2nthrt (problem 3.4.6)

Percentage Accurate: 53.0% → 91.8%

Time: 1.6min

Alternatives: 18

Speedup: 1.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.0% 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: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 380000:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.5, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 380000.0)
   (/
    (log
     (/
      x
      (exp
       (+
        (log1p x)
        (/
         (fma
          (- (pow (log1p x) 2.0) (pow (log x) 2.0))
          0.5
          (*
           0.16666666666666666
           (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)))
         n)))))
    (- n))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 380000.0) {
		tmp = log((x / exp((log1p(x) + (fma((pow(log1p(x), 2.0) - pow(log(x), 2.0)), 0.5, (0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n))) / n))))) / -n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 380000.0)
		tmp = Float64(log(Float64(x / exp(Float64(log1p(x) + Float64(fma(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)), 0.5, Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n))) / n))))) / Float64(-n));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 380000.0], N[(N[Log[N[(x / N[Exp[N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $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 < 3.8e5

   1. Initial program 43.4%

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

   3. Taylor expanded in n around -inf 75.8%

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

   5. Simplified 75.9%

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

      1. add-log-exp 87.4%

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

      2. diff-log 87.4%

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

   7. Applied egg-rr 87.4%

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

   if 3.8e5 < x

   1. Initial program 69.3%

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

   3. Taylor expanded in x around inf 98.4%

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

      1. associate-/r* 99.4%

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

      2. mul-1-neg 99.4%

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

      3. log-rec 99.4%

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

      4. mul-1-neg 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. mul-1-neg 99.4%

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

      7. remove-double-neg 99.4%

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

      8. *-rgt-identity 99.4%

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

      9. associate-/l* 99.4%

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

      10. exp-to-pow 99.4%

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

   5. Simplified 99.4%

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

Alternative 2: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0225:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n} + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0225)
   (/
    (-
     (+
      (log1p x)
      (/
       (+
        (* 0.16666666666666666 (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n))
        (* (- (pow (log1p x) 2.0) (pow (log x) 2.0)) 0.5))
       n))
     (log x))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0225) {
		tmp = ((log1p(x) + (((0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n)) + ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) * 0.5)) / 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 <= 0.0225) {
		tmp = ((Math.log1p(x) + (((0.16666666666666666 * ((Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0)) / n)) + ((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) * 0.5)) / 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 <= 0.0225:
		tmp = ((math.log1p(x) + (((0.16666666666666666 * ((math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0)) / n)) + ((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) * 0.5)) / 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 <= 0.0225)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n)) + Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) * 0.5)) / 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, 0.0225], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $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 < 0.022499999999999999

   1. Initial program 42.9%

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

   3. Taylor expanded in n around -inf 76.6%

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

   5. Simplified 76.6%

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

   if 0.022499999999999999 < x

   1. Initial program 69.3%

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

   3. Taylor expanded in x around inf 97.2%

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

      1. associate-/r* 98.2%

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

      2. mul-1-neg 98.2%

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

      3. log-rec 98.2%

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

      4. mul-1-neg 98.2%

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

      5. distribute-neg-frac 98.2%

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

      6. mul-1-neg 98.2%

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

      7. remove-double-neg 98.2%

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

      8. *-rgt-identity 98.2%

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

      9. associate-/l* 98.2%

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

      10. exp-to-pow 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0225:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n} + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.014:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.014)
   (/
    (-
     (/
      (+
       (* -0.16666666666666666 (/ (pow (log x) 3.0) n))
       (* (pow (log x) 2.0) -0.5))
      n)
     (log x))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.014) {
		tmp = ((((-0.16666666666666666 * (pow(log(x), 3.0) / n)) + (pow(log(x), 2.0) * -0.5)) / n) - log(x)) / 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 <= 0.014d0) then
        tmp = (((((-0.16666666666666666d0) * ((log(x) ** 3.0d0) / n)) + ((log(x) ** 2.0d0) * (-0.5d0))) / n) - log(x)) / 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 <= 0.014) {
		tmp = ((((-0.16666666666666666 * (Math.pow(Math.log(x), 3.0) / n)) + (Math.pow(Math.log(x), 2.0) * -0.5)) / 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 <= 0.014:
		tmp = ((((-0.16666666666666666 * (math.pow(math.log(x), 3.0) / n)) + (math.pow(math.log(x), 2.0) * -0.5)) / 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 <= 0.014)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) / n)) + Float64((log(x) ^ 2.0) * -0.5)) / n) - log(x)) / 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 <= 0.014)
		tmp = ((((-0.16666666666666666 * ((log(x) ^ 3.0) / n)) + ((log(x) ^ 2.0) * -0.5)) / n) - log(x)) / n;
	else
		tmp = ((x ^ (1.0 / n)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.014], N[(N[(N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / n), $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 < 0.0140000000000000003

