2nthrt (problem 3.4.6)

Percentage Accurate: 53.3% → 85.9%

Time: 49.2s

Alternatives: 21

Speedup: 1.8×

• 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: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-65)
   (/ (exp (/ (log x) n)) (* x n))
   (if (<= (/ 1.0 n) 1e-13)
     (/ (log (/ x (+ x 1.0))) (- n))
     (exp (log (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = exp((log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = log((x / (x + 1.0))) / -n;
	} else {
		tmp = exp(log((exp((log1p(x) / n)) - pow(x, (1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else {
		tmp = Math.exp(Math.log((Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n)))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-65:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	elif (1.0 / n) <= 1e-13:
		tmp = math.log((x / (x + 1.0))) / -n
	else:
		tmp = math.exp(math.log((math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n)))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-65)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-13)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	else
		tmp = exp(log(Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)))));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-65], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-13], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[Exp[N[Log[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999983e-65

   1. Initial program 82.5%

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

   3. Taylor expanded in x around inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. log-rec 89.8%

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

      3. mul-1-neg 89.8%

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

      4. distribute-neg-frac 89.8%

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

      5. mul-1-neg 89.8%

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

      6. remove-double-neg 89.8%

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

      7. *-commutative 89.8%

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

   5. Simplified 89.8%

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

   if -4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.8%

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

   7. Applied egg-rr 80.8%

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

      1. +-commutative 80.8%

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

   9. Simplified 80.8%

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

       1. clear-num 80.8%

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

       2. log-div 80.9%

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

       3. metadata-eval 80.9%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   11. Applied egg-rr 80.9%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   12. Step-by-step derivation

       1. neg-sub0 80.9%

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

   13. Simplified 80.9%

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

   if 1e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 65.1%

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

      1. add-exp-log 65.1%

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

      2. pow-to-exp 65.4%

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

      3. un-div-inv 65.4%

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

      4. +-commutative 65.4%

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

      5. log1p-define 96.9%

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

   4. Applied egg-rr 96.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5500000:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5500000.0)
   (/
    (-
     (+
      (log1p x)
      (/
       (+
        (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
        (*
         0.16666666666666666
         (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)))
       n))
     (log x))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5500000.0) {
		tmp = ((log1p(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) + (0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n))) / n)) - 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 <= 5500000.0) {
		tmp = ((Math.log1p(x) + (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) + (0.16666666666666666 * ((Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0)) / n))) / 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 <= 5500000.0:
		tmp = ((math.log1p(x) + (((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) + (0.16666666666666666 * ((math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0)) / n))) / 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 <= 5500000.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) + Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n))) / 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, 5500000.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $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 < 5.5e6

   1. Initial program 44.9%

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

   3. Taylor expanded in n around -inf 76.9%

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

   5. Simplified 76.9%

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

   if 5.5e6 < x

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. associate-/r* 99.5%

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

      2. mul-1-neg 99.5%

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

      3. log-rec 99.5%

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

      4. mul-1-neg 99.5%

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

      5. distribute-neg-frac 99.5%

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

      6. mul-1-neg 99.5%

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

      7. remove-double-neg 99.5%

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

      8. *-rgt-identity 99.5%

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

      9. associate-/l* 99.5%

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

      10. exp-to-pow 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5500000:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.27:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.27)
   (/
    (-
     (/
      (+
       (* -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 <= 0.27) {
		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 <= 0.27d0) 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 <= 0.27) {
		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 <= 0.27:
		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 <= 0.27)
		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 <= 0.27)
		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, 0.27], 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 < 0.27000000000000002

   1. Initial program 45.6%

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

   3. Taylor expanded in x around 0 44.3%

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

      1. *-rgt-identity 44.3%

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

      2. associate-*l/ 44.3%

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

      3. associate-/l* 44.3%

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

      4. exp-to-pow 44.3%

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

   5. Simplified 44.3%

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

   6. Taylor expanded in n around -inf 76.4%

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

      1. mul-1-neg 76.4%

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

   8. Simplified 76.4%

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

   if 0.27000000000000002 < x

   1. Initial program 64.4%

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

   3. Taylor expanded in x around inf 96.3%

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

      1. associate-/r* 96.9%

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

      2. mul-1-neg 96.9%

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

      3. log-rec 96.9%

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

      4. mul-1-neg 96.9%

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

      5. distribute-neg-frac 96.9%

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

      6. mul-1-neg 96.9%

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

      7. remove-double-neg 96.9%

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

      8. *-rgt-identity 96.9%

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

      9. associate-/l* 96.9%

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

      10. exp-to-pow 96.9%

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

   5. Simplified 96.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.27:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-65)
   (/ (exp (/ (log x) n)) (* x n))
   (if (<= (/ 1.0 n) 1e-13)
     (/ (log (/ x (+ x 1.0))) (- n))
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = exp((log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = log((x / (x + 1.0))) / -n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-65:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	elif (1.0 / n) <= 1e-13:
		tmp = math.log((x / (x + 1.0))) / -n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-65)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-13)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-65], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-13], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999983e-65

   1. Initial program 82.5%

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

   3. Taylor expanded in x around inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. log-rec 89.8%

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

      3. mul-1-neg 89.8%

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

      4. distribute-neg-frac 89.8%

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

      5. mul-1-neg 89.8%

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

      6. remove-double-neg 89.8%

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

      7. *-commutative 89.8%

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

   5. Simplified 89.8%

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

   if -4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.8%

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

   7. Applied egg-rr 80.8%

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

      1. +-commutative 80.8%

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

   9. Simplified 80.8%

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

       1. clear-num 80.8%

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

       2. log-div 80.9%

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

       3. metadata-eval 80.9%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   11. Applied egg-rr 80.9%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   12. Step-by-step derivation

