2nthrt (problem 3.4.6)

Percentage Accurate: 53.9% → 86.5%

Time: 27.1s

Alternatives: 18

Speedup: 1.9×

• could not determine a ground truth

  1. x = -1.493708010266056e+62
  2. n = 3.494300375282173e+186

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: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-9)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
     (if (<= (/ 1.0 n) 2e-50)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 0.02)
         (/ (exp (/ (log x) n)) (* x n))
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = exp((log(x) / n)) / (x * n);
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-9:
		tmp = math.pow((x + 1.0), (1.0 / n)) - t_0
	elif (1.0 / n) <= 2e-50:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 0.02:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-9)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-9], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000006e-9

   1. Initial program 98.6%

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

   if -1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.9

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

   5. Simplified 80.9%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 81.0

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

   7. Applied egg-rr 81.0%

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

   if 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 14.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.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. Simplified 68.8%

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

   6. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 70.1

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

   8. Simplified 70.1%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 68.5%

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

      1. pow-to-exp N/A

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

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

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

      3. un-div-inv N/A

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

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

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

      5. +-commutative N/A

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

      6. accelerator-lowering-log1p.f64 100.0

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, \log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right), 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n} - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 20.0)
   (/
    (-
     (/
      (fma
       0.5
       (* (log (* x (+ x 1.0))) (log (/ (+ x 1.0) x)))
       (* 0.16666666666666666 (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)))
      n)
     (log (/ x (+ x 1.0))))
    n)
   (* (/ 1.0 n) (/ (pow x (/ 1.0 n)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 20.0) {
		tmp = ((fma(0.5, (log((x * (x + 1.0))) * log(((x + 1.0) / x))), (0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n))) / n) - log((x / (x + 1.0)))) / n;
	} else {
		tmp = (1.0 / n) * (pow(x, (1.0 / n)) / x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 20.0)
		tmp = Float64(Float64(Float64(fma(0.5, Float64(log(Float64(x * Float64(x + 1.0))) * log(Float64(Float64(x + 1.0) / x))), Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n))) / n) - log(Float64(x / Float64(x + 1.0)))) / n);
	else
		tmp = Float64(Float64(1.0 / n) * Float64((x ^ Float64(1.0 / n)) / x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 20.0], N[(N[(N[(N[(0.5 * N[(N[Log[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 20

   1. Initial program 42.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 76.4%

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

   7. Applied egg-rr 76.4%

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

   if 20 < x

   1. Initial program 64.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 97.9

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

   5. Simplified 97.9%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f64 98.7

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

   7. Applied egg-rr 98.7%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.5, \log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right), 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n} - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 -0.1) t_2 (if (<= t_1 5e-13) (/ (log (/ (+ x 1.0) x)) n) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -0.1) {
		tmp = t_2;
	} else if (t_1 <= 5e-13) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    t_2 = 1.0d0 - t_0
    if (t_1 <= (-0.1d0)) then
        tmp = t_2
    else if (t_1 <= 5d-13) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -0.1) {
		tmp = t_2;
	} else if (t_1 <= 5e-13) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -0.1:
		tmp = t_2
	elif t_1 <= 5e-13:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = t_2;
	elseif (t_1 <= 5e-13)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = t_2;
	elseif (t_1 <= 5e-13)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], t$95$2, If[LessEqual[t$95$1, 5e-13], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 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))) < -0.10000000000000001 or 4.9999999999999999e-13 < (-.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 84.7%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

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

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

      17. /-lowering-/.f64 84.7

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

   5. Simplified 84.7%

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

   if -0.10000000000000001 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 4.9999999999999999e-13

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 79.4

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

   5. Simplified 79.4%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 79.4

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

   7. Applied egg-rr 79.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -0.1:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-9)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
     (if (<= (/ 1.0 n) 2e-50)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 0.02)
         (/ (exp (/ (log x) n)) (* x n))
         (if (<= (/ 1.0 n) 2e+239)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 0.3333333333333333 (* x (* x (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = exp((log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 2e+239) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-9)) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    else if ((1.0d0 / n) <= 2d-50) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 0.02d0) then
        tmp = exp((log(x) / n)) / (x * n)
    else if ((1.0d0 / n) <= 2d+239) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 2e+239) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-9:
		tmp = math.pow((x + 1.0), (1.0 / n)) - t_0
	elif (1.0 / n) <= 2e-50:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 0.02:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	elif (1.0 / n) <= 2e+239:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-9)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e+239)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-9)
		tmp = ((x + 1.0) ^ (1.0 / n)) - t_0;
	elseif ((1.0 / n) <= 2e-50)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 0.02)
		tmp = exp((log(x) / n)) / (x * n);
	elseif ((1.0 / n) <= 2e+239)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-9], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+239], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000006e-9