   1. Initial program 42.9%

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

   3. Taylor expanded in x around 0 40.7%

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

      1. *-rgt-identity 40.7%

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

      2. associate-*l/ 40.7%

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

      3. associate-/l* 40.7%

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

      4. exp-to-pow 40.7%

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

   5. Simplified 40.7%

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

   6. Taylor expanded in n around -inf 76.5%

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

      1. mul-1-neg 76.5%

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

   8. Simplified 76.5%

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

   if 0.0140000000000000003 < x

   1. Initial program 69.3%

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

   3. Taylor expanded in x around inf 97.2%

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

      1. associate-/r* 98.2%

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

      2. mul-1-neg 98.2%

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

      3. log-rec 98.2%

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

      4. mul-1-neg 98.2%

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

      5. distribute-neg-frac 98.2%

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

      6. mul-1-neg 98.2%

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

      7. remove-double-neg 98.2%

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

      8. *-rgt-identity 98.2%

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

      9. associate-/l* 98.2%

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

      10. exp-to-pow 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.014:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-35)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-14)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-14) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-14) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-35:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-14:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-35)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-14)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-35], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-14], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000002e-35

   1. Initial program 92.3%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. associate-/r* 97.6%

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

      2. mul-1-neg 97.6%

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

      3. log-rec 97.6%

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

      4. mul-1-neg 97.6%

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

      5. distribute-neg-frac 97.6%

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

      6. mul-1-neg 97.6%

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

      7. remove-double-neg 97.6%

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

      8. *-rgt-identity 97.6%

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

      9. associate-/l* 97.6%

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

      10. exp-to-pow 97.6%

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

   5. Simplified 97.6%

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

   if -2.00000000000000002e-35 < (/.f64 #s(literal 1 binary64) n) < 2e-14

   1. Initial program 35.2%

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

   3. Taylor expanded in n around inf 78.7%

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

      1. log1p-define 78.7%

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

   5. Simplified 78.7%

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

      1. log1p-undefine 78.7%

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

      2. diff-log 78.8%

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

   7. Applied egg-rr 78.8%

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

      1. +-commutative 78.8%

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

   9. Simplified 78.8%

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

   if 2e-14 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 48.9%

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

   3. Taylor expanded in n around 0 48.9%

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

      1. log1p-define 90.2%

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

      2. *-rgt-identity 90.2%

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

      3. associate-*l/ 90.2%

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

      4. associate-/l* 90.2%

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

      5. exp-to-pow 90.2%

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

   5. Simplified 90.2%

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

Alternative 5: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-35)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-10)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-35)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2d-10) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-35:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-10:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-35)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-35)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e-10)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-35], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000002e-35

   1. Initial program 92.3%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. associate-/r* 97.6%

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

      2. mul-1-neg 97.6%

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

      3. log-rec 97.6%

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

      4. mul-1-neg 97.6%

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

      5. distribute-neg-frac 97.6%

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

      6. mul-1-neg 97.6%

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

      7. remove-double-neg 97.6%

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

      8. *-rgt-identity 97.6%

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

      9. associate-/l* 97.6%

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

      10. exp-to-pow 97.6%

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

   5. Simplified 97.6%

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

   if -2.00000000000000002e-35 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 35.0%