       1. neg-sub0 80.9%

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

   13. Simplified 80.9%

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

   if 1e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 65.1%

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

   3. Taylor expanded in n around 0 65.1%

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

      1. log1p-define 96.6%

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

      2. *-rgt-identity 96.6%

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

      3. associate-*l/ 96.6%

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

      4. associate-/l* 96.6%

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

      5. exp-to-pow 96.9%

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

   5. Simplified 96.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-65)
   (/ (exp (/ (log x) n)) (* x n))
   (if (<= (/ 1.0 n) 1e-13)
     (/ (log (/ x (+ x 1.0))) (- n))
     (if (<= (/ 1.0 n) 4e+191)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (log1p (expm1 (/ 1.0 (* x n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = exp((log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log1p(expm1((1.0 / (x * n))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p(Math.expm1((1.0 / (x * n))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-65:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	elif (1.0 / n) <= 1e-13:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 4e+191:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p(math.expm1((1.0 / (x * n))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-65)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-13)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e+191)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = log1p(expm1(Float64(1.0 / Float64(x * n))));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-65], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-13], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+191], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999983e-65

   1. Initial program 82.5%

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

   3. Taylor expanded in x around inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. log-rec 89.8%

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

      3. mul-1-neg 89.8%

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

      4. distribute-neg-frac 89.8%

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

      5. mul-1-neg 89.8%

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

      6. remove-double-neg 89.8%

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

      7. *-commutative 89.8%

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

   5. Simplified 89.8%

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

   if -4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.8%

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

   7. Applied egg-rr 80.8%

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

      1. +-commutative 80.8%

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

   9. Simplified 80.8%

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

       1. clear-num 80.8%

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

       2. log-div 80.9%

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

       3. metadata-eval 80.9%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   11. Applied egg-rr 80.9%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   12. Step-by-step derivation

       1. neg-sub0 80.9%

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

   13. Simplified 80.9%

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

   if 1e-13 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000029e191

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0 72.2%

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

   if 4.00000000000000029e191 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 6.7%

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

      1. log1p-define 6.7%

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

   5. Simplified 6.7%

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

      1. log1p-undefine 6.7%

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

      2. diff-log 6.7%

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

   7. Applied egg-rr 6.7%

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

      1. +-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in x around inf 57.7%

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

       1. *-commutative 57.7%

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

   12. Simplified 57.7%

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

       1. log1p-expm1-u 78.5%

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

   14. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-65)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-13)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 4e+191)
         (- (+ 1.0 (/ x n)) t_0)
         (log1p (expm1 (/ 1.0 (* x n)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(expm1((1.0 / (x * n))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((1.0 / (x * n))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-65:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 1e-13:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 4e+191:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((1.0 / (x * n))))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-65)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-13)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e+191)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(1.0 / Float64(x * n))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-65], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-13], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+191], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999983e-65

   1. Initial program 82.5%

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

   3. Taylor expanded in x around inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. log-rec 89.8%

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

      3. mul-1-neg 89.8%

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

      4. distribute-neg-frac 89.8%

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

      5. mul-1-neg 89.8%

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

      6. remove-double-neg 89.8%

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

      7. *-commutative 89.8%

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

   5. Simplified 89.8%

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

      1. *-un-lft-identity 89.8%

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

      2. div-inv 89.8%

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

      3. exp-to-pow 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. *-lft-identity 89.8%

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

   9. Simplified 89.8%

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

   if -4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.8%

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

   7. Applied egg-rr 80.8%

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

      1. +-commutative 80.8%

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

   9. Simplified 80.8%

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

       1. clear-num 80.8%

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

       2. log-div 80.9%

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

       3. metadata-eval 80.9%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   11. Applied egg-rr 80.9%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   12. Step-by-step derivation

       1. neg-sub0 80.9%

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

   13. Simplified 80.9%

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

   if 1e-13 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000029e191

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0 72.2%

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

   if 4.00000000000000029e191 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 6.7%

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

      1. log1p-define 6.7%

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

   5. Simplified 6.7%

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

      1. log1p-undefine 6.7%

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

      2. diff-log 6.7%

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

   7. Applied egg-rr 6.7%

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

      1. +-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in x around inf 57.7%

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

       1. *-commutative 57.7%

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

   12. Simplified 57.7%

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

       1. log1p-expm1-u 78.5%

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

   14. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-65)
   (/ (exp (/ (log x) n)) (* x n))
   (if (<= (/ 1.0 n) 1e-13)
     (/ (log (/ x (+ x 1.0))) (- n))
     (- (exp (/ x n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = exp((log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = log((x / (x + 1.0))) / -n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-5d-65)) then
        tmp = exp((log(x) / n)) / (x * n)
    else if ((1.0d0 / n) <= 1d-13) then
        tmp = log((x / (x + 1.0d0))) / -n
    else
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-65:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	elif (1.0 / n) <= 1e-13:
		tmp = math.log((x / (x + 1.0))) / -n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-65)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-13)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5e-65)
		tmp = exp((log(x) / n)) / (x * n);
	elseif ((1.0 / n) <= 1e-13)
		tmp = log((x / (x + 1.0))) / -n;
	else
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-65], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-13], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999983e-65

   1. Initial program 82.5%

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

   3. Taylor expanded in x around inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. log-rec 89.8%

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

      3. mul-1-neg 89.8%

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

      4. distribute-neg-frac 89.8%

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

      5. mul-1-neg 89.8%

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

      6. remove-double-neg 89.8%

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

      7. *-commutative 89.8%

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

   5. Simplified 89.8%

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

   if -4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.8%

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

   7. Applied egg-rr 80.8%

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

      1. +-commutative 80.8%

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

   9. Simplified 80.8%

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

       1. clear-num 80.8%

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

       2. log-div 80.9%

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

       3. metadata-eval 80.9%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   11. Applied egg-rr 80.9%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   12. Step-by-step derivation

