2nthrt (problem 3.4.6)

Percentage Accurate: 54.4% → 85.3%

Time: 1.4min

Alternatives: 18

Speedup: 1.6×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{t\_0}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{t\_1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{n} \cdot \left(\log x - \mathsf{log1p}\left(x\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_1 + t\_0 \cdot \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ t_0 n)))
   (if (<= (/ 1.0 n) -2e-77)
     (/ t_1 x)
     (if (<= (/ 1.0 n) 5e-80)
       (* (/ -1.0 n) (- (log x) (log1p x)))
       (if (<= (/ 1.0 n) 2e-5)
         (/ (+ t_1 (* t_0 (/ (+ (/ 0.5 (pow n 2.0)) (/ -0.5 n)) x))) x)
         (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = t_0 / n;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_1 / x;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (-1.0 / n) * (log(x) - log1p(x));
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (t_1 + (t_0 * (((0.5 / pow(n, 2.0)) + (-0.5 / n)) / x))) / x;
	} else {
		tmp = exp((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 t_1 = t_0 / n;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_1 / x;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (-1.0 / n) * (Math.log(x) - Math.log1p(x));
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (t_1 + (t_0 * (((0.5 / Math.pow(n, 2.0)) + (-0.5 / n)) / x))) / x;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = t_0 / n
	tmp = 0
	if (1.0 / n) <= -2e-77:
		tmp = t_1 / x
	elif (1.0 / n) <= 5e-80:
		tmp = (-1.0 / n) * (math.log(x) - math.log1p(x))
	elif (1.0 / n) <= 2e-5:
		tmp = (t_1 + (t_0 * (((0.5 / math.pow(n, 2.0)) + (-0.5 / n)) / x))) / x
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(t_0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-77)
		tmp = Float64(t_1 / x);
	elseif (Float64(1.0 / n) <= 5e-80)
		tmp = Float64(Float64(-1.0 / n) * Float64(log(x) - log1p(x)));
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(Float64(t_1 + Float64(t_0 * Float64(Float64(Float64(0.5 / (n ^ 2.0)) + Float64(-0.5 / n)) / x))) / x);
	else
		tmp = Float64(exp(Float64(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]}, Block[{t$95$1 = N[(t$95$0 / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-77], N[(t$95$1 / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-80], N[(N[(-1.0 / n), $MachinePrecision] * N[(N[Log[x], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[(t$95$1 + N[(t$95$0 * N[(N[(N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf 91.8%

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

      1. mul-1-neg 91.8%

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

      2. log-rec 91.8%

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

      3. mul-1-neg 91.8%

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

      4. distribute-neg-frac 91.8%

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

      5. mul-1-neg 91.8%

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

      6. remove-double-neg 91.8%

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

      7. *-commutative 91.8%

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

   5. Simplified 91.8%

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

   6. Taylor expanded in x around 0 91.8%

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

      1. associate-/r* 92.6%

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

      2. *-rgt-identity 92.6%

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

      3. associate-*l/ 92.6%

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

      4. associate-*r/ 92.6%

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

      5. exp-to-pow 92.6%

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

   8. Simplified 92.6%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 5e-80

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 88.2%

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

   5. Simplified 88.2%

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

      1. add-sqr-sqrt 87.8%

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

      2. pow2 87.8%

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

      3. fma-neg 87.8%

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

   7. Applied egg-rr 87.8%

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

   8. Taylor expanded in n around inf 87.8%

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

      1. log1p-define 87.8%

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

   10. Simplified 87.8%

       \[\leadsto \frac{{\left(\sqrt{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}\right)}^{2}}{n} \]
   11. Step-by-step derivation

       1. div-inv 88.0%

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

       2. unpow2 88.0%

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

       3. add-sqr-sqrt 88.2%

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

   12. Applied egg-rr 88.2%

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

   if 5e-80 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 13.2%

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

      1. add-log-exp 12.9%

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

      2. pow-to-exp 12.9%

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

      3. un-div-inv 12.9%

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

      4. +-commutative 12.9%

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

      5. log1p-define 12.9%

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

   4. Applied egg-rr 12.9%

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

   5. Taylor expanded in x around inf 85.0%

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

   7. Simplified 85.0%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.8%

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

   3. Taylor expanded in n around 0 67.8%

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

      1. log1p-define 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{n} \cdot \left(\log x - \mathsf{log1p}\left(x\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) - \frac{\frac{-0.16666666666666666}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.5)
   (/
    (-
     (-
      (log1p x)
      (/
       (-
        (*
         (/ -0.16666666666666666 n)
         (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
        (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))))
       n))
     (log x))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.5) {
		tmp = ((log1p(x) - ((((-0.16666666666666666 / n) * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) - (0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0)))) / n)) - log(x)) / n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.5) {
		tmp = ((Math.log1p(x) - ((((-0.16666666666666666 / n) * (Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0))) - (0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)))) / n)) - Math.log(x)) / n;
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.5:
		tmp = ((math.log1p(x) - ((((-0.16666666666666666 / n) * (math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0))) - (0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)))) / n)) - math.log(x)) / n
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.5)
		tmp = Float64(Float64(Float64(log1p(x) - Float64(Float64(Float64(Float64(-0.16666666666666666 / n) * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) - Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)))) / n)) - log(x)) / n);
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.5], N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[(N[(N[(N[(-0.16666666666666666 / n), $MachinePrecision] * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 40.8%

