2nthrt (problem 3.4.6)

Percentage Accurate: 54.0% → 91.8%

Time: 50.5s

Alternatives: 18

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: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 15.6:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.5, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 15.6)
   (/
    (log
     (/
      x
      (exp
       (+
        (log1p x)
        (/
         (fma
          (- (pow (log1p x) 2.0) (pow (log x) 2.0))
          0.5
          (/
           (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
           n))
         n)))))
    (- n))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 15.6) {
		tmp = log((x / exp((log1p(x) + (fma((pow(log1p(x), 2.0) - pow(log(x), 2.0)), 0.5, ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n)) / n))))) / -n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 15.6)
		tmp = Float64(log(Float64(x / exp(Float64(log1p(x) + Float64(fma(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)), 0.5, Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n)) / n))))) / Float64(-n));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 15.6], N[(N[Log[N[(x / N[Exp[N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 15.5999999999999996

   1. Initial program 41.5%

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

   3. Taylor expanded in n around -inf 79.9%

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

   5. Simplified 79.9%

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

      1. add-log-exp 86.8%

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

      2. diff-log 86.8%

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

   7. Applied egg-rr 86.8%

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

   if 15.5999999999999996 < x

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf 97.4%

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

      1. associate-/r* 98.7%

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

      2. mul-1-neg 98.7%

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

      3. log-rec 98.7%

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

      4. mul-1-neg 98.7%

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

      5. distribute-neg-frac 98.7%

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

      6. mul-1-neg 98.7%

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

      7. remove-double-neg 98.7%

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

      8. *-rgt-identity 98.7%

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

      9. associate-/l* 98.7%

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

      10. exp-to-pow 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 15.6:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.5, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -1.115 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-45}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -1.115e-74)
     t_1
     (if (<= (/ 1.0 n) 1e-45)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 50000.0) t_1 (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1.115e-74) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-45) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 50000.0) {
		tmp = t_1;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1.115e-74) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-45) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 50000.0) {
		tmp = t_1;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -1.115e-74:
		tmp = t_1
	elif (1.0 / n) <= 1e-45:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 50000.0:
		tmp = t_1
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.115e-74)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-45)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 50000.0)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.115e-74], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-45], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50000.0], t$95$1, N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.115e-74 or 9.99999999999999984e-46 < (/.f64 #s(literal 1 binary64) n) < 5e4

   1. Initial program 77.7%

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

   3. Taylor expanded in x around inf 89.5%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 -1.115e-74 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999984e-46

   1. Initial program 38.5%

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

   3. Taylor expanded in n around inf 88.8%

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

      1. log1p-define 88.8%

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

   5. Simplified 88.8%

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

      1. log1p-undefine 88.8%

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

      2. diff-log 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. log-div 89.0%

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

       3. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. neg-sub0 89.0%

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

   13. Simplified 89.0%

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

   if 5e4 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 57.3%

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

   3. Taylor expanded in n around 0 57.3%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.115 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-45}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -1.115 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-45}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -1.115e-74)
     t_1
     (if (<= (/ 1.0 n) 1e-45)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 50000.0)
         t_1
         (-
          (+ 1.0 (* x (+ (/ 1.0 n) (/ (+ (* x -0.5) (* 0.5 (/ x n))) n))))
          t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1.115e-74) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-45) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 50000.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / n) / x
    if ((1.0d0 / n) <= (-1.115d-74)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d-45) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 50000.0d0) then
        tmp = t_1
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (((x * (-0.5d0)) + (0.5d0 * (x / n))) / n)))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1.115e-74) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-45) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 50000.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -1.115e-74:
		tmp = t_1
	elif (1.0 / n) <= 1e-45:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 50000.0:
		tmp = t_1
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.115e-74)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-45)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 50000.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n))) / n)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / n) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -1.115e-74)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e-45)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 50000.0)
		tmp = t_1;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.115e-74], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-45], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 50000.0], t$95$1, N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.115e-74 or 9.99999999999999984e-46 < (/.f64 #s(literal 1 binary64) n) < 5e4

   1. Initial program 77.7%

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

   3. Taylor expanded in x around inf 89.5%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 -1.115e-74 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999984e-46