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

   3. Taylor expanded in n around inf 78.1%

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

      1. log1p-define 78.1%

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

   5. Simplified 78.1%

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

      1. log1p-undefine 78.1%

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

      2. diff-log 78.2%

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

   7. Applied egg-rr 78.2%

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

      1. +-commutative 78.2%

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

   9. Simplified 78.2%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 50.0%

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

   3. Taylor expanded in n around 0 50.0%

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

      1. log1p-define 92.3%

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

      2. *-rgt-identity 92.3%

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

      3. associate-*l/ 92.3%

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

      4. associate-/l* 92.3%

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

      5. exp-to-pow 92.3%

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

   5. Simplified 92.3%

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

   6. Taylor expanded in x around 0 90.3%

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

Alternative 6: 68.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+98}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -100000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+162}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{{n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+98)
     (/ 0.0 n)
     (if (<= (/ 1.0 n) -100000000000.0)
       t_0
       (if (<= (/ 1.0 n) 2e-10)
         (/ (log (/ (+ x 1.0) x)) n)
         (if (<= (/ 1.0 n) 2e+162) t_0 (/ (/ n x) (pow n 2.0))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+98) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -100000000000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+162) {
		tmp = t_0;
	} else {
		tmp = (n / x) / pow(n, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-2d+98)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-100000000000.0d0)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-10) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+162) then
        tmp = t_0
    else
        tmp = (n / x) / (n ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+98) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -100000000000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+162) {
		tmp = t_0;
	} else {
		tmp = (n / x) / Math.pow(n, 2.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+98:
		tmp = 0.0 / n
	elif (1.0 / n) <= -100000000000.0:
		tmp = t_0
	elif (1.0 / n) <= 2e-10:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+162:
		tmp = t_0
	else:
		tmp = (n / x) / math.pow(n, 2.0)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+98)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -100000000000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+162)
		tmp = t_0;
	else
		tmp = Float64(Float64(n / x) / (n ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -2e+98)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -100000000000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-10)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+162)
		tmp = t_0;
	else
		tmp = (n / x) / (n ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+98], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -100000000000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+162], t$95$0, N[(N[(n / x), $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e98

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 67.5%

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

      1. log1p-define 67.5%

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

   5. Simplified 67.5%

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

      1. log1p-undefine 67.5%

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

      2. diff-log 67.5%

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

   7. Applied egg-rr 67.5%

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

      1. +-commutative 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in x around inf 67.8%

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

   if -2e98 < (/.f64 #s(literal 1 binary64) n) < -1e11 or 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e162

   1. Initial program 83.1%

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

   3. Taylor expanded in x around 0 72.3%

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

      1. *-rgt-identity 72.3%

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

      2. associate-*l/ 72.3%

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

      3. associate-/l* 72.3%

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

      4. exp-to-pow 72.3%

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

   5. Simplified 72.3%

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

   if -1e11 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 34.7%

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

   3. Taylor expanded in n around inf 75.5%

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

      1. log1p-define 75.5%

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

   5. Simplified 75.5%

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

      1. log1p-undefine 75.5%

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

      2. diff-log 75.6%

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

   7. Applied egg-rr 75.6%

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

      1. +-commutative 75.6%

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

   9. Simplified 75.6%

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

   if 1.9999999999999999e162 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 13.0%

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

      1. log1p-define 13.0%

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

   5. Simplified 13.0%

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

      1. div-inv 13.0%

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

   7. Applied egg-rr 13.0%

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

      1. div-inv 13.0%

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

      2. div-sub 13.0%

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

      3. frac-sub 94.6%

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

      4. pow2 94.6%

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

   9. Applied egg-rr 94.6%

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

       1. *-commutative 94.6%

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

       2. distribute-lft-out-- 94.6%

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

   11. Simplified 94.6%

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

   12. Taylor expanded in x around inf 94.6%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+98}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -100000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+162}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{{n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \left(x \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-35)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-14)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+119)
         (- (+ 1.0 (/ x n)) t_0)
         (/ 1.0 (* t_0 (* x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-14) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+119) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (t_0 * (x * n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-35)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2d-14) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+119) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (t_0 * (x * n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-14) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+119) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (t_0 * (x * n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-35:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-14:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+119:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (t_0 * (x * n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-35)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-14)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+119)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(t_0 * Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-35)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e-14)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+119)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (t_0 * (x * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-35], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-14], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+119], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(t$95$0 * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000002e-35

   1. Initial program 92.3%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. associate-/r* 97.6%

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

      2. mul-1-neg 97.6%

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

      3. log-rec 97.6%

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

      4. mul-1-neg 97.6%

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

      5. distribute-neg-frac 97.6%

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

      6. mul-1-neg 97.6%

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

      7. remove-double-neg 97.6%

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

      8. *-rgt-identity 97.6%

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

      9. associate-/l* 97.6%

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

      10. exp-to-pow 97.6%

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

   5. Simplified 97.6%

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

   if -2.00000000000000002e-35 < (/.f64 #s(literal 1 binary64) n) < 2e-14

   1. Initial program 35.2%

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

   3. Taylor expanded in n around inf 78.7%

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

      1. log1p-define 78.7%

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

   5. Simplified 78.7%

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

      1. log1p-undefine 78.7%

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

      2. diff-log 78.8%

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

   7. Applied egg-rr 78.8%

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

      1. +-commutative 78.8%

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

   9. Simplified 78.8%

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

   if 2e-14 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e119

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0 70.6%

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

   if 4.9999999999999999e119 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 28.4%