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

   3. Taylor expanded in n around -inf 82.0%

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

   5. Simplified 82.0%

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

   if 3.5 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0 97.7%

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

      1. associate-/r* 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.3%

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

      5. exp-to-pow 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) - \frac{\frac{-0.16666666666666666}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) - 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.52:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(\frac{1}{n} \cdot \frac{{\log x}^{3}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.52)
   (/
    (-
     (* -0.16666666666666666 (* (/ 1.0 n) (/ (pow (log x) 3.0) n)))
     (+ (log x) (* 0.5 (/ (pow (log x) 2.0) n))))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.52) {
		tmp = ((-0.16666666666666666 * ((1.0 / n) * (pow(log(x), 3.0) / n))) - (log(x) + (0.5 * (pow(log(x), 2.0) / n)))) / n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.52d0) then
        tmp = (((-0.16666666666666666d0) * ((1.0d0 / n) * ((log(x) ** 3.0d0) / n))) - (log(x) + (0.5d0 * ((log(x) ** 2.0d0) / n)))) / n
    else
        tmp = ((x ** (1.0d0 / n)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.52) {
		tmp = ((-0.16666666666666666 * ((1.0 / n) * (Math.pow(Math.log(x), 3.0) / n))) - (Math.log(x) + (0.5 * (Math.pow(Math.log(x), 2.0) / n)))) / n;
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.52:
		tmp = ((-0.16666666666666666 * ((1.0 / n) * (math.pow(math.log(x), 3.0) / n))) - (math.log(x) + (0.5 * (math.pow(math.log(x), 2.0) / n)))) / n
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.52)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * Float64(Float64(1.0 / n) * Float64((log(x) ^ 3.0) / n))) - Float64(log(x) + Float64(0.5 * Float64((log(x) ^ 2.0) / n)))) / n);
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.52)
		tmp = ((-0.16666666666666666 * ((1.0 / n) * ((log(x) ^ 3.0) / n))) - (log(x) + (0.5 * ((log(x) ^ 2.0) / n)))) / n;
	else
		tmp = ((x ^ (1.0 / n)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.52], N[(N[(N[(-0.16666666666666666 * N[(N[(1.0 / n), $MachinePrecision] * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] + N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.52000000000000002

   1. Initial program 40.8%

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

   3. Taylor expanded in x around 0 40.1%

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

   4. Taylor expanded in n around inf 81.4%

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

      1. *-un-lft-identity 81.4%

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

      2. unpow2 81.4%

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

      3. times-frac 81.4%

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

   6. Applied egg-rr 81.4%

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

   if 0.52000000000000002 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0 97.7%

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

      1. associate-/r* 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.3%

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

      5. exp-to-pow 99.3%

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

   8. Simplified 99.3%

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

Alternative 4: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{n} \cdot \left(\log x - \mathsf{log1p}\left(x\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -2e-77)
     t_1
     (if (<= (/ 1.0 n) 5e-80)
       (* (/ -1.0 n) (- (log x) (log1p x)))
       (if (<= (/ 1.0 n) 2e-5) t_1 (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (-1.0 / n) * (log(x) - log1p(x));
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_1;
	} else {
		tmp = exp((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 t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (-1.0 / n) * (Math.log(x) - Math.log1p(x));
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_1;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -2e-77:
		tmp = t_1
	elif (1.0 / n) <= 5e-80:
		tmp = (-1.0 / n) * (math.log(x) - math.log1p(x))
	elif (1.0 / n) <= 2e-5:
		tmp = t_1
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-77)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-80)
		tmp = Float64(Float64(-1.0 / n) * Float64(log(x) - log1p(x)));
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(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]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-77], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-80], N[(N[(-1.0 / n), $MachinePrecision] * N[(N[Log[x], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], t$95$1, N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77 or 5e-80 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf 90.0%

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

      1. mul-1-neg 90.0%

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

      2. log-rec 90.0%

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

      3. mul-1-neg 90.0%

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

      4. distribute-neg-frac 90.0%

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

      5. mul-1-neg 90.0%

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

      6. remove-double-neg 90.0%

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

      7. *-commutative 90.0%

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

   5. Simplified 90.0%

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

   6. Taylor expanded in x around 0 90.0%

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

      1. associate-/r* 91.0%

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

      2. *-rgt-identity 91.0%

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

      3. associate-*l/ 91.0%

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

      4. associate-*r/ 91.0%

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

      5. exp-to-pow 91.0%

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

   8. Simplified 91.0%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 5e-80

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 88.2%

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

   5. Simplified 88.2%

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

      1. add-sqr-sqrt 87.8%

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

      2. pow2 87.8%

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

      3. fma-neg 87.8%

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

   7. Applied egg-rr 87.8%

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

   8. Taylor expanded in n around inf 87.8%

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

      1. log1p-define 87.8%

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

   10. Simplified 87.8%

       \[\leadsto \frac{{\left(\sqrt{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}\right)}^{2}}{n} \]
   11. Step-by-step derivation

       1. div-inv 88.0%

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

       2. unpow2 88.0%

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

       3. add-sqr-sqrt 88.2%

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

   12. Applied egg-rr 88.2%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.8%

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

   3. Taylor expanded in n around 0 67.8%

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

      1. log1p-define 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{n} \cdot \left(\log x - \mathsf{log1p}\left(x\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{n} \cdot \left(\log x - \mathsf{log1p}\left(x\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -2e-77)
     t_1
     (if (<= (/ 1.0 n) 5e-80)
       (* (/ -1.0 n) (- (log x) (log1p x)))
       (if (<= (/ 1.0 n) 2e-5) t_1 (- (+ 1.0 (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (-1.0 / n) * (log(x) - log1p(x));
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (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 t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (-1.0 / n) * (Math.log(x) - Math.log1p(x));
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -2e-77:
		tmp = t_1
	elif (1.0 / n) <= 5e-80:
		tmp = (-1.0 / n) * (math.log(x) - math.log1p(x))
	elif (1.0 / n) <= 2e-5:
		tmp = t_1
	else:
		tmp = (1.0 + (x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-77)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-80)
		tmp = Float64(Float64(-1.0 / n) * Float64(log(x) - log1p(x)));
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 + Float64(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]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-77], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-80], N[(N[(-1.0 / n), $MachinePrecision] * N[(N[Log[x], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], t$95$1, N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77 or 5e-80 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf 90.0%

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

      1. mul-1-neg 90.0%

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

      2. log-rec 90.0%

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

      3. mul-1-neg 90.0%

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

      4. distribute-neg-frac 90.0%

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

      5. mul-1-neg 90.0%

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

      6. remove-double-neg 90.0%

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

      7. *-commutative 90.0%

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

   5. Simplified 90.0%

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

   6. Taylor expanded in x around 0 90.0%

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

      1. associate-/r* 91.0%

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

      2. *-rgt-identity 91.0%

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

      3. associate-*l/ 91.0%

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

      4. associate-*r/ 91.0%

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

      5. exp-to-pow 91.0%

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

   8. Simplified 91.0%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 5e-80

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 88.2%

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

   5. Simplified 88.2%

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

      1. add-sqr-sqrt 87.8%

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

      2. pow2 87.8%

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

      3. fma-neg 87.8%

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

   7. Applied egg-rr 87.8%

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

   8. Taylor expanded in n around inf 87.8%

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

      1. log1p-define 87.8%

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

   10. Simplified 87.8%

       \[\leadsto \frac{{\left(\sqrt{\color{blue}{\mathsf{log1p}\left(x\right) - \log x}}\right)}^{2}}{n} \]
   11. Step-by-step derivation