       1. neg-sub0 80.9%

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

   13. Simplified 80.9%

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

   if 1e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 65.1%

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

   3. Taylor expanded in n around 0 65.1%

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

      1. log1p-define 96.6%

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

      2. *-rgt-identity 96.6%

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

      3. associate-*l/ 96.6%

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

      4. associate-/l* 96.6%

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

      5. exp-to-pow 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in x around 0 96.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+275}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+275)
     (/ 1.0 (* x n))
     (if (<= (/ 1.0 n) -5e+204)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) -2e+46)
         t_0
         (if (<= (/ 1.0 n) 1e-13)
           (/ (log (/ x (+ x 1.0))) (- n))
           (if (<= (/ 1.0 n) 4e+191)
             t_0
             (/
              (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x))
              n))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+275) {
		tmp = 1.0 / (x * n);
	} else if ((1.0 / n) <= -5e+204) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= -2e+46) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-13) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = t_0;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-2d+275)) then
        tmp = 1.0d0 / (x * n)
    else if ((1.0d0 / n) <= (-5d+204)) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= (-2d+46)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d-13) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 4d+191) then
        tmp = t_0
    else
        tmp = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -x) / n
    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+275) {
		tmp = 1.0 / (x * n);
	} else if ((1.0 / n) <= -5e+204) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= -2e+46) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-13) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = t_0;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+275:
		tmp = 1.0 / (x * n)
	elif (1.0 / n) <= -5e+204:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= -2e+46:
		tmp = t_0
	elif (1.0 / n) <= 1e-13:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 4e+191:
		tmp = t_0
	else:
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n
	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+275)
		tmp = Float64(1.0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e+204)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= -2e+46)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-13)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e+191)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / Float64(-x)) / n);
	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+275)
		tmp = 1.0 / (x * n);
	elseif ((1.0 / n) <= -5e+204)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= -2e+46)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e-13)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 4e+191)
		tmp = t_0;
	else
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	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+275], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+204], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+46], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-13], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+191], t$95$0, N[(N[(N[(N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999992e275

   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 10.3%

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

      1. log1p-define 10.3%

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

   5. Simplified 10.3%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1.99999999999999992e275 < (/.f64 #s(literal 1 binary64) n) < -5.00000000000000008e204

   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 74.2%

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

      1. log1p-define 74.2%

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

   5. Simplified 74.2%

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

      1. log1p-undefine 74.2%

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

      2. diff-log 74.2%

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

   7. Applied egg-rr 74.2%

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

      1. +-commutative 74.2%

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

   9. Simplified 74.2%

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

   if -5.00000000000000008e204 < (/.f64 #s(literal 1 binary64) n) < -2e46 or 1e-13 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000029e191

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0 68.9%

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

      1. *-rgt-identity 68.9%

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

      2. associate-*l/ 68.9%

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

      3. associate-/l* 68.9%

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

      4. exp-to-pow 68.9%

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

   5. Simplified 68.9%

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

   if -2e46 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 34.5%

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

   3. Taylor expanded in n around inf 74.4%

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

      1. log1p-define 74.4%

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

   5. Simplified 74.4%

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

      1. log1p-undefine 74.4%

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

      2. diff-log 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. +-commutative 74.5%

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

   9. Simplified 74.5%

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

       1. clear-num 74.5%

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

       2. log-div 74.5%

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

       3. metadata-eval 74.5%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   11. Applied egg-rr 74.5%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   12. Step-by-step derivation

       1. neg-sub0 74.5%

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

   13. Simplified 74.5%

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

   if 4.00000000000000029e191 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 6.7%

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

      1. log1p-define 6.7%

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

   5. Simplified 6.7%

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

      1. log1p-undefine 6.7%

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

      2. diff-log 6.7%

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

   7. Applied egg-rr 6.7%

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

      1. +-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in x around -inf 78.5%

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

       1. mul-1-neg 78.5%

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

       2. distribute-neg-frac2 78.5%

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

       3. sub-neg 78.5%

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

       4. associate-*r/ 78.5%

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

       5. sub-neg 78.5%

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

       6. metadata-eval 78.5%

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

       7. distribute-lft-in 78.5%

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

       8. neg-mul-1 78.5%

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

       9. associate-*r/ 78.5%

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

       10. metadata-eval 78.5%

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

       11. distribute-neg-frac 78.5%

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

       12. metadata-eval 78.5%

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

       13. metadata-eval 78.5%

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

       14. metadata-eval 78.5%

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

   12. Simplified 78.5%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{-x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+275}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+46}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+275}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -2e+275)
     (/ 1.0 (* x n))
     (if (<= (/ 1.0 n) -5e+204)
       t_1
       (if (<= (/ 1.0 n) -2e+46)
         t_0
         (if (<= (/ 1.0 n) 1e-13)
           t_1
           (if (<= (/ 1.0 n) 4e+191)
             t_0
             (/
              (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x))
              n))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -2e+275) {
		tmp = 1.0 / (x * n);
	} else if ((1.0 / n) <= -5e+204) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e+46) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = t_0;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-2d+275)) then
        tmp = 1.0d0 / (x * n)
    else if ((1.0d0 / n) <= (-5d+204)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-2d+46)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d-13) then
        tmp = t_1
    else if ((1.0d0 / n) <= 4d+191) then
        tmp = t_0
    else
        tmp = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -x) / n
    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 t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -2e+275) {
		tmp = 1.0 / (x * n);
	} else if ((1.0 / n) <= -5e+204) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e+46) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = t_0;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -2e+275:
		tmp = 1.0 / (x * n)
	elif (1.0 / n) <= -5e+204:
		tmp = t_1
	elif (1.0 / n) <= -2e+46:
		tmp = t_0
	elif (1.0 / n) <= 1e-13:
		tmp = t_1
	elif (1.0 / n) <= 4e+191:
		tmp = t_0
	else:
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+275)
		tmp = Float64(1.0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e+204)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -2e+46)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-13)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e+191)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / Float64(-x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -2e+275)
		tmp = 1.0 / (x * n);
	elseif ((1.0 / n) <= -5e+204)
		tmp = t_1;
	elseif ((1.0 / n) <= -2e+46)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e-13)
		tmp = t_1;
	elseif ((1.0 / n) <= 4e+191)
		tmp = t_0;
	else
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	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]}, Block[{t$95$1 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+275], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+204], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+46], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-13], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+191], t$95$0, N[(N[(N[(N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999992e275

   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 10.3%

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

      1. log1p-define 10.3%

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

   5. Simplified 10.3%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1.99999999999999992e275 < (/.f64 #s(literal 1 binary64) n) < -5.00000000000000008e204 or -2e46 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 41.0%

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

   3. Taylor expanded in n around inf 74.4%

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

      1. log1p-define 74.4%

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

   5. Simplified 74.4%

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

      1. log1p-undefine 74.4%

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

      2. diff-log 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. +-commutative 74.5%