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

   3. Taylor expanded in x around inf 0.7%

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

      1. associate-/r* 1.8%

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

      2. mul-1-neg 1.8%

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

      3. log-rec 1.8%

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

      4. mul-1-neg 1.8%

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

      5. distribute-neg-frac 1.8%

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

      6. mul-1-neg 1.8%

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

      7. remove-double-neg 1.8%

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

      8. *-rgt-identity 1.8%

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

      9. associate-/l* 1.8%

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

      10. exp-to-pow 1.8%

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

   5. Simplified 1.8%

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

      1. clear-num 1.8%

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

      2. inv-pow 1.8%

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

   7. Applied egg-rr 87.4%

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

      1. unpow-1 87.4%

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

      2. associate-*r* 87.4%

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

   9. Simplified 87.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{\left(\frac{1}{n}\right)} \cdot \left(x \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+162}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{{n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-35)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-14)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e+162)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (/ n x) (pow n 2.0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-14) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+162) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (n / x) / pow(n, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-35)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2d-14) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+162) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (n / x) / (n ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-14) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+162) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (n / x) / Math.pow(n, 2.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-35:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-14:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+162:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (n / x) / math.pow(n, 2.0)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-35)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-14)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+162)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(n / x) / (n ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-35)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e-14)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+162)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (n / x) / (n ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-35], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-14], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+162], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000002e-35

   1. Initial program 92.3%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. associate-/r* 97.6%

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

      2. mul-1-neg 97.6%

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

      3. log-rec 97.6%

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

      4. mul-1-neg 97.6%

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

      5. distribute-neg-frac 97.6%

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

      6. mul-1-neg 97.6%

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

      7. remove-double-neg 97.6%

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

      8. *-rgt-identity 97.6%

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

      9. associate-/l* 97.6%

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

      10. exp-to-pow 97.6%

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

   5. Simplified 97.6%

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

   if -2.00000000000000002e-35 < (/.f64 #s(literal 1 binary64) n) < 2e-14

   1. Initial program 35.2%

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

   3. Taylor expanded in n around inf 78.7%

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

      1. log1p-define 78.7%

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

   5. Simplified 78.7%

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

      1. log1p-undefine 78.7%

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

      2. diff-log 78.8%

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

   7. Applied egg-rr 78.8%

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

      1. +-commutative 78.8%

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

   9. Simplified 78.8%

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

   if 2e-14 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e162

   1. Initial program 66.4%

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

   3. Taylor expanded in x around 0 64.9%

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

   if 1.9999999999999999e162 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 13.0%

