math.log(x) / n)) / n) / x
	elif math.pow(n, -1.0) <= 2e-20:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, math.pow(n, -1.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -5e-31)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	elseif ((n ^ -1.0) <= 2e-20)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-31], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e-20], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e-31

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 95.1%

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

   if -5e-31 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999989e-20

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if 1.99999999999999989e-20 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.8%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 94.4

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

   4. Applied rewrites 94.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{0.5}{n} - 0.5, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -5e-31)
   (/ (/ (exp (/ (log x) n)) n) x)
   (if (<= (pow n -1.0) 1e-10)
     (/ (- (log1p x) (log x)) n)
     (-
      (fma (/ (fma x (- (/ 0.5 n) 0.5) 1.0) n) x 1.0)
      (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -5e-31) {
		tmp = (exp((log(x) / n)) / n) / x;
	} else if (pow(n, -1.0) <= 1e-10) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma((fma(x, ((0.5 / n) - 0.5), 1.0) / n), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -5e-31)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	elseif ((n ^ -1.0) <= 1e-10)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(Float64(fma(x, Float64(Float64(0.5 / n) - 0.5), 1.0) / n), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-31], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-10], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(x * N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e-31

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 95.1%

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

   if -5e-31 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 25.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 57.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(\frac{1 + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(-0.5, x, 0.5\right)}{n} + \left(0.3333333333333333 \cdot x - 0.5\right), 1\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}, 1\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 78.9%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{0.5}{n} - 0.5, 1\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{0.5}{n} - 0.5, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{0.5}{n} - 0.5, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -5e-31)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (pow n -1.0) 1e-10)
     (/ (- (log1p x) (log x)) n)
     (-
      (fma (/ (fma x (- (/ 0.5 n) 0.5) 1.0) n) x 1.0)
      (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -5e-31) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if (pow(n, -1.0) <= 1e-10) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = fma((fma(x, ((0.5 / n) - 0.5), 1.0) / n), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -5e-31)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif ((n ^ -1.0) <= 1e-10)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(fma(Float64(fma(x, Float64(Float64(0.5 / n) - 0.5), 1.0) / n), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -5e-31], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-10], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(x * N[(N[(0.5 / n), $MachinePrecision] - 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e-31

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -5e-31 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 25.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 57.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(\frac{1 + \left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n}\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(-0.5, x, 0.5\right)}{n} + \left(0.3333333333333333 \cdot x - 0.5\right), 1\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{1}{2} \cdot \frac{1}{n} - \frac{1}{2}, 1\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 78.9%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{0.5}{n} - 0.5, 1\right)}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \frac{0.5}{n} - 0.5, 1\right)}{n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{-250}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-213}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}\right) - \frac{0.25}{{x}^{3}}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 2.6e-250)
     t_0
     (if (<= x 2.85e-213)
       (- (+ (/ x n) 1.0) (pow x (pow n -1.0)))
       (if (<= x 0.7)
         t_0
         (if (<= x 1.45e+201)
           (/
            (/
             (-
              (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x))
              (/ 0.25 (pow x 3.0)))
             x)
            n)
           (/ (- (log x) (log x)) n)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 2.6e-250) {
		tmp = t_0;
	} else if (x <= 2.85e-213) {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 1.45e+201) {
		tmp = (((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) - (0.25 / pow(x, 3.0))) / x) / n;
	} else {
		tmp = (log(x) - log(x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 2.6d-250) then
        tmp = t_0
    else if (x <= 2.85d-213) then
        tmp = ((x / n) + 1.0d0) - (x ** (n ** (-1.0d0)))
    else if (x <= 0.7d0) then
        tmp = t_0
    else if (x <= 1.45d+201) then
        tmp = (((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) - (0.25d0 / (x ** 3.0d0))) / x) / n
    else
        tmp = (log(x) - log(x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 2.6e-250) {
		tmp = t_0;
	} else if (x <= 2.85e-213) {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 1.45e+201) {
		tmp = (((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) - (0.25 / Math.pow(x, 3.0))) / x) / n;
	} else {
		tmp = (Math.log(x) - Math.log(x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 2.6e-250:
		tmp = t_0
	elif x <= 2.85e-213:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.7:
		tmp = t_0
	elif x <= 1.45e+201:
		tmp = (((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) - (0.25 / math.pow(x, 3.0))) / x) / n
	else:
		tmp = (math.log(x) - math.log(x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 2.6e-250)
		tmp = t_0;
	elseif (x <= 2.85e-213)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 1.45e+201)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) - Float64(0.25 / (x ^ 3.0))) / x) / n);
	else
		tmp = Float64(Float64(log(x) - log(x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 2.6e-250)
		tmp = t_0;
	elseif (x <= 2.85e-213)
		tmp = ((x / n) + 1.0) - (x ^ (n ^ -1.0));
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 1.45e+201)
		tmp = (((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) - (0.25 / (x ^ 3.0))) / x) / n;
	else
		tmp = (log(x) - log(x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2.6e-250], t$95$0, If[LessEqual[x, 2.85e-213], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.7], t$95$0, If[LessEqual[x, 1.45e+201], N[(N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.60000000000000008e-250 or 2.84999999999999997e-213 < x < 0.69999999999999996