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

   9. Simplified 74.5%

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

   if -5.00000000000000008e204 < (/.f64 #s(literal 1 binary64) n) < -2e46 or 1e-13 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000029e191

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0 68.9%

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

      1. *-rgt-identity 68.9%

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

      2. associate-*l/ 68.9%

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

      3. associate-/l* 68.9%

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

      4. exp-to-pow 68.9%

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

   5. Simplified 68.9%

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

   if 4.00000000000000029e191 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 6.7%

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

      1. log1p-define 6.7%

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

   5. Simplified 6.7%

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

      1. log1p-undefine 6.7%

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

      2. diff-log 6.7%

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

   7. Applied egg-rr 6.7%

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

      1. +-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in x around -inf 78.5%

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

       1. mul-1-neg 78.5%

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

       2. distribute-neg-frac2 78.5%

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

       3. sub-neg 78.5%

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

       4. associate-*r/ 78.5%

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

       5. sub-neg 78.5%

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

       6. metadata-eval 78.5%

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

       7. distribute-lft-in 78.5%

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

       8. neg-mul-1 78.5%

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

       9. associate-*r/ 78.5%

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

       10. metadata-eval 78.5%

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

       11. distribute-neg-frac 78.5%

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

       12. metadata-eval 78.5%

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

       13. metadata-eval 78.5%

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

       14. metadata-eval 78.5%

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

   12. Simplified 78.5%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{-x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+275}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+46}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-65)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-13)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 4e+191)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x)) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-65)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 1d-13) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 4d+191) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -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) <= -5e-65) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-65:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 1e-13:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 4e+191:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-65)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-13)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e+191)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / 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) <= -5e-65)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 1e-13)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 4e+191)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -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], -5e-65], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-13], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+191], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999983e-65

   1. Initial program 82.5%

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

   3. Taylor expanded in x around inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. log-rec 89.8%

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

      3. mul-1-neg 89.8%

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

      4. distribute-neg-frac 89.8%

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

      5. mul-1-neg 89.8%

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

      6. remove-double-neg 89.8%

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

      7. *-commutative 89.8%

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

   5. Simplified 89.8%

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

      1. *-un-lft-identity 89.8%

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

      2. div-inv 89.8%

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

      3. exp-to-pow 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. *-lft-identity 89.8%

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

   9. Simplified 89.8%

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

   if -4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.8%

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

   7. Applied egg-rr 80.8%

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

      1. +-commutative 80.8%

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

   9. Simplified 80.8%

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

       1. clear-num 80.8%

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

       2. log-div 80.9%

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

       3. metadata-eval 80.9%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   11. Applied egg-rr 80.9%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   12. Step-by-step derivation

       1. neg-sub0 80.9%

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

   13. Simplified 80.9%

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

   if 1e-13 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000029e191

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0 72.2%

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

   if 4.00000000000000029e191 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 6.7%

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

      1. log1p-define 6.7%

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

   5. Simplified 6.7%

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

      1. log1p-undefine 6.7%

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

      2. diff-log 6.7%

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

   7. Applied egg-rr 6.7%

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

      1. +-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in x around -inf 78.5%

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

       1. mul-1-neg 78.5%

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

       2. distribute-neg-frac2 78.5%

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

       3. sub-neg 78.5%

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

       4. associate-*r/ 78.5%

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

       5. sub-neg 78.5%

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

       6. metadata-eval 78.5%

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

       7. distribute-lft-in 78.5%

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

       8. neg-mul-1 78.5%

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

       9. associate-*r/ 78.5%

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

       10. metadata-eval 78.5%

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

       11. distribute-neg-frac 78.5%

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

       12. metadata-eval 78.5%

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

       13. metadata-eval 78.5%

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

       14. metadata-eval 78.5%

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

   12. Simplified 78.5%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{-x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-65)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 1e-13)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 4e+191)
         (- 1.0 t_0)
         (/ (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x)) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-65) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-65)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 1d-13) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 4d+191) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -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) <= -5e-65) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 1e-13) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-65:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 1e-13:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 4e+191:
		tmp = 1.0 - t_0
	else:
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-65)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-13)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e+191)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / 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) <= -5e-65)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 1e-13)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 4e+191)
		tmp = 1.0 - t_0;
	else
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -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], -5e-65], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-13], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+191], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999983e-65

   1. Initial program 82.5%

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

   3. Taylor expanded in x around inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. log-rec 89.8%

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

      3. mul-1-neg 89.8%

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

      4. distribute-neg-frac 89.8%

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

      5. mul-1-neg 89.8%

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

      6. remove-double-neg 89.8%

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

      7. *-commutative 89.8%

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

   5. Simplified 89.8%

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

      1. *-un-lft-identity 89.8%

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

      2. div-inv 89.8%

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

      3. exp-to-pow 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. *-lft-identity 89.8%

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

   9. Simplified 89.8%

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

   if -4.99999999999999983e-65 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. log1p-define 80.7%

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

   5. Simplified 80.7%

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

      1. log1p-undefine 80.7%

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

      2. diff-log 80.8%

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

   7. Applied egg-rr 80.8%

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

      1. +-commutative 80.8%

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

   9. Simplified 80.8%

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

       1. clear-num 80.8%

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

       2. log-div 80.9%

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

       3. metadata-eval 80.9%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   11. Applied egg-rr 80.9%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   12. Step-by-step derivation

       1. neg-sub0 80.9%

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

   13. Simplified 80.9%

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

   if 1e-13 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000029e191

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0 71.9%

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

      1. *-rgt-identity 71.9%

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

      2. associate-*l/ 71.9%

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

      3. associate-/l* 71.9%

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

      4. exp-to-pow 71.9%

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

   5. Simplified 71.9%

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

   if 4.00000000000000029e191 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 6.7%