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

      1. log1p-define 13.0%

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

   5. Simplified 13.0%

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

      1. div-inv 13.0%

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

   7. Applied egg-rr 13.0%

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

      1. div-inv 13.0%

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

      2. div-sub 13.0%

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

      3. frac-sub 94.6%

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

      4. pow2 94.6%

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

   9. Applied egg-rr 94.6%

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

       1. *-commutative 94.6%

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

       2. distribute-lft-out-- 94.6%

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

   11. Simplified 94.6%

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

   12. Taylor expanded in x around inf 94.6%

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

Alternative 9: 59.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := \frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{if}\;x \leq 1.2 \cdot 10^{-161}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n)))
        (t_1
         (/
          (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* x n)) (/ -0.5 n)) x))
          x)))
   (if (<= x 1.2e-161)
     t_0
     (if (<= x 5.4e-139)
       t_1
       (if (<= x 7e-131)
         t_0
         (if (<= x 9e-74)
           t_1
           (if (<= x 0.88)
             (/ (- x (log x)) n)
             (if (<= x 2.1e+151)
               (/
                (/
                 (+
                  1.0
                  (/
                   (- (/ (- 0.3333333333333333 (* 0.25 (/ 1.0 x))) x) 0.5)
                   x))
                 x)
                n)
               (/ 0.0 n)))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	double tmp;
	if (x <= 1.2e-161) {
		tmp = t_0;
	} else if (x <= 5.4e-139) {
		tmp = t_1;
	} else if (x <= 7e-131) {
		tmp = t_0;
	} else if (x <= 9e-74) {
		tmp = t_1;
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.1e+151) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0 / 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) + (((0.3333333333333333d0 / (x * n)) + ((-0.5d0) / n)) / x)) / x
    if (x <= 1.2d-161) then
        tmp = t_0
    else if (x <= 5.4d-139) then
        tmp = t_1
    else if (x <= 7d-131) then
        tmp = t_0
    else if (x <= 9d-74) then
        tmp = t_1
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.1d+151) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - (0.25d0 * (1.0d0 / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0 / 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) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	double tmp;
	if (x <= 1.2e-161) {
		tmp = t_0;
	} else if (x <= 5.4e-139) {
		tmp = t_1;
	} else if (x <= 7e-131) {
		tmp = t_0;
	} else if (x <= 9e-74) {
		tmp = t_1;
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.1e+151) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x
	tmp = 0
	if x <= 1.2e-161:
		tmp = t_0
	elif x <= 5.4e-139:
		tmp = t_1
	elif x <= 7e-131:
		tmp = t_0
	elif x <= 9e-74:
		tmp = t_1
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 2.1e+151:
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0 / 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(0.3333333333333333 / Float64(x * n)) + Float64(-0.5 / n)) / x)) / x)
	tmp = 0.0
	if (x <= 1.2e-161)
		tmp = t_0;
	elseif (x <= 5.4e-139)
		tmp = t_1;
	elseif (x <= 7e-131)
		tmp = t_0;
	elseif (x <= 9e-74)
		tmp = t_1;
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.1e+151)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(0.25 * Float64(1.0 / x))) / x) - 0.5) / x)) / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	tmp = 0.0;
	if (x <= 1.2e-161)
		tmp = t_0;
	elseif (x <= 5.4e-139)
		tmp = t_1;
	elseif (x <= 7e-131)
		tmp = t_0;
	elseif (x <= 9e-74)
		tmp = t_1;
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.1e+151)
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0 / 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[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, 1.2e-161], t$95$0, If[LessEqual[x, 5.4e-139], t$95$1, If[LessEqual[x, 7e-131], t$95$0, If[LessEqual[x, 9e-74], t$95$1, If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.1e+151], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(0.25 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.19999999999999999e-161 or 5.3999999999999997e-139 < x < 7.0000000000000004e-131

   1. Initial program 39.0%

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

   3. Taylor expanded in n around inf 63.9%

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

      1. log1p-define 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in x around 0 63.9%

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

      1. neg-mul-1 63.9%

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

   8. Simplified 63.9%

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

   if 1.19999999999999999e-161 < x < 5.3999999999999997e-139 or 7.0000000000000004e-131 < x < 8.9999999999999998e-74

   1. Initial program 55.3%

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

   3. Taylor expanded in n around inf 30.6%

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

      1. log1p-define 30.6%

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

   5. Simplified 30.6%

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

      1. div-inv 30.6%

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

   7. Applied egg-rr 30.6%

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

   8. Taylor expanded in x around inf 32.7%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
   9. Step-by-step derivation

   10. Simplified 67.0%

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

   if 8.9999999999999998e-74 < x < 0.880000000000000004

   1. Initial program 39.5%

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

   3. Taylor expanded in n around inf 51.2%

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

      1. log1p-define 51.2%

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

   5. Simplified 51.2%

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

   6. Taylor expanded in x around 0 51.1%

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

   if 0.880000000000000004 < x < 2.1000000000000001e151

   1. Initial program 46.9%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. log1p-define 44.8%

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

   5. Simplified 44.8%

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

   6. Taylor expanded in x around -inf 69.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   if 2.1000000000000001e151 < x