   1. Initial program 38.5%

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

   3. Taylor expanded in n around inf 88.8%

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

      1. log1p-define 88.8%

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

   5. Simplified 88.8%

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

      1. log1p-undefine 88.8%

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

      2. diff-log 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. log-div 89.0%

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

       3. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. neg-sub0 89.0%

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

   13. Simplified 89.0%

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

   if 5e4 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 57.3%

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

   3. Taylor expanded in x around 0 85.5%

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

   4. Taylor expanded in n around inf 85.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.115 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-45}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 50000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.115 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.002:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+150}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e+47)
   (/ 0.3333333333333333 (* n (pow x 3.0)))
   (if (<= (/ 1.0 n) -1e-40)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= (/ 1.0 n) -1.115e-74)
       (/ 1.0 (* x n))
       (if (<= (/ 1.0 n) 0.002)
         (/ (log (/ x (+ x 1.0))) (- n))
         (if (<= (/ 1.0 n) 4e+150)
           (+ 1.0 (pow x (/ 1.0 n)))
           (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) (* x n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e+47) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((1.0 / n) <= -1e-40) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= -1.115e-74) {
		tmp = 1.0 / (x * n);
	} else if ((1.0 / n) <= 0.002) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+150) {
		tmp = 1.0 + pow(x, (1.0 / n));
	} else {
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-5d+47)) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if ((1.0d0 / n) <= (-1d-40)) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= (-1.115d-74)) then
        tmp = 1.0d0 / (x * n)
    else if ((1.0d0 / n) <= 0.002d0) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 4d+150) then
        tmp = 1.0d0 + (x ** (1.0d0 / n))
    else
        tmp = (1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e+47) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((1.0 / n) <= -1e-40) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= -1.115e-74) {
		tmp = 1.0 / (x * n);
	} else if ((1.0 / n) <= 0.002) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 4e+150) {
		tmp = 1.0 + Math.pow(x, (1.0 / n));
	} else {
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e+47:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (1.0 / n) <= -1e-40:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= -1.115e-74:
		tmp = 1.0 / (x * n)
	elif (1.0 / n) <= 0.002:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 4e+150:
		tmp = 1.0 + math.pow(x, (1.0 / n))
	else:
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+47)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (Float64(1.0 / n) <= -1e-40)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= -1.115e-74)
		tmp = Float64(1.0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 0.002)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e+150)
		tmp = Float64(1.0 + (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5e+47)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif ((1.0 / n) <= -1e-40)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= -1.115e-74)
		tmp = 1.0 / (x * n);
	elseif ((1.0 / n) <= 0.002)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 4e+150)
		tmp = 1.0 + (x ^ (1.0 / n));
	else
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+47], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-40], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.115e-74], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.002], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+150], N[(1.0 + N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000022e47

   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 38.5%

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

      1. log1p-define 38.5%

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

   5. Simplified 38.5%

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

   6. Taylor expanded in x around -inf 48.6%

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

   7. Taylor expanded in x around 0 77.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. *-commutative 77.2%

         \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{3} \cdot n}} \]

   9. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{3} \cdot n}} \]

   if -5.00000000000000022e47 < (/.f64 #s(literal 1 binary64) n) < -9.9999999999999993e-41

   1. Initial program 69.9%

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

   3. Taylor expanded in n around inf 80.8%

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

      1. log1p-define 80.8%

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

   5. Simplified 80.8%

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

      1. log1p-undefine 80.8%

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

      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%

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

   if -9.9999999999999993e-41 < (/.f64 #s(literal 1 binary64) n) < -1.115e-74

   1. Initial program 6.9%

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

   3. Taylor expanded in n around inf 18.9%

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

      1. log1p-define 18.9%

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

   5. Simplified 18.9%

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

   6. Taylor expanded in x around inf 87.9%

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

   if -1.115e-74 < (/.f64 #s(literal 1 binary64) n) < 2e-3

   1. Initial program 38.3%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. diff-log 85.8%

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

   7. Applied egg-rr 85.8%

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

      1. +-commutative 85.8%

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

   9. Simplified 85.8%

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

       1. clear-num 85.8%

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

       2. log-div 85.8%

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

       3. metadata-eval 85.8%

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

   11. Applied egg-rr 85.8%

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

       1. neg-sub0 85.8%

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

   13. Simplified 85.8%

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

   if 2e-3 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999992e150

   1. Initial program 81.9%

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

   3. Taylor expanded in x around 0 77.5%

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

      1. *-rgt-identity 77.5%

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

      2. associate-*l/ 77.5%

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

      3. associate-/l* 77.5%

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

      4. exp-to-pow 77.5%

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

   5. Simplified 77.5%

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

      1. add-exp-log 77.5%

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

      2. sub-neg 77.5%

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

      3. log1p-define 77.5%

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

   7. Applied egg-rr 77.5%

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

      1. log1p-undefine 77.5%

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

      2. rem-exp-log 77.5%

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

      3. add-sqr-sqrt 72.9%

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

      4. sqrt-unprod 91.1%

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

      5. sqr-neg 91.1%

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

      6. sqrt-unprod 91.1%

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

      7. add-sqr-sqrt 91.1%

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

   9. Applied egg-rr 91.1%

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

   if 3.99999999999999992e150 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 12.7%