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

      1. log1p-define 6.7%

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

   5. Simplified 6.7%

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

      1. log1p-undefine 6.7%

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

      2. diff-log 6.7%

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

   7. Applied egg-rr 6.7%

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

      1. +-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in x around -inf 78.5%

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

       1. mul-1-neg 78.5%

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

       2. distribute-neg-frac2 78.5%

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

       3. sub-neg 78.5%

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

       4. associate-*r/ 78.5%

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

       5. sub-neg 78.5%

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

       6. metadata-eval 78.5%

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

       7. distribute-lft-in 78.5%

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

       8. neg-mul-1 78.5%

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

       9. associate-*r/ 78.5%

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

       10. metadata-eval 78.5%

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

       11. distribute-neg-frac 78.5%

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

       12. metadata-eval 78.5%

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

       13. metadata-eval 78.5%

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

       14. metadata-eval 78.5%

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

   12. Simplified 78.5%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{-x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-55)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-13)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 4e+191)
         (- 1.0 t_0)
         (/ (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x)) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-55) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-13) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-55)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 1d-13) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 4d+191) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -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) <= -5e-55) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-13) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+191) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-55:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-13:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 4e+191:
		tmp = 1.0 - t_0
	else:
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-55)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-13)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e+191)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / 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) <= -5e-55)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 1e-13)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 4e+191)
		tmp = 1.0 - t_0;
	else
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -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], -5e-55], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-13], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+191], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000002e-55

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf 90.4%

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

      1. associate-/r* 90.3%

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

      2. mul-1-neg 90.3%

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

      3. log-rec 90.3%

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

      4. mul-1-neg 90.3%

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

      5. distribute-neg-frac 90.3%

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

      6. mul-1-neg 90.3%

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

      7. remove-double-neg 90.3%

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

      8. *-rgt-identity 90.3%

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

      9. associate-/l* 90.3%

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

      10. exp-to-pow 90.3%

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

   5. Simplified 90.3%

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

   if -5.0000000000000002e-55 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 31.0%

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

   3. Taylor expanded in n around inf 80.6%

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

      1. log1p-define 80.6%

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

   5. Simplified 80.6%

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

      1. log1p-undefine 80.6%

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

      2. diff-log 80.7%

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

   7. Applied egg-rr 80.7%

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

      1. +-commutative 80.7%

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

   9. Simplified 80.7%

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

       1. clear-num 80.7%

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

       2. log-div 80.7%

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

       3. metadata-eval 80.7%

          \[\leadsto \frac{\color{blue}{0} - \log \left(\frac{x}{x + 1}\right)}{n} \]

   11. Applied egg-rr 80.7%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   12. Step-by-step derivation

       1. neg-sub0 80.7%

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

   13. Simplified 80.7%

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

   if 1e-13 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000029e191

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0 71.9%

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

      1. *-rgt-identity 71.9%

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

      2. associate-*l/ 71.9%

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

      3. associate-/l* 71.9%

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

      4. exp-to-pow 71.9%

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

   5. Simplified 71.9%

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

   if 4.00000000000000029e191 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 6.7%

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

      1. log1p-define 6.7%

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

   5. Simplified 6.7%

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

      1. log1p-undefine 6.7%

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

      2. diff-log 6.7%

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

   7. Applied egg-rr 6.7%

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

      1. +-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in x around -inf 78.5%

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

       1. mul-1-neg 78.5%

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

       2. distribute-neg-frac2 78.5%

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

       3. sub-neg 78.5%

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

       4. associate-*r/ 78.5%

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

       5. sub-neg 78.5%

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

       6. metadata-eval 78.5%

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

       7. distribute-lft-in 78.5%

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

       8. neg-mul-1 78.5%

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

       9. associate-*r/ 78.5%

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

       10. metadata-eval 78.5%

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

       11. distribute-neg-frac 78.5%

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

       12. metadata-eval 78.5%

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

       13. metadata-eval 78.5%

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

       14. metadata-eval 78.5%

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

   12. Simplified 78.5%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{-x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2.05 \cdot 10^{-277}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-146}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 2.05e-277)
     t_0
     (if (<= x 4.6e-146)
       (/ (log x) (- n))
       (if (<= x 1.35e-97)
         t_0
         (if (<= x 0.86)
           (/ (- x (log x)) n)
           (if (<= x 3.3e+75)
             (/
              (/
               (+
                1.0
                (/ (- (/ (- 0.3333333333333333 (* 0.25 (/ 1.0 x))) x) 0.5) x))
               x)
              n)
             0.0)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.05e-277) {
		tmp = t_0;
	} else if (x <= 4.6e-146) {
		tmp = log(x) / -n;
	} else if (x <= 1.35e-97) {
		tmp = t_0;
	} else if (x <= 0.86) {
		tmp = (x - log(x)) / n;
	} else if (x <= 3.3e+75) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 2.05d-277) then
        tmp = t_0
    else if (x <= 4.6d-146) then
        tmp = log(x) / -n
    else if (x <= 1.35d-97) then
        tmp = t_0
    else if (x <= 0.86d0) then
        tmp = (x - log(x)) / n
    else if (x <= 3.3d+75) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - (0.25d0 * (1.0d0 / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    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 (x <= 2.05e-277) {
		tmp = t_0;
	} else if (x <= 4.6e-146) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.35e-97) {
		tmp = t_0;
	} else if (x <= 0.86) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 3.3e+75) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 2.05e-277:
		tmp = t_0
	elif x <= 4.6e-146:
		tmp = math.log(x) / -n
	elif x <= 1.35e-97:
		tmp = t_0
	elif x <= 0.86:
		tmp = (x - math.log(x)) / n
	elif x <= 3.3e+75:
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 2.05e-277)
		tmp = t_0;
	elseif (x <= 4.6e-146)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.35e-97)
		tmp = t_0;
	elseif (x <= 0.86)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 3.3e+75)
		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 = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 2.05e-277)
		tmp = t_0;
	elseif (x <= 4.6e-146)
		tmp = log(x) / -n;
	elseif (x <= 1.35e-97)
		tmp = t_0;
	elseif (x <= 0.86)
		tmp = (x - log(x)) / n;
	elseif (x <= 3.3e+75)
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	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[x, 2.05e-277], t$95$0, If[LessEqual[x, 4.6e-146], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.35e-97], t$95$0, If[LessEqual[x, 0.86], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3.3e+75], 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], 0.0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.04999999999999994e-277 or 4.6000000000000001e-146 < x < 1.34999999999999993e-97