   1. Initial program 35.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. log-rec N/A

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

      12. mul-1-neg N/A

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

      13. associate-*r/ N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-exp.f64 N/A

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 59.4%

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

   if 2.60000000000000008e-250 < x < 2.84999999999999997e-213

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if 0.69999999999999996 < x < 1.4500000000000001e201

   1. Initial program 54.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \frac{11}{12} \cdot \frac{1}{n}}{{x}^{3}} + \left(\frac{1}{2} \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(\frac{1}{2} \cdot \frac{\frac{-2}{3} \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{\frac{1}{3}}{{x}^{2}}\right)\right)\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{4} \cdot \frac{1}{{x}^{3}}\right)}{x}}{n} \]

   8. Applied rewrites 68.5%

      \[\leadsto \frac{\frac{\left(\left(\mathsf{fma}\left(0.5, \frac{\frac{\mathsf{fma}\left(0.5, -\log x, 0.9166666666666666\right)}{n}}{{x}^{3}} + \frac{\frac{\mathsf{fma}\left(-1, \log x, 1\right)}{n}}{x}, \frac{\mathsf{fma}\left(0.5, -0.6666666666666666 \cdot \frac{\log x}{-n} - \frac{1}{n}, 0.3333333333333333\right)}{x \cdot x}\right) + \frac{\log x}{n}\right) + 1\right) - \left(\frac{0.25}{{x}^{3}} + \frac{0.5}{x}\right)}{x}}{n} \]

   9. Taylor expanded in n around inf

      \[\leadsto \frac{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{x}}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 67.2%

       \[\leadsto \frac{\frac{\left(\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}\right) - \frac{0.25}{{x}^{3}}}{x}}{n} \]

   if 1.4500000000000001e201 < x

   1. Initial program 89.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 89.9%

      \[\leadsto \frac{\log x - \log x}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-250}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-213}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}\right) - \frac{0.25}{{x}^{3}}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x - \log x}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{-250}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-213}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}\right) - \frac{0.25}{{x}^{3}}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 2.6e-250)
     t_0
     (if (<= x 2.85e-213)
       (- (+ (/ x n) 1.0) (pow x (pow n -1.0)))
       (if (<= x 0.7)
         t_0
         (/
          (/
           (-
            (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x))
            (/ 0.25 (pow x 3.0)))
           x)
          n))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 2.6e-250) {
		tmp = t_0;
	} else if (x <= 2.85e-213) {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	} else if (x <= 0.7) {
		tmp = t_0;
	} else {
		tmp = (((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) - (0.25 / pow(x, 3.0))) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 2.6d-250) then
        tmp = t_0
    else if (x <= 2.85d-213) then
        tmp = ((x / n) + 1.0d0) - (x ** (n ** (-1.0d0)))
    else if (x <= 0.7d0) then
        tmp = t_0
    else
        tmp = (((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) - (0.25d0 / (x ** 3.0d0))) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 2.6e-250) {
		tmp = t_0;
	} else if (x <= 2.85e-213) {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.7) {
		tmp = t_0;
	} else {
		tmp = (((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) - (0.25 / Math.pow(x, 3.0))) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 2.6e-250:
		tmp = t_0
	elif x <= 2.85e-213:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.7:
		tmp = t_0
	else:
		tmp = (((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) - (0.25 / math.pow(x, 3.0))) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 2.6e-250)
		tmp = t_0;
	elseif (x <= 2.85e-213)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	elseif (x <= 0.7)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) - Float64(0.25 / (x ^ 3.0))) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 2.6e-250)
		tmp = t_0;
	elseif (x <= 2.85e-213)
		tmp = ((x / n) + 1.0) - (x ^ (n ^ -1.0));
	elseif (x <= 0.7)
		tmp = t_0;
	else
		tmp = (((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) - (0.25 / (x ^ 3.0))) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2.6e-250], t$95$0, If[LessEqual[x, 2.85e-213], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.7], t$95$0, N[(N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.60000000000000008e-250 or 2.84999999999999997e-213 < x < 0.69999999999999996