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

   3. Taylor expanded in n around inf 7.2%

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

      1. log1p-define 7.2%

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

   5. Simplified 7.2%

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

   6. Taylor expanded in x around -inf 87.1%

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

   7. Taylor expanded in n around 0 87.1%

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

   9. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.115 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.002:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+150}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{\frac{t\_1}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -1.115 \cdot 10^{-74}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-45}:\\ \;\;\;\;\frac{\log t\_0}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.002:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+112}:\\ \;\;\;\;1 + t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(t\_0 + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (pow x (/ 1.0 n))) (t_2 (/ (/ t_1 n) x)))
   (if (<= (/ 1.0 n) -1.115e-74)
     t_2
     (if (<= (/ 1.0 n) 1e-45)
       (/ (log t_0) (- n))
       (if (<= (/ 1.0 n) 0.002)
         t_2
         (if (<= (/ 1.0 n) 5e+112)
           (+ 1.0 t_1)
           (/ (log1p (+ t_0 -1.0)) (- n))))))))

C

double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = pow(x, (1.0 / n));
	double t_2 = (t_1 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1.115e-74) {
		tmp = t_2;
	} else if ((1.0 / n) <= 1e-45) {
		tmp = log(t_0) / -n;
	} else if ((1.0 / n) <= 0.002) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e+112) {
		tmp = 1.0 + t_1;
	} else {
		tmp = log1p((t_0 + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = (t_1 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1.115e-74) {
		tmp = t_2;
	} else if ((1.0 / n) <= 1e-45) {
		tmp = Math.log(t_0) / -n;
	} else if ((1.0 / n) <= 0.002) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e+112) {
		tmp = 1.0 + t_1;
	} else {
		tmp = Math.log1p((t_0 + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / (x + 1.0)
	t_1 = math.pow(x, (1.0 / n))
	t_2 = (t_1 / n) / x
	tmp = 0
	if (1.0 / n) <= -1.115e-74:
		tmp = t_2
	elif (1.0 / n) <= 1e-45:
		tmp = math.log(t_0) / -n
	elif (1.0 / n) <= 0.002:
		tmp = t_2
	elif (1.0 / n) <= 5e+112:
		tmp = 1.0 + t_1
	else:
		tmp = math.log1p((t_0 + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(Float64(t_1 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.115e-74)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 1e-45)
		tmp = Float64(log(t_0) / Float64(-n));
	elseif (Float64(1.0 / n) <= 0.002)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 5e+112)
		tmp = Float64(1.0 + t_1);
	else
		tmp = Float64(log1p(Float64(t_0 + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.115e-74], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-45], N[(N[Log[t$95$0], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.002], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+112], N[(1.0 + t$95$1), $MachinePrecision], N[(N[Log[1 + N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.115e-74 or 9.99999999999999984e-46 < (/.f64 #s(literal 1 binary64) n) < 2e-3

   1. Initial program 80.2%

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

   3. Taylor expanded in x around inf 90.1%

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

      1. associate-/r* 91.0%

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

      2. mul-1-neg 91.0%

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

      3. log-rec 91.0%

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

      4. mul-1-neg 91.0%

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

      5. distribute-neg-frac 91.0%

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

      6. mul-1-neg 91.0%

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

      7. remove-double-neg 91.0%

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

      8. *-rgt-identity 91.0%

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

      9. associate-/l* 91.0%

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

      10. exp-to-pow 91.0%

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

   5. Simplified 91.0%

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

   if -1.115e-74 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999984e-46

   1. Initial program 38.5%

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

   3. Taylor expanded in n around inf 88.8%

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

      1. log1p-define 88.8%

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

   5. Simplified 88.8%

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

      1. log1p-undefine 88.8%

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

      2. diff-log 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. log-div 89.0%

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

       3. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. neg-sub0 89.0%

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

   13. Simplified 89.0%

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

   if 2e-3 < (/.f64 #s(literal 1 binary64) n) < 5e112

   1. Initial program 84.9%

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

   3. Taylor expanded in x around 0 80.1%

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

      1. *-rgt-identity 80.1%

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

      2. associate-*l/ 80.1%

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

      3. associate-/l* 80.1%

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

      4. exp-to-pow 80.1%

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

   5. Simplified 80.1%

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

      1. add-exp-log 80.1%

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

      2. sub-neg 80.1%

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

      3. log1p-define 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. log1p-undefine 80.1%