   1. Initial program 90.3%

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

   3. Taylor expanded in n around inf 90.3%

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

      1. log1p-define 90.3%

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

   5. Simplified 90.3%

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

      1. log1p-undefine 90.3%

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

      2. diff-log 90.3%

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

   7. Applied egg-rr 90.3%

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

      1. +-commutative 90.3%

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

   9. Simplified 90.3%

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

   10. Taylor expanded in x around inf 90.3%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-131}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := \frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{if}\;x \leq 1.15 \cdot 10^{-161}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n)))
        (t_1
         (/
          (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* x n)) (/ -0.5 n)) x))
          x)))
   (if (<= x 1.15e-161)
     t_0
     (if (<= x 4.2e-139)
       t_1
       (if (<= x 3.3e-131)
         t_0
         (if (<= x 7.5e-74)
           t_1
           (if (<= x 0.7)
             t_0
             (if (<= x 3.3e+150)
               (/
                (/
                 (+
                  1.0
                  (/
                   (- (/ (- 0.3333333333333333 (* 0.25 (/ 1.0 x))) x) 0.5)
                   x))
                 x)
                n)
               (/ 0.0 n)))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	double tmp;
	if (x <= 1.15e-161) {
		tmp = t_0;
	} else if (x <= 4.2e-139) {
		tmp = t_1;
	} else if (x <= 3.3e-131) {
		tmp = t_0;
	} else if (x <= 7.5e-74) {
		tmp = t_1;
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 3.3e+150) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0 / 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) + (((0.3333333333333333d0 / (x * n)) + ((-0.5d0) / n)) / x)) / x
    if (x <= 1.15d-161) then
        tmp = t_0
    else if (x <= 4.2d-139) then
        tmp = t_1
    else if (x <= 3.3d-131) then
        tmp = t_0
    else if (x <= 7.5d-74) then
        tmp = t_1
    else if (x <= 0.7d0) then
        tmp = t_0
    else if (x <= 3.3d+150) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - (0.25d0 * (1.0d0 / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0 / 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) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	double tmp;
	if (x <= 1.15e-161) {
		tmp = t_0;
	} else if (x <= 4.2e-139) {
		tmp = t_1;
	} else if (x <= 3.3e-131) {
		tmp = t_0;
	} else if (x <= 7.5e-74) {
		tmp = t_1;
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 3.3e+150) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x
	tmp = 0
	if x <= 1.15e-161:
		tmp = t_0
	elif x <= 4.2e-139:
		tmp = t_1
	elif x <= 3.3e-131:
		tmp = t_0
	elif x <= 7.5e-74:
		tmp = t_1
	elif x <= 0.7:
		tmp = t_0
	elif x <= 3.3e+150:
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0 / 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(0.3333333333333333 / Float64(x * n)) + Float64(-0.5 / n)) / x)) / x)
	tmp = 0.0
	if (x <= 1.15e-161)
		tmp = t_0;
	elseif (x <= 4.2e-139)
		tmp = t_1;
	elseif (x <= 3.3e-131)
		tmp = t_0;
	elseif (x <= 7.5e-74)
		tmp = t_1;
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 3.3e+150)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(0.25 * Float64(1.0 / x))) / x) - 0.5) / x)) / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	tmp = 0.0;
	if (x <= 1.15e-161)
		tmp = t_0;
	elseif (x <= 4.2e-139)
		tmp = t_1;
	elseif (x <= 3.3e-131)
		tmp = t_0;
	elseif (x <= 7.5e-74)
		tmp = t_1;
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 3.3e+150)
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0 / 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[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, 1.15e-161], t$95$0, If[LessEqual[x, 4.2e-139], t$95$1, If[LessEqual[x, 3.3e-131], t$95$0, If[LessEqual[x, 7.5e-74], t$95$1, If[LessEqual[x, 0.7], t$95$0, If[LessEqual[x, 3.3e+150], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(0.25 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.15e-161 or 4.20000000000000016e-139 < x < 3.3000000000000002e-131 or 7.5e-74 < x < 0.69999999999999996

   1. Initial program 38.6%

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

   3. Taylor expanded in n around inf 59.3%

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

      1. log1p-define 59.3%

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

   5. Simplified 59.3%

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

   6. Taylor expanded in x around 0 59.1%

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

      1. neg-mul-1 59.1%

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

   8. Simplified 59.1%

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

   if 1.15e-161 < x < 4.20000000000000016e-139 or 3.3000000000000002e-131 < x < 7.5e-74

   1. Initial program 55.3%

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

   3. Taylor expanded in n around inf 30.6%

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

      1. log1p-define 30.6%

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

   5. Simplified 30.6%

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

      1. div-inv 30.6%

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

   7. Applied egg-rr 30.6%

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

   8. Taylor expanded in x around inf 32.7%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
   9. Step-by-step derivation

   10. Simplified 67.0%

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

   if 0.69999999999999996 < x < 3.29999999999999981e150

   1. Initial program 47.8%

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

   3. Taylor expanded in n around inf 44.4%

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

      1. log1p-define 44.4%

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

   5. Simplified 44.4%

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

   6. Taylor expanded in x around -inf 68.8%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   if 3.29999999999999981e150 < x