       1. div-inv 88.0%

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

       2. unpow2 88.0%

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

       3. add-sqr-sqrt 88.2%

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

   12. Applied egg-rr 88.2%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0 66.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-1}{n} \cdot \left(\log x - \mathsf{log1p}\left(x\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -2e-77)
     t_1
     (if (<= (/ 1.0 n) 5e-80)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e-5) t_1 (- (+ 1.0 (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (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 t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -2e-77:
		tmp = t_1
	elif (1.0 / n) <= 5e-80:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-5:
		tmp = t_1
	else:
		tmp = (1.0 + (x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-77)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-80)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 + Float64(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]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-77], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-80], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], t$95$1, N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77 or 5e-80 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf 90.0%

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

      1. mul-1-neg 90.0%

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

      2. log-rec 90.0%

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

      3. mul-1-neg 90.0%

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

      4. distribute-neg-frac 90.0%

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

      5. mul-1-neg 90.0%

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

      6. remove-double-neg 90.0%

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

      7. *-commutative 90.0%

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

   5. Simplified 90.0%

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

   6. Taylor expanded in x around 0 90.0%

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

      1. associate-/r* 91.0%

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

      2. *-rgt-identity 91.0%

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

      3. associate-*l/ 91.0%

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

      4. associate-*r/ 91.0%

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

      5. exp-to-pow 91.0%

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

   8. Simplified 91.0%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 5e-80

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 88.2%

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

      1. log1p-define 88.2%

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

   5. Simplified 88.2%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0 66.1%

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

Alternative 7: 71.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 3.4 \cdot 10^{-226}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-215}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.125:\\ \;\;\;\;x \cdot \left(\frac{1}{n} - x \cdot \left(0.5 \cdot \frac{1}{n} - \frac{x}{n} \cdot 0.3333333333333333\right)\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log x) (- n))))
   (if (<= x 3.4e-226)
     t_1
     (if (<= x 5e-215)
       (- (+ 1.0 (/ x n)) t_0)
       (if (<= x 2.25e-129)
         t_1
         (if (<= x 4.5e-125)
           (/
            (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* x n))) x))
            x)
           (if (<= x 0.125)
             (-
              (*
               x
               (-
                (/ 1.0 n)
                (* x (- (* 0.5 (/ 1.0 n)) (* (/ x n) 0.3333333333333333)))))
              (/ (log x) n))
             (/ (/ t_0 n) x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(x) / -n;
	double tmp;
	if (x <= 3.4e-226) {
		tmp = t_1;
	} else if (x <= 5e-215) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 2.25e-129) {
		tmp = t_1;
	} else if (x <= 4.5e-125) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 0.125) {
		tmp = (x * ((1.0 / n) - (x * ((0.5 * (1.0 / n)) - ((x / n) * 0.3333333333333333))))) - (log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(x) / -n
    if (x <= 3.4d-226) then
        tmp = t_1
    else if (x <= 5d-215) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 2.25d-129) then
        tmp = t_1
    else if (x <= 4.5d-125) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (x * n))) / x)) / x
    else if (x <= 0.125d0) then
        tmp = (x * ((1.0d0 / n) - (x * ((0.5d0 * (1.0d0 / n)) - ((x / n) * 0.3333333333333333d0))))) - (log(x) / n)
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(x) / -n;
	double tmp;
	if (x <= 3.4e-226) {
		tmp = t_1;
	} else if (x <= 5e-215) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 2.25e-129) {
		tmp = t_1;
	} else if (x <= 4.5e-125) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 0.125) {
		tmp = (x * ((1.0 / n) - (x * ((0.5 * (1.0 / n)) - ((x / n) * 0.3333333333333333))))) - (Math.log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(x) / -n
	tmp = 0
	if x <= 3.4e-226:
		tmp = t_1
	elif x <= 5e-215:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 2.25e-129:
		tmp = t_1
	elif x <= 4.5e-125:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x
	elif x <= 0.125:
		tmp = (x * ((1.0 / n) - (x * ((0.5 * (1.0 / n)) - ((x / n) * 0.3333333333333333))))) - (math.log(x) / n)
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 3.4e-226)
		tmp = t_1;
	elseif (x <= 5e-215)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 2.25e-129)
		tmp = t_1;
	elseif (x <= 4.5e-125)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(x * n))) / x)) / x);
	elseif (x <= 0.125)
		tmp = Float64(Float64(x * Float64(Float64(1.0 / n) - Float64(x * Float64(Float64(0.5 * Float64(1.0 / n)) - Float64(Float64(x / n) * 0.3333333333333333))))) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(x) / -n;
	tmp = 0.0;
	if (x <= 3.4e-226)
		tmp = t_1;
	elseif (x <= 5e-215)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 2.25e-129)
		tmp = t_1;
	elseif (x <= 4.5e-125)
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	elseif (x <= 0.125)
		tmp = (x * ((1.0 / n) - (x * ((0.5 * (1.0 / n)) - ((x / n) * 0.3333333333333333))))) - (log(x) / n);
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 3.4e-226], t$95$1, If[LessEqual[x, 5e-215], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 2.25e-129], t$95$1, If[LessEqual[x, 4.5e-125], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.125], N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] - N[(x * N[(N[(0.5 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision] - N[(N[(x / n), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 3.40000000000000007e-226 or 4.99999999999999956e-215 < x < 2.25000000000000015e-129