   1. Initial program 98.6%

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

   if -1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.9

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

   5. Simplified 80.9%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 81.0

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

   7. Applied egg-rr 81.0%

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

   if 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 14.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.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. Simplified 68.8%

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

   6. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 70.1

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

   8. Simplified 70.1%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999998e239

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 82.8

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

   5. Simplified 82.8%

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

   if 1.99999999999999998e239 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 9.7

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

   5. Simplified 9.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-9)
     (* t_0 (/ 1.0 (* x n)))
     (if (<= (/ 1.0 n) 2e-50)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 0.02)
         (/ (exp (/ (log x) n)) (* x n))
         (if (<= (/ 1.0 n) 2e+239)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 0.3333333333333333 (* x (* x (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = t_0 * (1.0 / (x * n));
	} else if ((1.0 / n) <= 2e-50) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = exp((log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 2e+239) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-9)) then
        tmp = t_0 * (1.0d0 / (x * n))
    else if ((1.0d0 / n) <= 2d-50) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 0.02d0) then
        tmp = exp((log(x) / n)) / (x * n)
    else if ((1.0d0 / n) <= 2d+239) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = t_0 * (1.0 / (x * n));
	} else if ((1.0 / n) <= 2e-50) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 2e+239) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-9:
		tmp = t_0 * (1.0 / (x * n))
	elif (1.0 / n) <= 2e-50:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 0.02:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	elif (1.0 / n) <= 2e+239:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-9)
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * n)));
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e+239)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-9)
		tmp = t_0 * (1.0 / (x * n));
	elseif ((1.0 / n) <= 2e-50)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 0.02)
		tmp = exp((log(x) / n)) / (x * n);
	elseif ((1.0 / n) <= 2e+239)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-9], N[(t$95$0 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+239], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000006e-9

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 97.5

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

   5. Simplified 97.5%

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

      1. div-inv N/A

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 97.5

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

   7. Applied egg-rr 97.5%

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

   if -1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.9

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

   5. Simplified 80.9%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 81.0

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

   7. Applied egg-rr 81.0%

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

   if 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 14.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.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. Simplified 68.8%

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

   6. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 70.1

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

   8. Simplified 70.1%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999998e239

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 82.8

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

   5. Simplified 82.8%

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

   if 1.99999999999999998e239 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 9.7

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

   5. Simplified 9.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-9)
     (* t_0 (/ 1.0 (* x n)))
     (if (<= (/ 1.0 n) 2e-50)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 0.02)
         (/ t_0 (* x n))
         (if (<= (/ 1.0 n) 2e+239)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 0.3333333333333333 (* x (* x (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = t_0 * (1.0 / (x * n));
	} else if ((1.0 / n) <= 2e-50) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e+239) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-9)) then
        tmp = t_0 * (1.0d0 / (x * n))
    else if ((1.0d0 / n) <= 2d-50) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 0.02d0) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 2d+239) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = t_0 * (1.0 / (x * n));
	} else if ((1.0 / n) <= 2e-50) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e+239) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-9:
		tmp = t_0 * (1.0 / (x * n))
	elif (1.0 / n) <= 2e-50:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 0.02:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 2e+239:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-9)
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * n)));
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e+239)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-9)
		tmp = t_0 * (1.0 / (x * n));
	elseif ((1.0 / n) <= 2e-50)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 0.02)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 2e+239)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-9], N[(t$95$0 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+239], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000006e-9

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 97.5

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

   5. Simplified 97.5%

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

      1. div-inv N/A

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 97.5

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

   7. Applied egg-rr 97.5%

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

   if -1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.9

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

   5. Simplified 80.9%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 81.0

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

   7. Applied egg-rr 81.0%

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

   if 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 14.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 69.9

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

   5. Simplified 69.9%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999998e239

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 82.8

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

   5. Simplified 82.8%

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

   if 1.99999999999999998e239 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 9.7