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

      2. rem-exp-log 80.1%

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

      3. add-sqr-sqrt 75.1%

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

      4. sqrt-unprod 95.1%

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

      5. sqr-neg 95.1%

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

      6. sqrt-unprod 95.1%

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

      7. add-sqr-sqrt 95.1%

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

   9. Applied egg-rr 95.1%

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

   if 5e112 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.3%

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

   3. Taylor expanded in n around inf 6.9%

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

      1. log1p-define 6.9%

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

   5. Simplified 6.9%

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

      1. log1p-undefine 6.9%

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

      2. diff-log 6.9%

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

   7. Applied egg-rr 6.9%

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

      1. +-commutative 6.9%

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

   9. Simplified 6.9%

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

       1. clear-num 6.9%

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

       2. log-div 6.9%

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

       3. metadata-eval 6.9%

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

   11. Applied egg-rr 6.9%

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

       1. neg-sub0 6.9%

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

   13. Simplified 6.9%

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

       1. log1p-expm1-u 82.9%

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

       2. expm1-undefine 82.9%

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

       3. add-exp-log 82.9%

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

   15. Applied egg-rr 82.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.115 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-45}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.002:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+112}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{x}{x + 1} + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -1.115 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-45}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+150}:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -1.115e-74)
     t_1
     (if (<= (/ 1.0 n) 1e-45)
       (/ (log (/ x (+ x 1.0))) (- n))
       (if (<= (/ 1.0 n) 0.002)
         t_1
         (if (<= (/ 1.0 n) 4e+150)
           (+ 1.0 t_0)
           (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) (* x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1.115e-74) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-45) {
		tmp = log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 0.002) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+150) {
		tmp = 1.0 + t_0;
	} else {
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / n) / x
    if ((1.0d0 / n) <= (-1.115d-74)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d-45) then
        tmp = log((x / (x + 1.0d0))) / -n
    else if ((1.0d0 / n) <= 0.002d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 4d+150) then
        tmp = 1.0d0 + t_0
    else
        tmp = (1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -1.115e-74) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-45) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else if ((1.0 / n) <= 0.002) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+150) {
		tmp = 1.0 + t_0;
	} else {
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -1.115e-74:
		tmp = t_1
	elif (1.0 / n) <= 1e-45:
		tmp = math.log((x / (x + 1.0))) / -n
	elif (1.0 / n) <= 0.002:
		tmp = t_1
	elif (1.0 / n) <= 4e+150:
		tmp = 1.0 + t_0
	else:
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.115e-74)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-45)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 0.002)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e+150)
		tmp = Float64(1.0 + t_0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / n) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -1.115e-74)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e-45)
		tmp = log((x / (x + 1.0))) / -n;
	elseif ((1.0 / n) <= 0.002)
		tmp = t_1;
	elseif ((1.0 / n) <= 4e+150)
		tmp = 1.0 + t_0;
	else
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.115e-74], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-45], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.002], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+150], N[(1.0 + t$95$0), $MachinePrecision], N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.115e-74 or 9.99999999999999984e-46 < (/.f64 #s(literal 1 binary64) n) < 2e-3