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 35.8%

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

   4. Taylor expanded in n around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-frac-neg2 66.7%

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

   6. Simplified 66.7%

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

   if 3.40000000000000007e-226 < x < 4.99999999999999956e-215

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 83.9%

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

   if 2.25000000000000015e-129 < x < 4.50000000000000012e-125

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 51.8%

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

   5. Simplified 51.8%

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

      1. add-sqr-sqrt 51.8%

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

      2. pow2 51.8%

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

      3. fma-neg 51.8%

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

   7. Applied egg-rr 51.8%

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

   8. Taylor expanded in n around inf 6.0%

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

      1. log1p-define 6.0%

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

   10. Simplified 6.0%

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

   11. Taylor expanded in x around -inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   13. Simplified 100.0%

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

   if 4.50000000000000012e-125 < x < 0.125

   1. Initial program 37.7%

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

   3. Taylor expanded in n around inf 79.2%

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

   5. Simplified 79.2%

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

      1. add-sqr-sqrt 78.8%

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

      2. pow2 78.8%

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

      3. fma-neg 78.8%

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

   7. Applied egg-rr 78.8%

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

   8. Taylor expanded in n around inf 60.7%

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

      1. log1p-define 60.7%

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

   10. Simplified 60.7%

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

   11. Taylor expanded in x around 0 60.1%

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

   if 0.125 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0 97.7%

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

      1. associate-/r* 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.3%

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

      5. exp-to-pow 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-226}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-215}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.125:\\ \;\;\;\;x \cdot \left(\frac{1}{n} - x \cdot \left(0.5 \cdot \frac{1}{n} - \frac{x}{n} \cdot 0.3333333333333333\right)\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 4.5 \cdot 10^{-226}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-215}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.043:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log x) (- n))))
   (if (<= x 4.5e-226)
     t_1
     (if (<= x 4.4e-215)
       (- (+ 1.0 (/ x n)) t_0)
       (if (<= x 2.3e-129)
         t_1
         (if (<= x 5.2e-125)
           (/
            (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* x n))) x))
            x)
           (if (<= x 0.043)
             (- (* x (+ (/ 1.0 n) (* -0.5 (/ x n)))) (/ (log x) n))
             (/ (/ t_0 n) x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(x) / -n;
	double tmp;
	if (x <= 4.5e-226) {
		tmp = t_1;
	} else if (x <= 4.4e-215) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 2.3e-129) {
		tmp = t_1;
	} else if (x <= 5.2e-125) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 0.043) {
		tmp = (x * ((1.0 / n) + (-0.5 * (x / n)))) - (log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(x) / -n
    if (x <= 4.5d-226) then
        tmp = t_1
    else if (x <= 4.4d-215) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 2.3d-129) then
        tmp = t_1
    else if (x <= 5.2d-125) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (x * n))) / x)) / x
    else if (x <= 0.043d0) then
        tmp = (x * ((1.0d0 / n) + ((-0.5d0) * (x / n)))) - (log(x) / n)
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(x) / -n;
	double tmp;
	if (x <= 4.5e-226) {
		tmp = t_1;
	} else if (x <= 4.4e-215) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 2.3e-129) {
		tmp = t_1;
	} else if (x <= 5.2e-125) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 0.043) {
		tmp = (x * ((1.0 / n) + (-0.5 * (x / n)))) - (Math.log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(x) / -n
	tmp = 0
	if x <= 4.5e-226:
		tmp = t_1
	elif x <= 4.4e-215:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 2.3e-129:
		tmp = t_1
	elif x <= 5.2e-125:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x
	elif x <= 0.043:
		tmp = (x * ((1.0 / n) + (-0.5 * (x / n)))) - (math.log(x) / n)
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 4.5e-226)
		tmp = t_1;
	elseif (x <= 4.4e-215)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 2.3e-129)
		tmp = t_1;
	elseif (x <= 5.2e-125)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(x * n))) / x)) / x);
	elseif (x <= 0.043)
		tmp = Float64(Float64(x * Float64(Float64(1.0 / n) + Float64(-0.5 * Float64(x / n)))) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(x) / -n;
	tmp = 0.0;
	if (x <= 4.5e-226)
		tmp = t_1;
	elseif (x <= 4.4e-215)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 2.3e-129)
		tmp = t_1;
	elseif (x <= 5.2e-125)
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	elseif (x <= 0.043)
		tmp = (x * ((1.0 / n) + (-0.5 * (x / n)))) - (log(x) / n);
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 4.5e-226], t$95$1, If[LessEqual[x, 4.4e-215], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 2.3e-129], t$95$1, If[LessEqual[x, 5.2e-125], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.043], N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(-0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 4.50000000000000011e-226 or 4.39999999999999993e-215 < x < 2.3e-129

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 35.8%

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

   4. Taylor expanded in n around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-frac-neg2 66.7%

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

   6. Simplified 66.7%

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

   if 4.50000000000000011e-226 < x < 4.39999999999999993e-215

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 83.9%

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

   if 2.3e-129 < x < 5.20000000000000011e-125

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 51.8%

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

   5. Simplified 51.8%

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

      1. add-sqr-sqrt 51.8%

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

      2. pow2 51.8%

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

      3. fma-neg 51.8%

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

   7. Applied egg-rr 51.8%

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

   8. Taylor expanded in n around inf 6.0%

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

      1. log1p-define 6.0%

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

   10. Simplified 6.0%

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

   11. Taylor expanded in x around -inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   13. Simplified 100.0%

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

   if 5.20000000000000011e-125 < x < 0.042999999999999997

   1. Initial program 37.7%

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

   3. Taylor expanded in n around inf 79.2%

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

   5. Simplified 79.2%

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

      1. add-sqr-sqrt 78.8%

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

      2. pow2 78.8%

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

      3. fma-neg 78.8%

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

   7. Applied egg-rr 78.8%

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

   8. Taylor expanded in n around inf 60.7%

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

      1. log1p-define 60.7%

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

   10. Simplified 60.7%

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

   11. Taylor expanded in x around 0 59.9%

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

   if 0.042999999999999997 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0 97.7%