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

   5. Simplified 9.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{x \cdot n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* x n))))
   (if (<= (/ 1.0 n) -1e-9)
     t_1
     (if (<= (/ 1.0 n) 2e-50)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 0.02)
         t_1
         (if (<= (/ 1.0 n) 2e+239)
           (- (+ 1.0 (/ x n)) t_0)
           (/ 0.3333333333333333 (* x (* x (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (x * n);
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+239) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (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) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (x * n)
    if ((1.0d0 / n) <= (-1d-9)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-50) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 0.02d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d+239) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (x * n);
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+239) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (x * n)
	tmp = 0
	if (1.0 / n) <= -1e-9:
		tmp = t_1
	elif (1.0 / n) <= 2e-50:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 0.02:
		tmp = t_1
	elif (1.0 / n) <= 2e+239:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(x * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-9)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+239)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (x * n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-9)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-50)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 0.02)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e+239)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-9], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+239], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000006e-9 or 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 86.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 93.4

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

   5. Simplified 93.4%

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

   if -1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.9

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

   5. Simplified 80.9%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 81.0

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

   7. Applied egg-rr 81.0%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999998e239

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. /-lowering-/.f64 82.8

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

   5. Simplified 82.8%

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

   if 1.99999999999999998e239 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 9.7

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

   5. Simplified 9.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{x \cdot n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 (* x n))))
   (if (<= (/ 1.0 n) -1e-9)
     t_1
     (if (<= (/ 1.0 n) 2e-50)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 0.02)
         t_1
         (if (<= (/ 1.0 n) 2e+239)
           (- 1.0 t_0)
           (/ 0.3333333333333333 (* x (* x (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / (x * n);
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+239) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (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) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = t_0 / (x * n)
    if ((1.0d0 / n) <= (-1d-9)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-50) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 0.02d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d+239) then
        tmp = 1.0d0 - t_0
    else
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = t_0 / (x * n);
	double tmp;
	if ((1.0 / n) <= -1e-9) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-50) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 0.02) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+239) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / (x * n)
	tmp = 0
	if (1.0 / n) <= -1e-9:
		tmp = t_1
	elif (1.0 / n) <= 2e-50:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 0.02:
		tmp = t_1
	elif (1.0 / n) <= 2e+239:
		tmp = 1.0 - t_0
	else:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / Float64(x * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-9)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-50)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+239)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = t_0 / (x * n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-9)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-50)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 0.02)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e+239)
		tmp = 1.0 - t_0;
	else
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-9], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-50], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+239], N[(1.0 - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000006e-9 or 2.00000000000000002e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 86.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. exp-to-pow N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 93.4

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

   5. Simplified 93.4%

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

   if -1.00000000000000006e-9 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000002e-50

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 80.9

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

   5. Simplified 80.9%

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

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

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

      2. diff-log N/A

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

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

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

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

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

      5. +-lowering-+.f64 81.0

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

   7. Applied egg-rr 81.0%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999998e239

   1. Initial program 80.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

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

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

      17. /-lowering-/.f64 80.1

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

   5. Simplified 80.1%

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

   if 1.99999999999999998e239 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 3.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 9.7

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

   5. Simplified 9.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+239}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.86)
   (/ (- x (log x)) n)
   (if (<= x 3.4e+159)
     (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.86) {
		tmp = (x - log(x)) / n;
	} else if (x <= 3.4e+159) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.86d0) then
        tmp = (x - log(x)) / n
    else if (x <= 3.4d+159) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.86) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 3.4e+159) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.86:
		tmp = (x - math.log(x)) / n
	elif x <= 3.4e+159:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.86)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 3.4e+159)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.86)
		tmp = (x - log(x)) / n;
	elseif (x <= 3.4e+159)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.86], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3.4e+159], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.859999999999999987

   1. Initial program 42.2%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 57.7

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

   5. Simplified 57.7%

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

   6. Taylor expanded in x around 0

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

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

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

      2. log-lowering-log.f64 57.3

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

   8. Simplified 57.3%

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

   if 0.859999999999999987 < x < 3.39999999999999991e159

   1. Initial program 44.6%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 41.5

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

   5. Simplified 41.5%

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

   6. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

         \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{x}}{n} \]