   1. Initial program 80.2%

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

   3. Taylor expanded in x around inf 90.1%

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

      1. associate-/r* 91.0%

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

      2. mul-1-neg 91.0%

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

      3. log-rec 91.0%

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

      4. mul-1-neg 91.0%

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

      5. distribute-neg-frac 91.0%

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

      6. mul-1-neg 91.0%

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

      7. remove-double-neg 91.0%

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

      8. *-rgt-identity 91.0%

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

      9. associate-/l* 91.0%

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

      10. exp-to-pow 91.0%

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

   5. Simplified 91.0%

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

   if -1.115e-74 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999984e-46

   1. Initial program 38.5%

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

   3. Taylor expanded in n around inf 88.8%

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

      1. log1p-define 88.8%

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

   5. Simplified 88.8%

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

      1. log1p-undefine 88.8%

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

      2. diff-log 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. log-div 89.0%

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

       3. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. neg-sub0 89.0%

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

   13. Simplified 89.0%

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

   if 2e-3 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999992e150

   1. Initial program 81.9%

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

   3. Taylor expanded in x around 0 77.5%

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

      1. *-rgt-identity 77.5%

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

      2. associate-*l/ 77.5%

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

      3. associate-/l* 77.5%

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

      4. exp-to-pow 77.5%

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

   5. Simplified 77.5%

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

      1. add-exp-log 77.5%

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

      2. sub-neg 77.5%

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

      3. log1p-define 77.5%

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

   7. Applied egg-rr 77.5%

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

      1. log1p-undefine 77.5%

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

      2. rem-exp-log 77.5%

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

      3. add-sqr-sqrt 72.9%

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

      4. sqrt-unprod 91.1%

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

      5. sqr-neg 91.1%

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

      6. sqrt-unprod 91.1%

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

      7. add-sqr-sqrt 91.1%

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

   9. Applied egg-rr 91.1%

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

   if 3.99999999999999992e150 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 12.7%

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

   3. Taylor expanded in n around inf 7.2%

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

      1. log1p-define 7.2%

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

   5. Simplified 7.2%

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

   6. Taylor expanded in x around -inf 87.1%

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

   7. Taylor expanded in n around 0 87.1%

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

   9. Simplified 87.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.115 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-45}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.002:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+150}:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2.05 \cdot 10^{-234}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-200}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 2.05e-234)
     t_0
     (if (<= x 5.8e-200)
       t_1
       (if (<= x 1.15e-150)
         t_0
         (if (<= x 6e-123)
           t_1
           (if (<= x 0.86)
             (- (/ x n) (/ (log x) n))
             (if (<= x 9.5e+153)
               (/
                (/
                 (- 1.0 (/ (- 0.5 (/ (+ 0.3333333333333333 (/ -0.25 x)) x)) x))
                 x)
                n)
               (/ 0.0 n)))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.05e-234) {
		tmp = t_0;
	} else if (x <= 5.8e-200) {
		tmp = t_1;
	} else if (x <= 1.15e-150) {
		tmp = t_0;
	} else if (x <= 6e-123) {
		tmp = t_1;
	} else if (x <= 0.86) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 9.5e+153) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 2.05d-234) then
        tmp = t_0
    else if (x <= 5.8d-200) then
        tmp = t_1
    else if (x <= 1.15d-150) then
        tmp = t_0
    else if (x <= 6d-123) then
        tmp = t_1
    else if (x <= 0.86d0) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 9.5d+153) then
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 + ((-0.25d0) / x)) / x)) / x)) / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.05e-234) {
		tmp = t_0;
	} else if (x <= 5.8e-200) {
		tmp = t_1;
	} else if (x <= 1.15e-150) {
		tmp = t_0;
	} else if (x <= 6e-123) {
		tmp = t_1;
	} else if (x <= 0.86) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 9.5e+153) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 2.05e-234:
		tmp = t_0
	elif x <= 5.8e-200:
		tmp = t_1
	elif x <= 1.15e-150:
		tmp = t_0
	elif x <= 6e-123:
		tmp = t_1
	elif x <= 0.86:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 9.5e+153:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 2.05e-234)
		tmp = t_0;
	elseif (x <= 5.8e-200)
		tmp = t_1;
	elseif (x <= 1.15e-150)
		tmp = t_0;
	elseif (x <= 6e-123)
		tmp = t_1;
	elseif (x <= 0.86)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 9.5e+153)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x)) / x)) / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 2.05e-234)
		tmp = t_0;
	elseif (x <= 5.8e-200)
		tmp = t_1;
	elseif (x <= 1.15e-150)
		tmp = t_0;
	elseif (x <= 6e-123)
		tmp = t_1;
	elseif (x <= 0.86)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 9.5e+153)
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.05e-234], t$95$0, If[LessEqual[x, 5.8e-200], t$95$1, If[LessEqual[x, 1.15e-150], t$95$0, If[LessEqual[x, 6e-123], t$95$1, If[LessEqual[x, 0.86], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+153], 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], N[(0.0 / n), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.05000000000000005e-234 or 5.8e-200 < x < 1.15000000000000001e-150

   1. Initial program 33.6%

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

   3. Taylor expanded in n around inf 65.4%

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

      1. log1p-define 65.4%

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

   5. Simplified 65.4%

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

   6. Taylor expanded in x around 0 65.4%

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

      1. neg-mul-1 65.4%

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

   8. Simplified 65.4%

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

   if 2.05000000000000005e-234 < x < 5.8e-200 or 1.15000000000000001e-150 < x < 5.99999999999999968e-123