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

      1. associate-/r* 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.3%

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

      5. exp-to-pow 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-215}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.043:\\ \;\;\;\;x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 5.1 \cdot 10^{-226}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-215}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.043:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log x) (- n))))
   (if (<= x 5.1e-226)
     t_1
     (if (<= x 4e-215)
       (- (+ 1.0 (/ x n)) t_0)
       (if (<= x 2.3e-129)
         t_1
         (if (<= x 4.5e-125)
           (/
            (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* x n))) x))
            x)
           (if (<= x 0.043) (- (/ x n) (/ (log x) n)) (/ (/ t_0 n) x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(x) / -n;
	double tmp;
	if (x <= 5.1e-226) {
		tmp = t_1;
	} else if (x <= 4e-215) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 2.3e-129) {
		tmp = t_1;
	} else if (x <= 4.5e-125) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 0.043) {
		tmp = (x / n) - (log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(x) / -n
    if (x <= 5.1d-226) then
        tmp = t_1
    else if (x <= 4d-215) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 2.3d-129) then
        tmp = t_1
    else if (x <= 4.5d-125) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (x * n))) / x)) / x
    else if (x <= 0.043d0) then
        tmp = (x / n) - (log(x) / n)
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(x) / -n;
	double tmp;
	if (x <= 5.1e-226) {
		tmp = t_1;
	} else if (x <= 4e-215) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 2.3e-129) {
		tmp = t_1;
	} else if (x <= 4.5e-125) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 0.043) {
		tmp = (x / n) - (Math.log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(x) / -n
	tmp = 0
	if x <= 5.1e-226:
		tmp = t_1
	elif x <= 4e-215:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 2.3e-129:
		tmp = t_1
	elif x <= 4.5e-125:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x
	elif x <= 0.043:
		tmp = (x / n) - (math.log(x) / n)
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 5.1e-226)
		tmp = t_1;
	elseif (x <= 4e-215)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 2.3e-129)
		tmp = t_1;
	elseif (x <= 4.5e-125)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(x * n))) / x)) / x);
	elseif (x <= 0.043)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(x) / -n;
	tmp = 0.0;
	if (x <= 5.1e-226)
		tmp = t_1;
	elseif (x <= 4e-215)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 2.3e-129)
		tmp = t_1;
	elseif (x <= 4.5e-125)
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	elseif (x <= 0.043)
		tmp = (x / n) - (log(x) / n);
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 5.1e-226], t$95$1, If[LessEqual[x, 4e-215], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 2.3e-129], t$95$1, If[LessEqual[x, 4.5e-125], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.043], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 5.09999999999999973e-226 or 4.00000000000000017e-215 < x < 2.3e-129

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 35.8%

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

   4. Taylor expanded in n around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-frac-neg2 66.7%

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

   6. Simplified 66.7%

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

   if 5.09999999999999973e-226 < x < 4.00000000000000017e-215

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 83.9%

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

   if 2.3e-129 < x < 4.50000000000000012e-125

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 51.8%

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

   5. Simplified 51.8%

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

      1. add-sqr-sqrt 51.8%

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

      2. pow2 51.8%

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

      3. fma-neg 51.8%

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

   7. Applied egg-rr 51.8%

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

   8. Taylor expanded in n around inf 6.0%

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

      1. log1p-define 6.0%

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

   10. Simplified 6.0%

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

   11. Taylor expanded in x around -inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   13. Simplified 100.0%

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

   if 4.50000000000000012e-125 < x < 0.042999999999999997

   1. Initial program 37.7%

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

   3. Taylor expanded in n around inf 79.2%

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

   5. Simplified 79.2%

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

      1. add-sqr-sqrt 78.8%

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

      2. pow2 78.8%

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

      3. fma-neg 78.8%

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

   7. Applied egg-rr 78.8%

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

   8. Taylor expanded in n around inf 60.7%

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

      1. log1p-define 60.7%

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

   10. Simplified 60.7%

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

   11. Taylor expanded in x around 0 59.8%

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

   if 0.042999999999999997 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0 97.7%

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

      1. associate-/r* 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.3%

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

      5. exp-to-pow 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-215}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.043:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2 \cdot 10^{-226}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-215}:\\ \;\;\;\;1 - t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.055:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (pow x (/ 1.0 n))))
   (if (<= x 2e-226)
     t_0
     (if (<= x 4.1e-215)
       (- 1.0 t_1)
       (if (<= x 2.3e-129)
         t_0
         (if (<= x 1.25e-124)
           (/
            (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* x n))) x))
            x)
           (if (<= x 0.055) (- (/ x n) (/ (log x) n)) (/ (/ t_1 n) x))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 2e-226) {
		tmp = t_0;
	} else if (x <= 4.1e-215) {
		tmp = 1.0 - t_1;
	} else if (x <= 2.3e-129) {
		tmp = t_0;
	} else if (x <= 1.25e-124) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 0.055) {
		tmp = (x / n) - (log(x) / n);
	} else {
		tmp = (t_1 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = x ** (1.0d0 / n)
    if (x <= 2d-226) then
        tmp = t_0
    else if (x <= 4.1d-215) then
        tmp = 1.0d0 - t_1
    else if (x <= 2.3d-129) then
        tmp = t_0
    else if (x <= 1.25d-124) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (x * n))) / x)) / x
    else if (x <= 0.055d0) then
        tmp = (x / n) - (log(x) / n)
    else
        tmp = (t_1 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 2e-226) {
		tmp = t_0;
	} else if (x <= 4.1e-215) {
		tmp = 1.0 - t_1;
	} else if (x <= 2.3e-129) {
		tmp = t_0;
	} else if (x <= 1.25e-124) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 0.055) {
		tmp = (x / n) - (Math.log(x) / n);
	} else {
		tmp = (t_1 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 2e-226:
		tmp = t_0
	elif x <= 4.1e-215:
		tmp = 1.0 - t_1
	elif x <= 2.3e-129:
		tmp = t_0
	elif x <= 1.25e-124:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x
	elif x <= 0.055:
		tmp = (x / n) - (math.log(x) / n)
	else:
		tmp = (t_1 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 2e-226)
		tmp = t_0;
	elseif (x <= 4.1e-215)
		tmp = Float64(1.0 - t_1);
	elseif (x <= 2.3e-129)
		tmp = t_0;
	elseif (x <= 1.25e-124)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(x * n))) / x)) / x);
	elseif (x <= 0.055)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(t_1 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 2e-226)
		tmp = t_0;
	elseif (x <= 4.1e-215)
		tmp = 1.0 - t_1;
	elseif (x <= 2.3e-129)
		tmp = t_0;
	elseif (x <= 1.25e-124)
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	elseif (x <= 0.055)
		tmp = (x / n) - (log(x) / n);
	else
		tmp = (t_1 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2e-226], t$95$0, If[LessEqual[x, 4.1e-215], N[(1.0 - t$95$1), $MachinePrecision], If[LessEqual[x, 2.3e-129], t$95$0, If[LessEqual[x, 1.25e-124], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.055], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.99999999999999984e-226 or 4.09999999999999985e-215 < x < 2.3e-129