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

          \[\leadsto \frac{\frac{\color{blue}{1 + \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{x}}{n} \]

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

          \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{x}}{n} \]

      12. sub-neg N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\frac{1}{3} \cdot \frac{1}{x} + \color{blue}{\frac{-1}{2}}}{x}}{x}}{n} \]

      14. +-commutative N/A

          \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{1}{3} \cdot \frac{1}{x}}}{x}}{x}}{n} \]

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

          \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{1}{3} \cdot \frac{1}{x}}}{x}}{x}}{n} \]

      16. associate-*r/ N/A

          \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{3} \cdot 1}{x}}}{x}}{x}}{n} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \frac{\color{blue}{\frac{1}{3}}}{x}}{x}}{x}}{n} \]

      18. /-lowering-/.f64 66.0

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

   8. Simplified 66.0%

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

   if 3.39999999999999991e159 < x

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

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

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

      17. /-lowering-/.f64 43.2

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

   5. Simplified 43.2%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 83.0%

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

      1. metadata-eval 83.0

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

   10. Applied egg-rr 83.0%

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

Alternative 10: 61.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.58)
   (- 0.0 (/ (log x) n))
   (if (<= x 3e+160)
     (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.58) {
		tmp = 0.0 - (log(x) / n);
	} else if (x <= 3e+160) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.58d0) then
        tmp = 0.0d0 - (log(x) / n)
    else if (x <= 3d+160) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.58) {
		tmp = 0.0 - (Math.log(x) / n);
	} else if (x <= 3e+160) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.58:
		tmp = 0.0 - (math.log(x) / n)
	elif x <= 3e+160:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.58)
		tmp = Float64(0.0 - Float64(log(x) / n));
	elseif (x <= 3e+160)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.58)
		tmp = 0.0 - (log(x) / n);
	elseif (x <= 3e+160)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.58], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+160], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.57999999999999996

   1. Initial program 42.2%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

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

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

      17. /-lowering-/.f64 42.2

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

   5. Simplified 42.2%

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

   6. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. log-lowering-log.f64 57.1

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

   8. Simplified 57.1%

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

   if 0.57999999999999996 < x < 2.9999999999999999e160

   1. Initial program 44.6%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 41.5

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

   5. Simplified 41.5%

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

   6. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

         \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{x}}{n} \]

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

          \[\leadsto \frac{\frac{\color{blue}{1 + \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{x}}{n} \]

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

          \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{x}}{n} \]

      12. sub-neg N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\frac{1}{3} \cdot \frac{1}{x} + \color{blue}{\frac{-1}{2}}}{x}}{x}}{n} \]

      14. +-commutative N/A

          \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{1}{3} \cdot \frac{1}{x}}}{x}}{x}}{n} \]

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

          \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{1}{3} \cdot \frac{1}{x}}}{x}}{x}}{n} \]

      16. associate-*r/ N/A

          \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{3} \cdot 1}{x}}}{x}}{x}}{n} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \frac{\color{blue}{\frac{1}{3}}}{x}}{x}}{x}}{n} \]

      18. /-lowering-/.f64 66.0

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

   8. Simplified 66.0%

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

   if 2.9999999999999999e160 < x

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. exp-to-pow N/A

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

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

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

      17. /-lowering-/.f64 43.2

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

   5. Simplified 43.2%

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

   6. Taylor expanded in n around inf

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

   8. Simplified 83.0%

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

      1. metadata-eval 83.0

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

   10. Applied egg-rr 83.0%

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

Alternative 11: 52.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.32 \cdot 10^{+24}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.32e+24)
   (/ 0.3333333333333333 (* x (* x (* x n))))
   (/ (+ (/ 1.0 n) (/ (+ -0.5 (/ 0.3333333333333333 x)) (* x n))) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.32e+24) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1.32d+24)) then
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    else
        tmp = ((1.0d0 / n) + (((-0.5d0) + (0.3333333333333333d0 / x)) / (x * n))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.32e+24) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1.32e+24:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	else:
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.32e+24)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / Float64(x * n))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1.32e+24)
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	else
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.32e+24], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.32000000000000012e24

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 54.1

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

   5. Simplified 54.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 43.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 76.7

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

   11. Simplified 76.7%

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

   if -1.32000000000000012e24 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 32.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 62.2