   1. Initial program 69.7%

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

   3. Taylor expanded in x around 0 69.7%

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

      1. *-rgt-identity 69.7%

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

      2. associate-*l/ 69.7%

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

      3. associate-/l* 69.7%

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

      4. exp-to-pow 69.7%

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

   5. Simplified 69.7%

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

   if 5.99999999999999968e-123 < x < 0.859999999999999987

   1. Initial program 34.0%

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

   3. Taylor expanded in n around inf 57.9%

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

      1. log1p-define 57.8%

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

   5. Simplified 57.8%

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

   6. Taylor expanded in x around 0 56.3%

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

      1. neg-mul-1 56.3%

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

      2. distribute-neg-frac 56.3%

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

      3. log-rec 56.3%

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

      4. +-commutative 56.3%

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

      5. log-rec 56.3%

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

      6. distribute-neg-frac 56.3%

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

      7. unsub-neg 56.3%

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

   8. Simplified 56.3%

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

   if 0.859999999999999987 < x < 9.4999999999999995e153

   1. Initial program 52.7%

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

   3. Taylor expanded in n around inf 50.5%

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

      1. log1p-define 50.5%

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

   5. Simplified 50.5%

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

      1. log1p-undefine 50.5%

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

      2. diff-log 50.8%

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

   7. Applied egg-rr 50.8%

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

      1. +-commutative 50.8%

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

   9. Simplified 50.8%

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

       1. clear-num 50.8%

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

       2. log-div 50.9%

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

       3. metadata-eval 50.9%

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

   11. Applied egg-rr 50.9%

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

       1. neg-sub0 50.9%

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

   13. Simplified 50.9%

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

   14. Taylor expanded in x around -inf 76.4%

       \[\leadsto \frac{-\color{blue}{-1 \cdot \frac{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x}}{x}}{x}}}{n} \]
   15. Step-by-step derivation

       1. mul-1-neg 76.4%

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

       2. distribute-neg-frac2 76.4%

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

   16. Simplified 76.4%

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

   if 9.4999999999999995e153 < x

   1. Initial program 89.0%

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

   3. Taylor expanded in n around inf 89.0%

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

      1. log1p-define 89.0%

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

   5. Simplified 89.0%

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

      1. log1p-undefine 89.0%

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

      2. diff-log 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. log-div 89.0%

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

       3. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. neg-sub0 89.0%

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

   13. Simplified 89.0%

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

   14. Taylor expanded in x around inf 89.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-234}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-200}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-150}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-123}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;n \leq -8.95 \cdot 10^{+73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;n \leq -1.65 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 175:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= n -8.95e+73)
     t_0
     (if (<= n -1.35e+44)
       (/ 1.0 (* x n))
       (if (<= n -1.65e-37)
         t_0
         (if (<= n 3.05e-152)
           (/ 0.3333333333333333 (* n (pow x 3.0)))
           (if (<= n 175.0) (+ 1.0 (pow x (/ 1.0 n))) t_0)))))))

C

double code(double x, double n) {
	double t_0 = log(((x + 1.0) / x)) / n;
	double tmp;
	if (n <= -8.95e+73) {
		tmp = t_0;
	} else if (n <= -1.35e+44) {
		tmp = 1.0 / (x * n);
	} else if (n <= -1.65e-37) {
		tmp = t_0;
	} else if (n <= 3.05e-152) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 175.0) {
		tmp = 1.0 + pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(((x + 1.0d0) / x)) / n
    if (n <= (-8.95d+73)) then
        tmp = t_0
    else if (n <= (-1.35d+44)) then
        tmp = 1.0d0 / (x * n)
    else if (n <= (-1.65d-37)) then
        tmp = t_0
    else if (n <= 3.05d-152) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 175.0d0) then
        tmp = 1.0d0 + (x ** (1.0d0 / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if (n <= -8.95e+73) {
		tmp = t_0;
	} else if (n <= -1.35e+44) {
		tmp = 1.0 / (x * n);
	} else if (n <= -1.65e-37) {
		tmp = t_0;
	} else if (n <= 3.05e-152) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 175.0) {
		tmp = 1.0 + Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if n <= -8.95e+73:
		tmp = t_0
	elif n <= -1.35e+44:
		tmp = 1.0 / (x * n)
	elif n <= -1.65e-37:
		tmp = t_0
	elif n <= 3.05e-152:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 175.0:
		tmp = 1.0 + math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (n <= -8.95e+73)
		tmp = t_0;
	elseif (n <= -1.35e+44)
		tmp = Float64(1.0 / Float64(x * n));
	elseif (n <= -1.65e-37)
		tmp = t_0;
	elseif (n <= 3.05e-152)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 175.0)
		tmp = Float64(1.0 + (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if (n <= -8.95e+73)
		tmp = t_0;
	elseif (n <= -1.35e+44)
		tmp = 1.0 / (x * n);
	elseif (n <= -1.65e-37)
		tmp = t_0;
	elseif (n <= 3.05e-152)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 175.0)
		tmp = 1.0 + (x ^ (1.0 / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -8.95e+73], t$95$0, If[LessEqual[n, -1.35e+44], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.65e-37], t$95$0, If[LessEqual[n, 3.05e-152], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 175.0], N[(1.0 + N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -8.9500000000000001e73 or -1.35e44 < n < -1.64999999999999991e-37 or 175 < n