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 35.8%

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

   4. Taylor expanded in n around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-frac-neg2 66.7%

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

   6. Simplified 66.7%

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

   if 1.99999999999999984e-226 < x < 4.09999999999999985e-215

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 83.9%

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

   if 2.3e-129 < x < 1.2500000000000001e-124

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 51.8%

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

   5. Simplified 51.8%

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

      1. add-sqr-sqrt 51.8%

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

      2. pow2 51.8%

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

      3. fma-neg 51.8%

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

   7. Applied egg-rr 51.8%

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

   8. Taylor expanded in n around inf 6.0%

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

      1. log1p-define 6.0%

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

   10. Simplified 6.0%

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

   11. Taylor expanded in x around -inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   13. Simplified 100.0%

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

   if 1.2500000000000001e-124 < x < 0.0550000000000000003

   1. Initial program 37.7%

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

   3. Taylor expanded in n around inf 79.2%

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

   5. Simplified 79.2%

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

      1. add-sqr-sqrt 78.8%

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

      2. pow2 78.8%

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

      3. fma-neg 78.8%

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

   7. Applied egg-rr 78.8%

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

   8. Taylor expanded in n around inf 60.7%

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

      1. log1p-define 60.7%

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

   10. Simplified 60.7%

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

   11. Taylor expanded in x around 0 59.8%

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

   if 0.0550000000000000003 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0 97.7%

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

      1. associate-/r* 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.3%

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

      5. exp-to-pow 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-226}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-215}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.055:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 4.5 \cdot 10^{-226}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-215}:\\ \;\;\;\;1 - t\_1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.055:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (pow x (/ 1.0 n))))
   (if (<= x 4.5e-226)
     t_0
     (if (<= x 6.8e-215)
       (- 1.0 t_1)
       (if (<= x 1.85e-129)
         t_0
         (if (<= x 4.8e-125)
           (/
            (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* x n))) x))
            x)
           (if (<= x 0.055) (/ (- x (log x)) n) (/ (/ t_1 n) x))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 4.5e-226) {
		tmp = t_0;
	} else if (x <= 6.8e-215) {
		tmp = 1.0 - t_1;
	} else if (x <= 1.85e-129) {
		tmp = t_0;
	} else if (x <= 4.8e-125) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 0.055) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (t_1 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = x ** (1.0d0 / n)
    if (x <= 4.5d-226) then
        tmp = t_0
    else if (x <= 6.8d-215) then
        tmp = 1.0d0 - t_1
    else if (x <= 1.85d-129) then
        tmp = t_0
    else if (x <= 4.8d-125) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (x * n))) / x)) / x
    else if (x <= 0.055d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (t_1 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 4.5e-226) {
		tmp = t_0;
	} else if (x <= 6.8e-215) {
		tmp = 1.0 - t_1;
	} else if (x <= 1.85e-129) {
		tmp = t_0;
	} else if (x <= 4.8e-125) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 0.055) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (t_1 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 4.5e-226:
		tmp = t_0
	elif x <= 6.8e-215:
		tmp = 1.0 - t_1
	elif x <= 1.85e-129:
		tmp = t_0
	elif x <= 4.8e-125:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x
	elif x <= 0.055:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (t_1 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 4.5e-226)
		tmp = t_0;
	elseif (x <= 6.8e-215)
		tmp = Float64(1.0 - t_1);
	elseif (x <= 1.85e-129)
		tmp = t_0;
	elseif (x <= 4.8e-125)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(x * n))) / x)) / x);
	elseif (x <= 0.055)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(t_1 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 4.5e-226)
		tmp = t_0;
	elseif (x <= 6.8e-215)
		tmp = 1.0 - t_1;
	elseif (x <= 1.85e-129)
		tmp = t_0;
	elseif (x <= 4.8e-125)
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	elseif (x <= 0.055)
		tmp = (x - log(x)) / n;
	else
		tmp = (t_1 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4.5e-226], t$95$0, If[LessEqual[x, 6.8e-215], N[(1.0 - t$95$1), $MachinePrecision], If[LessEqual[x, 1.85e-129], t$95$0, If[LessEqual[x, 4.8e-125], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.055], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 4.50000000000000011e-226 or 6.80000000000000003e-215 < x < 1.8500000000000001e-129

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 35.8%

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

   4. Taylor expanded in n around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-frac-neg2 66.7%

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

   6. Simplified 66.7%

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

   if 4.50000000000000011e-226 < x < 6.80000000000000003e-215

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 83.9%

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

   if 1.8500000000000001e-129 < x < 4.8000000000000003e-125

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 51.8%

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

   5. Simplified 51.8%

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

      1. add-sqr-sqrt 51.8%

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

      2. pow2 51.8%

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

      3. fma-neg 51.8%

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

   7. Applied egg-rr 51.8%

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

   8. Taylor expanded in n around inf 6.0%

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

      1. log1p-define 6.0%

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

   10. Simplified 6.0%

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

   11. Taylor expanded in x around -inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   13. Simplified 100.0%

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

   if 4.8000000000000003e-125 < x < 0.0550000000000000003

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 36.1%

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

   4. Taylor expanded in n around inf 59.8%

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

   if 0.0550000000000000003 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0 97.7%

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

      1. associate-/r* 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.3%

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

      5. exp-to-pow 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-215}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.055:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 2 \cdot 10^{-226}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-215}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125} \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 2e-226)
     t_0
     (if (<= x 4e-215)
       (- 1.0 (pow x (/ 1.0 n)))
       (if (<= x 2.3e-129)
         t_0
         (if (or (<= x 4.5e-125) (not (<= x 0.85)))
           (/
            (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* x n))) x))
            x)
           (/ (- x (log x)) n)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 2e-226) {
		tmp = t_0;
	} else if (x <= 4e-215) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 2.3e-129) {
		tmp = t_0;
	} else if ((x <= 4.5e-125) || !(x <= 0.85)) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else {
		tmp = (x - log(x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 2d-226) then
        tmp = t_0
    else if (x <= 4d-215) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 2.3d-129) then
        tmp = t_0
    else if ((x <= 4.5d-125) .or. (.not. (x <= 0.85d0))) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (x * n))) / x)) / x
    else
        tmp = (x - log(x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 2e-226) {
		tmp = t_0;
	} else if (x <= 4e-215) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 2.3e-129) {
		tmp = t_0;
	} else if ((x <= 4.5e-125) || !(x <= 0.85)) {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	} else {
		tmp = (x - Math.log(x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 2e-226:
		tmp = t_0
	elif x <= 4e-215:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 2.3e-129:
		tmp = t_0
	elif (x <= 4.5e-125) or not (x <= 0.85):
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x
	else:
		tmp = (x - math.log(x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 2e-226)
		tmp = t_0;
	elseif (x <= 4e-215)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 2.3e-129)
		tmp = t_0;
	elseif ((x <= 4.5e-125) || !(x <= 0.85))
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(x * n))) / x)) / x);
	else
		tmp = Float64(Float64(x - log(x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 2e-226)
		tmp = t_0;
	elseif (x <= 4e-215)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 2.3e-129)
		tmp = t_0;
	elseif ((x <= 4.5e-125) || ~((x <= 0.85)))
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	else
		tmp = (x - log(x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2e-226], t$95$0, If[LessEqual[x, 4e-215], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-129], t$95$0, If[Or[LessEqual[x, 4.5e-125], N[Not[LessEqual[x, 0.85]], $MachinePrecision]], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.99999999999999984e-226 or 4.00000000000000017e-215 < x < 2.3e-129