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

   5. Simplified 62.2%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 40.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot n}, -0.5 + \frac{0.3333333333333333}{x}, \frac{1}{n}\right)}{x}} \]
   9. Step-by-step derivation

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

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

      2. associate-*l/ N/A

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

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

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

      4. *-lft-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}}{x \cdot n} + \frac{1}{n}}{x} \]

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

         \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}}{x \cdot n} + \frac{1}{n}}{x} \]

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

         \[\leadsto \frac{\frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{3}}{x}}}{x \cdot n} + \frac{1}{n}}{x} \]

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

         \[\leadsto \frac{\frac{\frac{-1}{2} + \frac{\frac{1}{3}}{x}}{\color{blue}{x \cdot n}} + \frac{1}{n}}{x} \]

      8. /-lowering-/.f64 40.3

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

   10. Applied egg-rr 40.3%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.32 \cdot 10^{+24}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x \cdot n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.32 \cdot 10^{+24}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.32e+24)
   (/ 0.3333333333333333 (* x (* x (* x n))))
   (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.32e+24) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / 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 ((1.0d0 / n) <= (-1.32d+24)) then
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    else
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.32e+24) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1.32e+24:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	else:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.32e+24)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1.32e+24)
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	else
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.32e+24], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.32000000000000012e24

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 54.1

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

   5. Simplified 54.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 43.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

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

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 76.7

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

   11. Simplified 76.7%

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

   if -1.32000000000000012e24 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 32.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

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

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

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. log-lowering-log.f64 62.2

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

   5. Simplified 62.2%

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

   6. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. div-sub N/A

         \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{x}}{n} \]

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

          \[\leadsto \frac{\frac{\color{blue}{1 + \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{x}}{n} \]

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

          \[\leadsto \frac{\frac{1 + \color{blue}{\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{x}}{n} \]

      12. sub-neg N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\frac{1}{3} \cdot \frac{1}{x} + \color{blue}{\frac{-1}{2}}}{x}}{x}}{n} \]

      14. +-commutative N/A

          \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{1}{3} \cdot \frac{1}{x}}}{x}}{x}}{n} \]

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

          \[\leadsto \frac{\frac{1 + \frac{\color{blue}{\frac{-1}{2} + \frac{1}{3} \cdot \frac{1}{x}}}{x}}{x}}{n} \]

      16. associate-*r/ N/A

          \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{3} \cdot 1}{x}}}{x}}{x}}{n} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\frac{1 + \frac{\frac{-1}{2} + \frac{\color{blue}{\frac{1}{3}}}{x}}{x}}{x}}{n} \]

      18. /-lowering-/.f64 40.3

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

   8. Simplified 40.3%

      \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.32 \cdot 10^{+24}:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.32e+24)
   (/ 0.3333333333333333 (* x (* x (* x n))))
   (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.32e+24) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / 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 ((1.0d0 / n) <= (-1.32d+24)) then
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    else
        tmp = (1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.32e+24) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1.32e+24:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	else:
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.32e+24)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1.32e+24)
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	else
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.32e+24], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.32000000000000012e24

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. log-lowering-log.f64 54.1

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{\frac{1}{3}}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{\frac{1}{2}}{n \cdot x}}{x}} \]

   8. Simplified 43.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot n}, -0.5 + \frac{0.3333333333333333}{x}, \frac{1}{n}\right)}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. *-lowering-*.f64 76.7

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   11. Simplified 76.7%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]

   if -1.32000000000000012e24 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 32.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. log-lowering-log.f64 62.2

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{\frac{1}{3}}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{\frac{1}{2}}{n \cdot x}}{x}} \]

   8. Simplified 40.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot n}, -0.5 + \frac{0.3333333333333333}{x}, \frac{1}{n}\right)}{x}} \]

   9. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n \cdot x}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\left(1 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{x}^{2}}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n \cdot x} \]

       2. metadata-eval N/A

          \[\leadsto \frac{\left(1 + \frac{\color{blue}{\frac{1}{3}}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n \cdot x} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n \cdot x}} \]