   1. Initial program 62.3%

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

   3. Taylor expanded in x around 0 62.3%

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

      1. *-rgt-identity 62.3%

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

      2. associate-*l/ 62.3%

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

      3. associate-/l* 62.3%

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

      4. exp-to-pow 62.3%

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

   5. Simplified 62.3%

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

   if 2.04999999999999994e-277 < x < 4.6000000000000001e-146

   1. Initial program 39.1%

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

   3. Taylor expanded in n around inf 65.5%

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

      1. log1p-define 65.5%

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

   5. Simplified 65.5%

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

   6. Taylor expanded in x around 0 65.5%

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

      1. neg-mul-1 65.5%

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

   8. Simplified 65.5%

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

   if 1.34999999999999993e-97 < x < 0.859999999999999987

   1. Initial program 37.5%

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

   3. Taylor expanded in n around inf 55.2%

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

      1. log1p-define 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in x around 0 52.0%

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

   if 0.859999999999999987 < x < 3.29999999999999998e75

   1. Initial program 23.9%

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

   3. Taylor expanded in n around inf 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. Taylor expanded in x around -inf 79.6%

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

   if 3.29999999999999998e75 < x

   1. Initial program 81.8%

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

      1. sub-neg 81.8%

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

      2. +-commutative 81.8%

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

      3. add-log-exp 81.8%

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

      4. add-log-exp 81.8%

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

      5. sum-log 81.8%

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

      6. pow-to-exp 81.8%

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

      7. un-div-inv 81.8%

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

      8. +-commutative 81.8%

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

      9. log1p-define 81.8%

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in n around inf 81.8%

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

      1. prod-exp 81.8%

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

      2. metadata-eval 81.8%

         \[\leadsto \log \left(e^{\color{blue}{0}}\right) \]

      3. 1-exp 81.8%

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

      4. metadata-eval 81.8%

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

   7. Simplified 81.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-277}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-146}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 9.2e-144)
   (/ (log x) (- n))
   (if (<= x 6.5e-100)
     (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x)) x)
     (if (<= x 0.88)
       (/ (- x (log x)) n)
       (if (<= x 2.25e+77)
         (/
          (/
           (+
            1.0
            (/ (- (/ (- 0.3333333333333333 (* 0.25 (/ 1.0 x))) x) 0.5) x))
           x)
          n)
         0.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 9.2e-144) {
		tmp = log(x) / -n;
	} else if (x <= 6.5e-100) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.25e+77) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 9.2d-144) then
        tmp = log(x) / -n
    else if (x <= 6.5d-100) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.25d+77) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - (0.25d0 * (1.0d0 / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 9.2e-144) {
		tmp = Math.log(x) / -n;
	} else if (x <= 6.5e-100) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.25e+77) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 9.2e-144:
		tmp = math.log(x) / -n
	elif x <= 6.5e-100:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	elif x <= 2.25e+77:
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 9.2e-144)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 6.5e-100)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.25e+77)
		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 = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 9.2e-144)
		tmp = log(x) / -n;
	elseif (x <= 6.5e-100)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.25e+77)
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 9.2e-144], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 6.5e-100], 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, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.25e+77], 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], 0.0]]]]

Derivation

1. Split input into 5 regimes

2. if x < 9.2e-144

   1. Initial program 47.6%

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

   3. Taylor expanded in n around inf 57.3%

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

      1. log1p-define 57.3%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in x around 0 57.3%

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

      1. neg-mul-1 57.3%

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

   8. Simplified 57.3%

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

   if 9.2e-144 < x < 6.50000000000000013e-100

   1. Initial program 56.0%

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

   3. Taylor expanded in n around inf 28.6%

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

      1. log1p-define 28.6%

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

   5. Simplified 28.6%

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

   6. Taylor expanded in x around -inf 51.5%

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

      1. associate-*r/ 51.5%

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

      2. mul-1-neg 51.5%

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

      3. associate-*r/ 51.5%

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

      4. mul-1-neg 51.5%

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

      5. associate-*r/ 51.5%

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

      6. metadata-eval 51.5%

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

      7. *-commutative 51.5%

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

      8. associate-*r/ 51.5%

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

      9. metadata-eval 51.5%

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

   8. Simplified 51.5%

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

   if 6.50000000000000013e-100 < x < 0.880000000000000004

   1. Initial program 37.5%

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

   3. Taylor expanded in n around inf 55.2%

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

      1. log1p-define 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in x around 0 52.0%

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

   if 0.880000000000000004 < x < 2.25000000000000012e77

   1. Initial program 23.9%

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

   3. Taylor expanded in n around inf 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. Taylor expanded in x around -inf 79.6%

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

   if 2.25000000000000012e77 < x

   1. Initial program 81.8%

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

      1. sub-neg 81.8%

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

      2. +-commutative 81.8%

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

      3. add-log-exp 81.8%

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

      4. add-log-exp 81.8%

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

      5. sum-log 81.8%

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

      6. pow-to-exp 81.8%

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

      7. un-div-inv 81.8%

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

      8. +-commutative 81.8%

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

      9. log1p-define 81.8%

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in n around inf 81.8%

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

      1. prod-exp 81.8%

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

      2. metadata-eval 81.8%

         \[\leadsto \log \left(e^{\color{blue}{0}}\right) \]

      3. 1-exp 81.8%

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

      4. metadata-eval 81.8%

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

   7. Simplified 81.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 1.6 \cdot 10^{-144}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 1.6e-144)
     t_0
     (if (<= x 6.5e-100)
       (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x)) x)
       (if (<= x 0.7)
         t_0
         (if (<= x 8.6e+71)
           (/
            (/
             (+
              1.0
              (/ (- (/ (- 0.3333333333333333 (* 0.25 (/ 1.0 x))) x) 0.5) x))
             x)
            n)
           0.0))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 1.6e-144) {
		tmp = t_0;
	} else if (x <= 6.5e-100) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 8.6e+71) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 1.6d-144) then
        tmp = t_0
    else if (x <= 6.5d-100) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else if (x <= 0.7d0) then
        tmp = t_0
    else if (x <= 8.6d+71) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - (0.25d0 * (1.0d0 / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 1.6e-144) {
		tmp = t_0;
	} else if (x <= 6.5e-100) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 8.6e+71) {
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 1.6e-144:
		tmp = t_0
	elif x <= 6.5e-100:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	elif x <= 0.7:
		tmp = t_0
	elif x <= 8.6e+71:
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 1.6e-144)
		tmp = t_0;
	elseif (x <= 6.5e-100)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 8.6e+71)
		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 = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 1.6e-144)
		tmp = t_0;
	elseif (x <= 6.5e-100)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 8.6e+71)
		tmp = ((1.0 + ((((0.3333333333333333 - (0.25 * (1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 1.6e-144], t$95$0, If[LessEqual[x, 6.5e-100], 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, 0.7], t$95$0, If[LessEqual[x, 8.6e+71], 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], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.59999999999999986e-144 or 6.50000000000000013e-100 < x < 0.69999999999999996