   1. Initial program 35.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. log-rec N/A

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

      12. mul-1-neg N/A

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

      13. associate-*r/ N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-exp.f64 N/A

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 59.4%

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

   if 2.60000000000000008e-250 < x < 2.84999999999999997e-213

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if 0.69999999999999996 < x

   1. Initial program 64.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \frac{11}{12} \cdot \frac{1}{n}}{{x}^{3}} + \left(\frac{1}{2} \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(\frac{1}{2} \cdot \frac{\frac{-2}{3} \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{\frac{1}{3}}{{x}^{2}}\right)\right)\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{4} \cdot \frac{1}{{x}^{3}}\right)}{x}}{n} \]

   8. Applied rewrites 69.2%

      \[\leadsto \frac{\frac{\left(\left(\mathsf{fma}\left(0.5, \frac{\frac{\mathsf{fma}\left(0.5, -\log x, 0.9166666666666666\right)}{n}}{{x}^{3}} + \frac{\frac{\mathsf{fma}\left(-1, \log x, 1\right)}{n}}{x}, \frac{\mathsf{fma}\left(0.5, -0.6666666666666666 \cdot \frac{\log x}{-n} - \frac{1}{n}, 0.3333333333333333\right)}{x \cdot x}\right) + \frac{\log x}{n}\right) + 1\right) - \left(\frac{0.25}{{x}^{3}} + \frac{0.5}{x}\right)}{x}}{n} \]

   9. Taylor expanded in n around inf

      \[\leadsto \frac{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{4} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)}{x}}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 68.0%

       \[\leadsto \frac{\frac{\left(\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}\right) - \frac{0.25}{{x}^{3}}}{x}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-250}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-213}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}\right) - \frac{0.25}{{x}^{3}}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{-250}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-213}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 2.6e-250)
     t_0
     (if (<= x 2.85e-213)
       (- (+ (/ x n) 1.0) (pow x (pow n -1.0)))
       (if (<= x 0.6)
         t_0
         (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) x) n))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 2.6e-250) {
		tmp = t_0;
	} else if (x <= 2.85e-213) {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	} else if (x <= 0.6) {
		tmp = t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 2.6d-250) then
        tmp = t_0
    else if (x <= 2.85d-213) then
        tmp = ((x / n) + 1.0d0) - (x ** (n ** (-1.0d0)))
    else if (x <= 0.6d0) then
        tmp = t_0
    else
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 2.6e-250) {
		tmp = t_0;
	} else if (x <= 2.85e-213) {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.6) {
		tmp = t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 2.6e-250:
		tmp = t_0
	elif x <= 2.85e-213:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.6:
		tmp = t_0
	else:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 2.6e-250)
		tmp = t_0;
	elseif (x <= 2.85e-213)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	elseif (x <= 0.6)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 2.6e-250)
		tmp = t_0;
	elseif (x <= 2.85e-213)
		tmp = ((x / n) + 1.0) - (x ^ (n ^ -1.0));
	elseif (x <= 0.6)
		tmp = t_0;
	else
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2.6e-250], t$95$0, If[LessEqual[x, 2.85e-213], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.6], t$95$0, N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.60000000000000008e-250 or 2.84999999999999997e-213 < x < 0.599999999999999978

   1. Initial program 35.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. log-rec N/A

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

      12. mul-1-neg N/A

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

      13. associate-*r/ N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-exp.f64 N/A

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 59.4%

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

   if 2.60000000000000008e-250 < x < 2.84999999999999997e-213

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if 0.599999999999999978 < x

   1. Initial program 64.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 68.9%

       \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\frac{-\log x}{n}, -1, \mathsf{fma}\left(\frac{1 + \left(-\log x\right)}{n \cdot x}, 0.5, \frac{\mathsf{fma}\left(-0.6666666666666666 \cdot \frac{-\log x}{n} - \frac{1}{n}, 0.5, 0.3333333333333333\right)}{x \cdot x}\right)\right) + 1\right) - \frac{0.5}{x}}{x}}{n} \]