   11. Simplified 39.9%

       \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 45.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{+220}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -1.32 \cdot 10^{+24}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x n))))
   (if (<= (/ 1.0 n) -4e+220) t_0 (if (<= (/ 1.0 n) -1.32e+24) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (x * n);
	double tmp;
	if ((1.0 / n) <= -4e+220) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1.32e+24) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (x * n)
    if ((1.0d0 / n) <= (-4d+220)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-1.32d+24)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (x * n);
	double tmp;
	if ((1.0 / n) <= -4e+220) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1.32e+24) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (x * n)
	tmp = 0
	if (1.0 / n) <= -4e+220:
		tmp = t_0
	elif (1.0 / n) <= -1.32e+24:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(x * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e+220)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -1.32e+24)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (x * n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e+220)
		tmp = t_0;
	elseif ((1.0 / n) <= -1.32e+24)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e+220], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.32e+24], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e220 or -1.32000000000000012e24 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 39.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. log-lowering-log.f64 60.7

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

      3. *-lowering-*.f64 40.3

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   8. Simplified 40.3%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if -4e220 < (/.f64 #s(literal 1 binary64) n) < -1.32000000000000012e24

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. --lowering--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. /-lowering-/.f64 40.7

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 40.7%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 61.8%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 61.8

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 61.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5000000:\\ \;\;\;\;\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5000000.0)
   (/ 0.3333333333333333 (* x (* x (* x n))))
   (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5000000.0) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-5000000.0d0)) then
        tmp = 0.3333333333333333d0 / (x * (x * (x * n)))
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5000000.0) {
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5000000.0:
		tmp = 0.3333333333333333 / (x * (x * (x * n)))
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5000000.0)
		tmp = Float64(0.3333333333333333 / Float64(x * Float64(x * Float64(x * n))));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5000000.0)
		tmp = 0.3333333333333333 / (x * (x * (x * n)));
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000000.0], N[(0.3333333333333333 / N[(x * N[(x * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e6

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. log-lowering-log.f64 52.7

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{\frac{1}{3}}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{\frac{1}{2}}{n \cdot x}}{x}} \]

   8. Simplified 43.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot n}, -0.5 + \frac{0.3333333333333333}{x}, \frac{1}{n}\right)}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3}}{n \cdot {x}^{3}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{3} \cdot n}} \]

       3. cube-mult N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot n} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot n} \]

       5. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left({x}^{2} \cdot n\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(n \cdot {x}^{2}\right)}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{\color{blue}{x \cdot \left(n \cdot {x}^{2}\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left({x}^{2} \cdot n\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot n\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot n\right)\right)}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(n \cdot x\right)}\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \color{blue}{\left(x \cdot \left(n \cdot x\right)\right)}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{1}{3}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

       14. *-lowering-*.f64 76.0

           \[\leadsto \frac{0.3333333333333333}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot n\right)}\right)} \]

   11. Simplified 76.0%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{x \cdot \left(x \cdot \left(x \cdot n\right)\right)}} \]

   if -5e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 32.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. log-lowering-log.f64 62.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 39.1

         \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   8. Simplified 39.1%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 45.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -1.18 \cdot 10^{-217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -7.5e-25)
   (/ (/ 1.0 x) n)
   (if (<= n -1.18e-217) 0.0 (/ (/ 1.0 n) x))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -7.5e-25) {
		tmp = (1.0 / x) / n;
	} else if (n <= -1.18e-217) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-7.5d-25)) then
        tmp = (1.0d0 / x) / n
    else if (n <= (-1.18d-217)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -7.5e-25) {
		tmp = (1.0 / x) / n;
	} else if (n <= -1.18e-217) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -7.5e-25:
		tmp = (1.0 / x) / n
	elif n <= -1.18e-217:
		tmp = 0.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -7.5e-25)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (n <= -1.18e-217)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -7.5e-25)
		tmp = (1.0 / x) / n;
	elseif (n <= -1.18e-217)
		tmp = 0.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -7.5e-25], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -1.18e-217], 0.0, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.49999999999999989e-25

   1. Initial program 26.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. log-lowering-log.f64 74.9

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 74.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 40.5

         \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   8. Simplified 40.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if -7.49999999999999989e-25 < n < -1.18000000000000001e-217

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. --lowering--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. /-lowering-/.f64 40.7

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 40.7%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 61.8%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 61.8

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 61.8%

       \[\leadsto \color{blue}{0} \]

   if -1.18000000000000001e-217 < n

   1. Initial program 46.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. log-lowering-log.f64 52.9

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{\frac{1}{3}}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{\frac{1}{2}}{n \cdot x}}{x}} \]