   1. Initial program 44.1%

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

   3. Taylor expanded in n around inf 56.6%

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

      1. log1p-define 56.6%

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

   5. Simplified 56.6%

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

   6. Taylor expanded in x around 0 55.3%

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

      1. neg-mul-1 55.3%

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

   8. Simplified 55.3%

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

   if 1.59999999999999986e-144 < x < 6.50000000000000013e-100

   1. Initial program 56.0%

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

   3. Taylor expanded in n around inf 28.6%

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

      1. log1p-define 28.6%

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

   5. Simplified 28.6%

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

   6. Taylor expanded in x around -inf 51.5%

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

      1. associate-*r/ 51.5%

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

      2. mul-1-neg 51.5%

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

      3. associate-*r/ 51.5%

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

      4. mul-1-neg 51.5%

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

      5. associate-*r/ 51.5%

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

      6. metadata-eval 51.5%

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

      7. *-commutative 51.5%

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

      8. associate-*r/ 51.5%

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

      9. metadata-eval 51.5%

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

   8. Simplified 51.5%

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

   if 0.69999999999999996 < x < 8.59999999999999967e71

   1. Initial program 23.9%

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

   3. Taylor expanded in n around inf 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. Taylor expanded in x around -inf 79.6%

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

   if 8.59999999999999967e71 < x

   1. Initial program 81.8%

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

      1. sub-neg 81.8%

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

      2. +-commutative 81.8%

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

      3. add-log-exp 81.8%

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

      4. add-log-exp 81.8%

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

      5. sum-log 81.8%

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

      6. pow-to-exp 81.8%

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

      7. un-div-inv 81.8%

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

      8. +-commutative 81.8%

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

      9. log1p-define 81.8%

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in n around inf 81.8%

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

      1. prod-exp 81.8%

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

      2. metadata-eval 81.8%

         \[\leadsto \log \left(e^{\color{blue}{0}}\right) \]

      3. 1-exp 81.8%

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

      4. metadata-eval 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.1e+76)
   (/ (/ (- 1.0 (/ (+ 0.5 (/ (+ 0.3333333333333333 (/ -0.25 x)) x)) x)) x) n)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.1e+76) {
		tmp = ((1.0 - ((0.5 + ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.1d+76) then
        tmp = ((1.0d0 - ((0.5d0 + ((0.3333333333333333d0 + ((-0.25d0) / x)) / x)) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.1e+76) {
		tmp = ((1.0 - ((0.5 + ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.1e+76:
		tmp = ((1.0 - ((0.5 + ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.1e+76)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x)) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.1e+76)
		tmp = ((1.0 - ((0.5 + ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.1e+76], N[(N[(N[(1.0 - N[(N[(0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000011e76

   1. Initial program 41.6%

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

   3. Taylor expanded in n around inf 49.0%

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

      1. log1p-define 49.0%

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

   5. Simplified 49.0%

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

   6. Taylor expanded in x around -inf 15.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.7%

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

      2. sqrt-unprod 15.0%

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

      3. sqr-neg 15.0%

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

      4. mul-1-neg 15.0%

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

      5. mul-1-neg 15.0%

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

      6. sqrt-unprod 14.4%

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

      7. add-sqr-sqrt 40.3%

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

      8. sub-neg 40.3%

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

      9. un-div-inv 40.3%

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

      10. metadata-eval 40.3%

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

   8. Applied egg-rr 40.3%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{\frac{0.25}{x} + -0.3333333333333333}{x}\right)} - 0.5}{x} - 1}{x}}{n} \]
   9. Step-by-step derivation

      1. neg-mul-1 40.3%

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

      2. distribute-neg-frac 40.3%

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

      3. +-commutative 40.3%

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

      4. distribute-neg-in 40.3%

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

      5. metadata-eval 40.3%

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

      6. distribute-neg-frac 40.3%

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

      7. metadata-eval 40.3%

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

   10. Simplified 40.3%

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

   if 3.10000000000000011e76 < x

   1. Initial program 81.8%

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

      1. sub-neg 81.8%

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

      2. +-commutative 81.8%

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

      3. add-log-exp 81.8%

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

      4. add-log-exp 81.8%

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

      5. sum-log 81.8%

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

      6. pow-to-exp 81.8%

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

      7. un-div-inv 81.8%

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

      8. +-commutative 81.8%

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

      9. log1p-define 81.8%

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in n around inf 81.8%

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

      1. prod-exp 81.8%

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

      2. metadata-eval 81.8%

         \[\leadsto \log \left(e^{\color{blue}{0}}\right) \]

      3. 1-exp 81.8%

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

      4. metadata-eval 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+75}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1e+75)
   (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x)) x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1e+75) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1d+75) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1e+75) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1e+75:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1e+75)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1e+75)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1e+75], 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], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999927e74

   1. Initial program 41.6%

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

   3. Taylor expanded in n around inf 49.0%

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

      1. log1p-define 49.0%

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

   5. Simplified 49.0%

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

   6. Taylor expanded in x around -inf 40.1%

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

      1. associate-*r/ 40.1%

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

      2. mul-1-neg 40.1%

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

      3. associate-*r/ 40.1%

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

      4. mul-1-neg 40.1%

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

      5. associate-*r/ 40.1%

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

      6. metadata-eval 40.1%

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

      7. *-commutative 40.1%

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

      8. associate-*r/ 40.1%

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

      9. metadata-eval 40.1%

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

   8. Simplified 40.1%

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

   if 9.99999999999999927e74 < x

   1. Initial program 81.8%

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

      1. sub-neg 81.8%

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

      2. +-commutative 81.8%

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

      3. add-log-exp 81.8%

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

      4. add-log-exp 81.8%

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

      5. sum-log 81.8%

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

      6. pow-to-exp 81.8%

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

      7. un-div-inv 81.8%

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

      8. +-commutative 81.8%

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

      9. log1p-define 81.8%

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in n around inf 81.8%

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

      1. prod-exp 81.8%

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

      2. metadata-eval 81.8%

         \[\leadsto \log \left(e^{\color{blue}{0}}\right) \]