   1. Initial program 41.2%

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

   3. Taylor expanded in n around inf 85.1%

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

      1. log1p-define 85.1%

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

   5. Simplified 85.1%

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

      1. log1p-undefine 85.1%

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

      2. diff-log 85.2%

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

   7. Applied egg-rr 85.2%

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

      1. +-commutative 85.2%

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

   9. Simplified 85.2%

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

   if -8.9500000000000001e73 < n < -1.35e44

   1. Initial program 6.9%

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

   3. Taylor expanded in n around inf 18.9%

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

      1. log1p-define 18.9%

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

   5. Simplified 18.9%

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

   6. Taylor expanded in x around inf 87.9%

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

   if -1.64999999999999991e-37 < n < 3.04999999999999991e-152

   1. Initial program 81.3%

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

   3. Taylor expanded in n around inf 32.7%

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

      1. log1p-define 32.7%

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

   5. Simplified 32.7%

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

   6. Taylor expanded in x around -inf 56.2%

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

   7. Taylor expanded in x around 0 79.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. *-commutative 79.7%

         \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{3} \cdot n}} \]

   9. Simplified 79.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{3} \cdot n}} \]

   if 3.04999999999999991e-152 < n < 175

   1. Initial program 81.9%

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

   3. Taylor expanded in x around 0 77.5%

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

      1. *-rgt-identity 77.5%

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

      2. associate-*l/ 77.5%

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

      3. associate-/l* 77.5%

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

      4. exp-to-pow 77.5%

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

   5. Simplified 77.5%

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

      1. add-exp-log 77.5%

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

      2. sub-neg 77.5%

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

      3. log1p-define 77.5%

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

   7. Applied egg-rr 77.5%

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

      1. log1p-undefine 77.5%

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

      2. rem-exp-log 77.5%

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

      3. add-sqr-sqrt 72.9%

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

      4. sqrt-unprod 91.1%

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

      5. sqr-neg 91.1%

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

      6. sqrt-unprod 91.1%

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

      7. add-sqr-sqrt 91.1%

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

   9. Applied egg-rr 91.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.95 \cdot 10^{+73}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;n \leq -1.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 3.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 175:\\ \;\;\;\;1 + {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2.3 \cdot 10^{-234}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-200}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 10^{+154}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 2.3e-234)
     t_0
     (if (<= x 5.8e-200)
       t_1
       (if (<= x 1.12e-150)
         t_0
         (if (<= x 5.7e-123)
           t_1
           (if (<= x 0.9)
             (/ (- x (log x)) n)
             (if (<= x 1e+154)
               (/
                (/
                 (- 1.0 (/ (- 0.5 (/ (+ 0.3333333333333333 (/ -0.25 x)) x)) x))
                 x)
                n)
               (/ 0.0 n)))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.3e-234) {
		tmp = t_0;
	} else if (x <= 5.8e-200) {
		tmp = t_1;
	} else if (x <= 1.12e-150) {
		tmp = t_0;
	} else if (x <= 5.7e-123) {
		tmp = t_1;
	} else if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1e+154) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 2.3d-234) then
        tmp = t_0
    else if (x <= 5.8d-200) then
        tmp = t_1
    else if (x <= 1.12d-150) then
        tmp = t_0
    else if (x <= 5.7d-123) then
        tmp = t_1
    else if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1d+154) then
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 + ((-0.25d0) / x)) / x)) / x)) / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.3e-234) {
		tmp = t_0;
	} else if (x <= 5.8e-200) {
		tmp = t_1;
	} else if (x <= 1.12e-150) {
		tmp = t_0;
	} else if (x <= 5.7e-123) {
		tmp = t_1;
	} else if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1e+154) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 2.3e-234:
		tmp = t_0
	elif x <= 5.8e-200:
		tmp = t_1
	elif x <= 1.12e-150:
		tmp = t_0
	elif x <= 5.7e-123:
		tmp = t_1
	elif x <= 0.9:
		tmp = (x - math.log(x)) / n
	elif x <= 1e+154:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 2.3e-234)
		tmp = t_0;
	elseif (x <= 5.8e-200)
		tmp = t_1;
	elseif (x <= 1.12e-150)
		tmp = t_0;
	elseif (x <= 5.7e-123)
		tmp = t_1;
	elseif (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1e+154)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x)) / x)) / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 2.3e-234)
		tmp = t_0;
	elseif (x <= 5.8e-200)
		tmp = t_1;
	elseif (x <= 1.12e-150)
		tmp = t_0;
	elseif (x <= 5.7e-123)
		tmp = t_1;
	elseif (x <= 0.9)
		tmp = (x - log(x)) / n;
	elseif (x <= 1e+154)
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.3e-234], t$95$0, If[LessEqual[x, 5.8e-200], t$95$1, If[LessEqual[x, 1.12e-150], t$95$0, If[LessEqual[x, 5.7e-123], t$95$1, If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1e+154], 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], N[(0.0 / n), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.2999999999999999e-234 or 5.8e-200 < x < 1.12e-150