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 35.8%

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

   4. Taylor expanded in n around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-frac-neg2 66.7%

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

   6. Simplified 66.7%

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

   if 1.99999999999999984e-226 < x < 4.00000000000000017e-215

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 83.9%

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

   if 2.3e-129 < x < 4.50000000000000012e-125 or 0.849999999999999978 < x

   1. Initial program 68.8%

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

   3. Taylor expanded in n around inf 65.8%

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

   5. Simplified 66.6%

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

      1. add-sqr-sqrt 66.5%

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

      2. pow2 66.5%

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

      3. fma-neg 66.5%

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

   7. Applied egg-rr 66.5%

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

   8. Taylor expanded in n around inf 64.4%

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

      1. log1p-define 64.4%

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

   10. Simplified 64.4%

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

   11. Taylor expanded in x around -inf 66.0%

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

       1. mul-1-neg 66.0%

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

       2. distribute-neg-frac2 66.0%

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

   13. Simplified 66.0%

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

   if 4.50000000000000012e-125 < x < 0.849999999999999978

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 36.1%

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

   4. Taylor expanded in n around inf 59.8%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-226}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-215}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125} \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := \frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{if}\;x \leq 5.8 \cdot 10^{-226}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125} \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n)))
        (t_1
         (/
          (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* x n))) x))
          x)))
   (if (<= x 5.8e-226)
     t_0
     (if (<= x 3.5e-215)
       t_1
       (if (<= x 2.2e-129)
         t_0
         (if (or (<= x 4.5e-125) (not (<= x 0.85)))
           t_1
           (/ (- x (log x)) n)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	double tmp;
	if (x <= 5.8e-226) {
		tmp = t_0;
	} else if (x <= 3.5e-215) {
		tmp = t_1;
	} else if (x <= 2.2e-129) {
		tmp = t_0;
	} else if ((x <= 4.5e-125) || !(x <= 0.85)) {
		tmp = t_1;
	} else {
		tmp = (x - log(x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (x * n))) / x)) / x
    if (x <= 5.8d-226) then
        tmp = t_0
    else if (x <= 3.5d-215) then
        tmp = t_1
    else if (x <= 2.2d-129) then
        tmp = t_0
    else if ((x <= 4.5d-125) .or. (.not. (x <= 0.85d0))) then
        tmp = t_1
    else
        tmp = (x - log(x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	double tmp;
	if (x <= 5.8e-226) {
		tmp = t_0;
	} else if (x <= 3.5e-215) {
		tmp = t_1;
	} else if (x <= 2.2e-129) {
		tmp = t_0;
	} else if ((x <= 4.5e-125) || !(x <= 0.85)) {
		tmp = t_1;
	} else {
		tmp = (x - Math.log(x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x
	tmp = 0
	if x <= 5.8e-226:
		tmp = t_0
	elif x <= 3.5e-215:
		tmp = t_1
	elif x <= 2.2e-129:
		tmp = t_0
	elif (x <= 4.5e-125) or not (x <= 0.85):
		tmp = t_1
	else:
		tmp = (x - math.log(x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(x * n))) / x)) / x)
	tmp = 0.0
	if (x <= 5.8e-226)
		tmp = t_0;
	elseif (x <= 3.5e-215)
		tmp = t_1;
	elseif (x <= 2.2e-129)
		tmp = t_0;
	elseif ((x <= 4.5e-125) || !(x <= 0.85))
		tmp = t_1;
	else
		tmp = Float64(Float64(x - log(x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	tmp = 0.0;
	if (x <= 5.8e-226)
		tmp = t_0;
	elseif (x <= 3.5e-215)
		tmp = t_1;
	elseif (x <= 2.2e-129)
		tmp = t_0;
	elseif ((x <= 4.5e-125) || ~((x <= 0.85)))
		tmp = t_1;
	else
		tmp = (x - log(x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, 5.8e-226], t$95$0, If[LessEqual[x, 3.5e-215], t$95$1, If[LessEqual[x, 2.2e-129], t$95$0, If[Or[LessEqual[x, 4.5e-125], N[Not[LessEqual[x, 0.85]], $MachinePrecision]], t$95$1, N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.80000000000000003e-226 or 3.5000000000000002e-215 < x < 2.20000000000000003e-129

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 35.8%

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

   4. Taylor expanded in n around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-frac-neg2 66.7%

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

   6. Simplified 66.7%

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

   if 5.80000000000000003e-226 < x < 3.5000000000000002e-215 or 2.20000000000000003e-129 < x < 4.50000000000000012e-125 or 0.849999999999999978 < x

   1. Initial program 69.5%

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

   3. Taylor expanded in n around inf 63.7%

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

   5. Simplified 64.4%

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

      1. add-sqr-sqrt 64.4%

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

      2. pow2 64.4%

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

      3. fma-neg 64.4%

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

   7. Applied egg-rr 64.4%

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

   8. Taylor expanded in n around inf 61.7%

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

      1. log1p-define 61.7%

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

   10. Simplified 61.7%

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

   11. Taylor expanded in x around -inf 66.8%

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

       1. mul-1-neg 66.8%

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

       2. distribute-neg-frac2 66.8%

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

   13. Simplified 66.8%

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

   if 4.50000000000000012e-125 < x < 0.849999999999999978

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 36.1%

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

   4. Taylor expanded in n around inf 59.8%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125} \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.1 \cdot 10^{-226} \lor \neg \left(x \leq 3.5 \cdot 10^{-215}\right) \land \left(x \leq 2.05 \cdot 10^{-129} \lor \neg \left(x \leq 1.2 \cdot 10^{-124}\right) \land x \leq 0.6\right):\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= x 6.1e-226)
         (and (not (<= x 3.5e-215))
              (or (<= x 2.05e-129) (and (not (<= x 1.2e-124)) (<= x 0.6)))))
   (/ (log x) (- n))
   (/ (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ 0.3333333333333333 (* x n))) x)) x)))