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 67.7%

       \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{x}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-250}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-213}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{-250}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-213}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 2.6e-250)
     t_0
     (if (<= x 2.85e-213)
       (- 1.0 (pow x (pow n -1.0)))
       (if (<= x 0.6)
         t_0
         (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) x) n))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 2.6e-250) {
		tmp = t_0;
	} else if (x <= 2.85e-213) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else if (x <= 0.6) {
		tmp = t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 2.6d-250) then
        tmp = t_0
    else if (x <= 2.85d-213) then
        tmp = 1.0d0 - (x ** (n ** (-1.0d0)))
    else if (x <= 0.6d0) then
        tmp = t_0
    else
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 2.6e-250) {
		tmp = t_0;
	} else if (x <= 2.85e-213) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else if (x <= 0.6) {
		tmp = t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 2.6e-250:
		tmp = t_0
	elif x <= 2.85e-213:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	elif x <= 0.6:
		tmp = t_0
	else:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 2.6e-250)
		tmp = t_0;
	elseif (x <= 2.85e-213)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	elseif (x <= 0.6)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 2.6e-250)
		tmp = t_0;
	elseif (x <= 2.85e-213)
		tmp = 1.0 - (x ^ (n ^ -1.0));
	elseif (x <= 0.6)
		tmp = t_0;
	else
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2.6e-250], t$95$0, If[LessEqual[x, 2.85e-213], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.6], t$95$0, N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.60000000000000008e-250 or 2.84999999999999997e-213 < x < 0.599999999999999978

   1. Initial program 35.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. log-rec N/A

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

      12. mul-1-neg N/A

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

      13. associate-*r/ N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-exp.f64 N/A

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 59.4%

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

   if 2.60000000000000008e-250 < x < 2.84999999999999997e-213

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.5%

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

   if 0.599999999999999978 < x

   1. Initial program 64.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 68.9%

       \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\frac{-\log x}{n}, -1, \mathsf{fma}\left(\frac{1 + \left(-\log x\right)}{n \cdot x}, 0.5, \frac{\mathsf{fma}\left(-0.6666666666666666 \cdot \frac{-\log x}{n} - \frac{1}{n}, 0.5, 0.3333333333333333\right)}{x \cdot x}\right)\right) + 1\right) - \frac{0.5}{x}}{x}}{n} \]

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 67.7%

       \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{x}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-250}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-213}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.6)
   (/ (log x) (- n))
   (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.6) {
		tmp = log(x) / -n;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = log(x) / -n
    else
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.6) {
		tmp = Math.log(x) / -n;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.6:
		tmp = math.log(x) / -n
	else:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(log(x) / Float64(-n));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.6)
		tmp = log(x) / -n;
	else
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.6], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.599999999999999978

   1. Initial program 41.1%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. log-rec N/A

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

      12. mul-1-neg N/A

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

      13. associate-*r/ N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-exp.f64 N/A

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 55.2%

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

   if 0.599999999999999978 < x

   1. Initial program 64.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 68.9%

       \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\frac{-\log x}{n}, -1, \mathsf{fma}\left(\frac{1 + \left(-\log x\right)}{n \cdot x}, 0.5, \frac{\mathsf{fma}\left(-0.6666666666666666 \cdot \frac{-\log x}{n} - \frac{1}{n}, 0.5, 0.3333333333333333\right)}{x \cdot x}\right)\right) + 1\right) - \frac{0.5}{x}}{x}}{n} \]

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 67.7%

       \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 41.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 35.3%

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

6. Taylor expanded in n around inf

   \[\leadsto \frac{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{n}}{x} \]
7. Step-by-step derivation

8. Applied rewrites 26.8%

   \[\leadsto \frac{\frac{1 - \frac{0.5}{x}}{n}}{x} \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 39.5%

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

12. Final simplification 39.5%

    \[\leadsto \frac{{n}^{-1}}{x} \]
13. Add Preprocessing

Alternative 12: 47.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 66.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 27.5%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 34.9%

    \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\frac{-\log x}{n}, -1, \mathsf{fma}\left(\frac{1 + \left(-\log x\right)}{n \cdot x}, 0.5, \frac{\mathsf{fma}\left(-0.6666666666666666 \cdot \frac{-\log x}{n} - \frac{1}{n}, 0.5, 0.3333333333333333\right)}{x \cdot x}\right)\right) + 1\right) - \frac{0.5}{x}}{x}}{n} \]

12. Taylor expanded in n around inf

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

14. Applied rewrites 46.1%

    \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{x}}{n} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))