   1. Initial program 90.3%

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

   3. Taylor expanded in n around inf 90.3%

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

      1. log1p-define 90.3%

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

   5. Simplified 90.3%

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

      1. log1p-undefine 90.3%

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

      2. diff-log 90.3%

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

   7. Applied egg-rr 90.3%

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

      1. +-commutative 90.3%

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

   9. Simplified 90.3%

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

   10. Taylor expanded in x around inf 90.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-161}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+162}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{{n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-35)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 2e-10)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e+162) (- 1.0 t_0) (/ (/ n x) (pow n 2.0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+162) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / pow(n, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-35)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 2d-10) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+162) then
        tmp = 1.0d0 - t_0
    else
        tmp = (n / x) / (n ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 2e-10) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+162) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / Math.pow(n, 2.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-35:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 2e-10:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+162:
		tmp = 1.0 - t_0
	else:
		tmp = (n / x) / math.pow(n, 2.0)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-35)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-10)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+162)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(n / x) / (n ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-35)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 2e-10)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+162)
		tmp = 1.0 - t_0;
	else
		tmp = (n / x) / (n ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-35], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-10], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+162], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000002e-35

   1. Initial program 92.3%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. associate-/r* 97.6%

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

      2. mul-1-neg 97.6%

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

      3. log-rec 97.6%

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

      4. mul-1-neg 97.6%

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

      5. distribute-neg-frac 97.6%

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

      6. mul-1-neg 97.6%

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

      7. remove-double-neg 97.6%

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

      8. *-rgt-identity 97.6%

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

      9. associate-/l* 97.6%

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

      10. exp-to-pow 97.6%

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

   5. Simplified 97.6%

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

   if -2.00000000000000002e-35 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000007e-10

   1. Initial program 35.0%

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

   3. Taylor expanded in n around inf 78.1%

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

      1. log1p-define 78.1%

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

   5. Simplified 78.1%

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

      1. log1p-undefine 78.1%

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

      2. diff-log 78.2%

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

   7. Applied egg-rr 78.2%

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

      1. +-commutative 78.2%

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

   9. Simplified 78.2%

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

   if 2.00000000000000007e-10 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e162

   1. Initial program 69.0%

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

   3. Taylor expanded in x around 0 65.4%

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

      1. *-rgt-identity 65.4%

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

      2. associate-*l/ 65.4%

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

      3. associate-/l* 65.4%

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

      4. exp-to-pow 65.4%

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

   5. Simplified 65.4%

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

   if 1.9999999999999999e162 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 13.0%

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

      1. log1p-define 13.0%

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

   5. Simplified 13.0%

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

      1. div-inv 13.0%

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

   7. Applied egg-rr 13.0%

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

      1. div-inv 13.0%

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

      2. div-sub 13.0%

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

      3. frac-sub 94.6%

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

      4. pow2 94.6%

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

   9. Applied egg-rr 94.6%

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

       1. *-commutative 94.6%

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

       2. distribute-lft-out-- 94.6%

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

   11. Simplified 94.6%

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

   12. Taylor expanded in x around inf 94.6%

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

Alternative 12: 46.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+98}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e+98)
   (/ 0.0 n)
   (/ (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* x n)) (/ -0.5 n)) x)) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+98) {
		tmp = 0.0 / n;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / 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 ((1.0d0 / n) <= (-2d+98)) then
        tmp = 0.0d0 / n
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+98) {
		tmp = 0.0 / n;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e+98:
		tmp = 0.0 / n
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+98)
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e+98)
		tmp = 0.0 / n;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+98], N[(0.0 / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e98

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 67.5%

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

      1. log1p-define 67.5%

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

   5. Simplified 67.5%

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

      1. log1p-undefine 67.5%

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

      2. diff-log 67.5%

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

   7. Applied egg-rr 67.5%

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

      1. +-commutative 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in x around inf 67.8%

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

   if -2e98 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.2%

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

   3. Taylor expanded in n around inf 56.7%

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

      1. log1p-define 56.7%

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

   5. Simplified 56.7%

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

      1. div-inv 56.6%

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

   7. Applied egg-rr 56.6%

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

   8. Taylor expanded in x around inf 47.0%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
   9. Step-by-step derivation

   10. Simplified 53.4%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+98}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7.8e+220)
   (/ (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* x n)) (/ -0.5 n)) x)) x)
   (/ (/ -0.5 (* x n)) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7.8e+220) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	} else {
		tmp = (-0.5 / (x * n)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 7.8d+220) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) + ((-0.5d0) / n)) / x)) / x
    else
        tmp = ((-0.5d0) / (x * n)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 7.8e+220) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	} else {
		tmp = (-0.5 / (x * n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7.8e+220:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x
	else:
		tmp = (-0.5 / (x * n)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7.8e+220)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) + Float64(-0.5 / n)) / x)) / x);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 7.8e+220)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
	else
		tmp = (-0.5 / (x * n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7.8e+220], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.5 / N[(x * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.80000000000000032e220