C

double code(double x, double n) {
	double tmp;
	if ((x <= 6.1e-226) || (!(x <= 3.5e-215) && ((x <= 2.05e-129) || (!(x <= 1.2e-124) && (x <= 0.6))))) {
		tmp = log(x) / -n;
	} else {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((x <= 6.1d-226) .or. (.not. (x <= 3.5d-215)) .and. (x <= 2.05d-129) .or. (.not. (x <= 1.2d-124)) .and. (x <= 0.6d0)) then
        tmp = log(x) / -n
    else
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + (0.3333333333333333d0 / (x * n))) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((x <= 6.1e-226) || (!(x <= 3.5e-215) && ((x <= 2.05e-129) || (!(x <= 1.2e-124) && (x <= 0.6))))) {
		tmp = Math.log(x) / -n;
	} else {
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (x <= 6.1e-226) or (not (x <= 3.5e-215) and ((x <= 2.05e-129) or (not (x <= 1.2e-124) and (x <= 0.6)))):
		tmp = math.log(x) / -n
	else:
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((x <= 6.1e-226) || (!(x <= 3.5e-215) && ((x <= 2.05e-129) || (!(x <= 1.2e-124) && (x <= 0.6)))))
		tmp = Float64(log(x) / Float64(-n));
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(0.3333333333333333 / Float64(x * n))) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((x <= 6.1e-226) || (~((x <= 3.5e-215)) && ((x <= 2.05e-129) || (~((x <= 1.2e-124)) && (x <= 0.6)))))
		tmp = log(x) / -n;
	else
		tmp = ((1.0 / n) + (((-0.5 / n) + (0.3333333333333333 / (x * n))) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[x, 6.1e-226], And[N[Not[LessEqual[x, 3.5e-215]], $MachinePrecision], Or[LessEqual[x, 2.05e-129], And[N[Not[LessEqual[x, 1.2e-124]], $MachinePrecision], LessEqual[x, 0.6]]]]], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.0999999999999998e-226 or 3.5000000000000002e-215 < x < 2.05e-129 or 1.19999999999999996e-124 < x < 0.599999999999999978

   1. Initial program 36.6%

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

   3. Taylor expanded in x around 0 35.8%

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

   4. Taylor expanded in n around inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. distribute-frac-neg2 63.7%

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

   6. Simplified 63.7%

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

   if 6.0999999999999998e-226 < x < 3.5000000000000002e-215 or 2.05e-129 < x < 1.19999999999999996e-124 or 0.599999999999999978 < x

   1. Initial program 69.5%

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

   3. Taylor expanded in n around inf 63.7%

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

   5. Simplified 64.4%

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

      1. add-sqr-sqrt 64.4%

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

      2. pow2 64.4%

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

      3. fma-neg 64.4%

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

   7. Applied egg-rr 64.4%

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

   8. Taylor expanded in n around inf 61.7%

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

      1. log1p-define 61.7%

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

   10. Simplified 61.7%

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

   11. Taylor expanded in x around -inf 66.8%

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

       1. mul-1-neg 66.8%

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

       2. distribute-neg-frac2 66.8%

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

   13. Simplified 66.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{-x} + \frac{-1}{n}}{-x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.1 \cdot 10^{-226} \lor \neg \left(x \leq 3.5 \cdot 10^{-215}\right) \land \left(x \leq 2.05 \cdot 10^{-129} \lor \neg \left(x \leq 1.2 \cdot 10^{-124}\right) \land x \leq 0.6\right):\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in n around inf 70.1%

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

5. Simplified 70.5%

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

   1. add-sqr-sqrt 70.2%

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

   2. pow2 70.2%

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

   3. fma-neg 70.2%

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

7. Applied egg-rr 70.2%

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

8. Taylor expanded in n around inf 62.8%

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

   1. log1p-define 62.8%

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

10. Simplified 62.8%

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

11. Taylor expanded in x around -inf 45.5%

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

    1. mul-1-neg 45.5%

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

    2. distribute-neg-frac2 45.5%

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

13. Simplified 45.5%

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

14. Final simplification 45.5%

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

Alternative 16: 40.6% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around inf 60.0%

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

   1. mul-1-neg 60.0%

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

   2. log-rec 60.0%

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

   3. mul-1-neg 60.0%

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

   4. distribute-neg-frac 60.0%

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

   5. mul-1-neg 60.0%

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

   6. remove-double-neg 60.0%

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

   7. *-commutative 60.0%

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

5. Simplified 60.0%

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

   1. associate-/r* 60.8%

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

   2. div-inv 60.8%

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

   3. pow-to-exp 60.8%

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

   4. pow1 60.8%

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

   5. pow-div 60.8%

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

7. Applied egg-rr 60.8%

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

8. Taylor expanded in n around inf 39.4%

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

   1. associate-/r* 40.0%

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

10. Simplified 40.0%

    \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]
11. Add Preprocessing

Alternative 17: 40.2% accurate, 42.2× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* x n)))

C

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

Fortran

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

Java

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

Python

def code(x, n):
	return 1.0 / (x * n)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around inf 60.0%

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

   1. mul-1-neg 60.0%

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

   2. log-rec 60.0%

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

   3. mul-1-neg 60.0%

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

   4. distribute-neg-frac 60.0%

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

   5. mul-1-neg 60.0%

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

   6. remove-double-neg 60.0%

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

   7. *-commutative 60.0%

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

5. Simplified 60.0%

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

6. Taylor expanded in n around inf 39.4%

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

   1. *-commutative 39.4%

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

8. Simplified 39.4%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
9. Add Preprocessing

Alternative 18: 4.5% accurate, 70.3× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

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

Fortran

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

Java

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

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around 0 28.5%

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

4. Taylor expanded in x around inf 4.2%

   \[\leadsto \color{blue}{\frac{x}{n}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))