      3. 1-exp 81.8%

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

      4. metadata-eval 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+75}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 49.5% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.9e+73)
   (/ (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x)) n)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.9e+73) {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.9d+73) then
        tmp = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.9e+73) {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.9e+73:
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.9e+73)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / Float64(-x)) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.9e+73)
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.9e+73], N[(N[(N[(N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.9000000000000002e73

   1. Initial program 41.6%

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

   3. Taylor expanded in n around inf 49.0%

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

      1. log1p-define 49.0%

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

   5. Simplified 49.0%

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

      1. log1p-undefine 49.0%

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

      2. diff-log 49.1%

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

   7. Applied egg-rr 49.1%

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

      1. +-commutative 49.1%

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

   9. Simplified 49.1%

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

   10. Taylor expanded in x around -inf 40.1%

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

       1. mul-1-neg 40.1%

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

       2. distribute-neg-frac2 40.1%

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

       3. sub-neg 40.1%

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

       4. associate-*r/ 40.1%

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

       5. sub-neg 40.1%

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

       6. metadata-eval 40.1%

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

       7. distribute-lft-in 40.1%

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

       8. neg-mul-1 40.1%

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

       9. associate-*r/ 40.1%

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

       10. metadata-eval 40.1%

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

       11. distribute-neg-frac 40.1%

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

       12. metadata-eval 40.1%

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

       13. metadata-eval 40.1%

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

       14. metadata-eval 40.1%

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

   12. Simplified 40.1%

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

   if 2.9000000000000002e73 < x

   1. Initial program 81.8%

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

      1. sub-neg 81.8%

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

      2. +-commutative 81.8%

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

      3. add-log-exp 81.8%

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

      4. add-log-exp 81.8%

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

      5. sum-log 81.8%

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

      6. pow-to-exp 81.8%

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

      7. un-div-inv 81.8%

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

      8. +-commutative 81.8%

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

      9. log1p-define 81.8%

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in n around inf 81.8%

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

      1. prod-exp 81.8%

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

      2. metadata-eval 81.8%

         \[\leadsto \log \left(e^{\color{blue}{0}}\right) \]

      3. 1-exp 81.8%

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

      4. metadata-eval 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 44.0% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 9.5e+76) (/ (/ 1.0 n) x) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 9.5e+76) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 9.5d+76) then
        tmp = (1.0d0 / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 9.5e+76) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 9.5e+76:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 9.5e+76)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 9.5e+76)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 9.5e+76], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000003e76

   1. Initial program 41.6%

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

   3. Taylor expanded in n around inf 49.0%

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

      1. log1p-define 49.0%

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

   5. Simplified 49.0%

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

   6. Taylor expanded in x around inf 32.2%

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

      1. associate-/r* 32.6%

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

   8. Simplified 32.6%

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

   if 9.5000000000000003e76 < x

   1. Initial program 81.8%

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

      1. sub-neg 81.8%

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

      2. +-commutative 81.8%

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

      3. add-log-exp 81.8%

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

      4. add-log-exp 81.8%

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

      5. sum-log 81.8%

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

      6. pow-to-exp 81.8%

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

      7. un-div-inv 81.8%

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

      8. +-commutative 81.8%

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

      9. log1p-define 81.8%

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in n around inf 81.8%

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

      1. prod-exp 81.8%

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

      2. metadata-eval 81.8%

         \[\leadsto \log \left(e^{\color{blue}{0}}\right) \]

      3. 1-exp 81.8%

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

      4. metadata-eval 81.8%

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

   7. Simplified 81.8%

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

Alternative 20: 43.9% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 6.5e+76) (/ 1.0 (* x n)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.5e+76) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 6.5d+76) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6.5e+76) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6.5e+76:
		tmp = 1.0 / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.5e+76)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6.5e+76)
		tmp = 1.0 / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.5e+76], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.5000000000000005e76

   1. Initial program 41.6%

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

   3. Taylor expanded in n around inf 49.0%

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

      1. log1p-define 49.0%

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

   5. Simplified 49.0%

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

   6. Taylor expanded in x around inf 32.2%

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

   if 6.5000000000000005e76 < x

   1. Initial program 81.8%

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

      1. sub-neg 81.8%

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

      2. +-commutative 81.8%

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

      3. add-log-exp 81.8%

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

      4. add-log-exp 81.8%

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

      5. sum-log 81.8%

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

      6. pow-to-exp 81.8%

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

      7. un-div-inv 81.8%

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

      8. +-commutative 81.8%

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

      9. log1p-define 81.8%

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in n around inf 81.8%

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

      1. prod-exp 81.8%

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

      2. metadata-eval 81.8%

         \[\leadsto \log \left(e^{\color{blue}{0}}\right) \]

      3. 1-exp 81.8%

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

      4. metadata-eval 81.8%

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

   7. Simplified 81.8%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 30.4% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 53.7%

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

   1. sub-neg 53.7%

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

   2. +-commutative 53.7%

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

   3. add-log-exp 53.3%

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

   4. add-log-exp 53.3%

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

   5. sum-log 53.2%

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

   6. pow-to-exp 53.2%

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

   7. un-div-inv 53.2%

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

   8. +-commutative 53.2%

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

   9. log1p-define 57.4%

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

4. Applied egg-rr 57.4%

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

5. Taylor expanded in n around inf 29.4%

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

   1. prod-exp 29.4%

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

   2. metadata-eval 29.4%

      \[\leadsto \log \left(e^{\color{blue}{0}}\right) \]

   3. 1-exp 29.4%

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

   4. metadata-eval 29.4%

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

7. Simplified 29.4%

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

Reproduce

herbie shell --seed 2024088 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))