   1. Initial program 48.6%

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

   3. Taylor expanded in n around inf 52.5%

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

      1. log1p-define 52.5%

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

   5. Simplified 52.5%

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

      1. div-inv 52.4%

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

   7. Applied egg-rr 52.4%

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

   8. Taylor expanded in x around inf 35.5%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
   9. Step-by-step derivation

   10. Simplified 47.6%

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

   if 7.80000000000000032e220 < x

   1. Initial program 92.3%

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

   3. Taylor expanded in n around inf 92.3%

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

      1. log1p-define 92.3%

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

   5. Simplified 92.3%

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

   6. Taylor expanded in x around inf 60.2%

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

      1. associate-*r/ 60.2%

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

      2. metadata-eval 60.2%

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

      3. *-commutative 60.2%

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

   8. Simplified 60.2%

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

   9. Taylor expanded in x around 0 78.7%

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

       1. *-commutative 78.7%

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

   11. Simplified 78.7%

       \[\leadsto \frac{\color{blue}{\frac{-0.5}{x \cdot n}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 41.7% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.6e+220) (/ (/ 1.0 x) n) (/ (/ -0.5 (* x n)) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.6e+220) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (-0.5 / (x * n)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.6d+220) then
        tmp = (1.0d0 / x) / n
    else
        tmp = ((-0.5d0) / (x * n)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.6e+220) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (-0.5 / (x * n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.6e+220:
		tmp = (1.0 / x) / n
	else:
		tmp = (-0.5 / (x * n)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.6e+220)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(-0.5 / Float64(x * n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.6e+220)
		tmp = (1.0 / x) / n;
	else
		tmp = (-0.5 / (x * n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.6e+220], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.5 / N[(x * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.59999999999999993e220

   1. Initial program 48.6%

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

   3. Taylor expanded in n around inf 52.5%

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

      1. log1p-define 52.5%

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

   5. Simplified 52.5%

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

   6. Taylor expanded in x around inf 39.7%

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

   if 4.59999999999999993e220 < x

   1. Initial program 92.3%

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

   3. Taylor expanded in n around inf 92.3%

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

      1. log1p-define 92.3%

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

   5. Simplified 92.3%

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

   6. Taylor expanded in x around inf 60.2%

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

      1. associate-*r/ 60.2%

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

      2. metadata-eval 60.2%

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

      3. *-commutative 60.2%

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

   8. Simplified 60.2%

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

   9. Taylor expanded in x around 0 78.7%

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

       1. *-commutative 78.7%

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

   11. Simplified 78.7%

       \[\leadsto \frac{\color{blue}{\frac{-0.5}{x \cdot n}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 39.9% accurate, 42.2× speedup

Math

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

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(Float64(1.0 / x) / n)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in n around inf 58.9%

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

   1. log1p-define 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around inf 43.0%

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

Alternative 16: 39.9% 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 55.6%

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

3. Taylor expanded in n around inf 58.9%

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

   1. log1p-define 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around inf 42.5%

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

   1. associate-/r* 43.0%

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

8. Simplified 43.0%

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

Alternative 17: 39.4% 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 55.6%

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

3. Taylor expanded in n around inf 58.9%

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

   1. log1p-define 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around inf 42.5%

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

7. Final simplification 42.5%

   \[\leadsto \frac{1}{x \cdot n} \]
8. Add Preprocessing

Alternative 18: 4.6% 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 55.6%

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

3. Taylor expanded in n around inf 58.9%

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

   1. log1p-define 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around inf 42.5%

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

   1. inv-pow 42.5%

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

   2. *-commutative 42.5%

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

   3. unpow-prod-down 43.0%

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

   4. inv-pow 43.0%

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

   5. add-exp-log 42.0%

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

   6. rec-exp 42.0%

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

   7. add-sqr-sqrt 12.6%

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

   8. sqrt-unprod 14.2%

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

   9. sqr-neg 14.2%

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

   10. sqrt-unprod 1.6%

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

   11. add-sqr-sqrt 4.6%

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

   12. add-exp-log 4.6%

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

   13. inv-pow 4.6%

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

8. Applied egg-rr 4.6%

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

   1. associate-*r/ 4.6%

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

   2. *-rgt-identity 4.6%

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

10. Simplified 4.6%

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

Reproduce

herbie shell --seed 2024102 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))