   8. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot n}, -0.5 + \frac{0.3333333333333333}{x}, \frac{1}{n}\right)}{x}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 40.8

          \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

   11. Simplified 40.8%

       \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 45.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ \mathbf{if}\;n \leq -7.5 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.05 \cdot 10^{-216}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 n) x)))
   (if (<= n -7.5e-25) t_0 (if (<= n -2.05e-216) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double tmp;
	if (n <= -7.5e-25) {
		tmp = t_0;
	} else if (n <= -2.05e-216) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / n) / x
    if (n <= (-7.5d-25)) then
        tmp = t_0
    else if (n <= (-2.05d-216)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double tmp;
	if (n <= -7.5e-25) {
		tmp = t_0;
	} else if (n <= -2.05e-216) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / n) / x
	tmp = 0
	if n <= -7.5e-25:
		tmp = t_0
	elif n <= -2.05e-216:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / n) / x)
	tmp = 0.0
	if (n <= -7.5e-25)
		tmp = t_0;
	elseif (n <= -2.05e-216)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / n) / x;
	tmp = 0.0;
	if (n <= -7.5e-25)
		tmp = t_0;
	elseif (n <= -2.05e-216)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[n, -7.5e-25], t$95$0, If[LessEqual[n, -2.05e-216], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -7.49999999999999989e-25 or -2.05000000000000012e-216 < n

   1. Initial program 39.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) - \log x}}{n} \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

      4. log-lowering-log.f64 60.7

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \color{blue}{\log x}}{n} \]

   5. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{\frac{1}{3}}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{\frac{1}{2}}{n \cdot x}}{x}} \]

   8. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{x \cdot n}, -0.5 + \frac{0.3333333333333333}{x}, \frac{1}{n}\right)}{x}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 40.7

          \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

   11. Simplified 40.7%

       \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

   if -7.49999999999999989e-25 < n < -2.05000000000000012e-216

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

      2. mul-1-neg N/A

         \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

      5. log-rec N/A

         \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

      6. mul-1-neg N/A

         \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      7. --lowering--.f64 N/A

         \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

      8. log-rec N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

      9. mul-1-neg N/A

         \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

      10. associate-*r/ N/A

          \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

      11. associate-*r* N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

      12. metadata-eval N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

      13. *-commutative N/A

          \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      14. associate-/l* N/A

          \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      15. exp-to-pow N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      16. pow-lowering-pow.f64 N/A

          \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

      17. /-lowering-/.f64 40.7

          \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   5. Simplified 40.7%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   6. Taylor expanded in n around inf

      \[\leadsto 1 - \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 61.8%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 61.8

         \[\leadsto \color{blue}{0} \]

   10. Applied egg-rr 61.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 31.9% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 51.3%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
4. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto 1 - e^{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)}}{n}} \]

   2. mul-1-neg N/A

      \[\leadsto 1 - e^{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot \log x}\right)}{n}} \]

   3. distribute-neg-frac N/A

      \[\leadsto 1 - e^{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}\right)} \]

   5. log-rec N/A

      \[\leadsto 1 - e^{\mathsf{neg}\left(\frac{\color{blue}{\log \left(\frac{1}{x}\right)}}{n}\right)} \]

   6. mul-1-neg N/A

      \[\leadsto 1 - e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   7. --lowering--.f64 N/A

      \[\leadsto \color{blue}{1 - e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}} \]

   8. log-rec N/A

      \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{\mathsf{neg}\left(\log x\right)}}{n}} \]

   9. mul-1-neg N/A

      \[\leadsto 1 - e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}} \]

   10. associate-*r/ N/A

       \[\leadsto 1 - e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}} \]

   11. associate-*r* N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}} \]

   12. metadata-eval N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{1} \cdot \log x}{n}} \]

   13. *-commutative N/A

       \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

   14. associate-/l* N/A

       \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

   15. exp-to-pow N/A

       \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

   16. pow-lowering-pow.f64 N/A

       \[\leadsto 1 - \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \]

   17. /-lowering-/.f64 36.8

       \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

5. Simplified 36.8%

   \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

6. Taylor expanded in n around inf

   \[\leadsto 1 - \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 28.4%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 28.4

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 28.4%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))