   1. Initial program 33.6%

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

   3. Taylor expanded in n around inf 65.4%

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

      1. log1p-define 65.4%

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

   5. Simplified 65.4%

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

   6. Taylor expanded in x around 0 65.4%

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

      1. neg-mul-1 65.4%

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

   8. Simplified 65.4%

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

   if 2.2999999999999999e-234 < x < 5.8e-200 or 1.12e-150 < x < 5.70000000000000027e-123

   1. Initial program 69.7%

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

   3. Taylor expanded in x around 0 69.7%

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

      1. *-rgt-identity 69.7%

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

      2. associate-*l/ 69.7%

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

      3. associate-/l* 69.7%

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

      4. exp-to-pow 69.7%

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

   5. Simplified 69.7%

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

   if 5.70000000000000027e-123 < x < 0.900000000000000022

   1. Initial program 34.0%

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

   3. Taylor expanded in n around inf 57.9%

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

      1. log1p-define 57.8%

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

   5. Simplified 57.8%

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

   6. Taylor expanded in x around 0 56.3%

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

   if 0.900000000000000022 < x < 1.00000000000000004e154

   1. Initial program 52.7%

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

   3. Taylor expanded in n around inf 50.5%

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

      1. log1p-define 50.5%

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

   5. Simplified 50.5%

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

      1. log1p-undefine 50.5%

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

      2. diff-log 50.8%

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

   7. Applied egg-rr 50.8%

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

      1. +-commutative 50.8%

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

   9. Simplified 50.8%

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

       1. clear-num 50.8%

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

       2. log-div 50.9%

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

       3. metadata-eval 50.9%

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

   11. Applied egg-rr 50.9%

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

       1. neg-sub0 50.9%

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

   13. Simplified 50.9%

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

   14. Taylor expanded in x around -inf 76.4%

       \[\leadsto \frac{-\color{blue}{-1 \cdot \frac{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x}}{x}}{x}}}{n} \]
   15. Step-by-step derivation

       1. mul-1-neg 76.4%

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

       2. distribute-neg-frac2 76.4%

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

   16. Simplified 76.4%

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

   if 1.00000000000000004e154 < x

   1. Initial program 89.0%

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

   3. Taylor expanded in n around inf 89.0%

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

      1. log1p-define 89.0%

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

   5. Simplified 89.0%

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

      1. log1p-undefine 89.0%

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

      2. diff-log 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. log-div 89.0%

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

       3. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. neg-sub0 89.0%

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

   13. Simplified 89.0%

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

   14. Taylor expanded in x around inf 89.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-234}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-200}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-150}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-123}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 10^{+154}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-234}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.8e-234)
   (/ (log x) (- n))
   (if (<= x 3.5e-196)
     (/ 1.0 (* x n))
     (if (<= x 0.9)
       (/ (- x (log x)) n)
       (if (<= x 2.1e+154)
         (/
          (/ (- 1.0 (/ (- 0.5 (/ (+ 0.3333333333333333 (/ -0.25 x)) x)) x)) x)
          n)
         (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.8e-234) {
		tmp = log(x) / -n;
	} else if (x <= 3.5e-196) {
		tmp = 1.0 / (x * n);
	} else if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.1e+154) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.8d-234) then
        tmp = log(x) / -n
    else if (x <= 3.5d-196) then
        tmp = 1.0d0 / (x * n)
    else if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.1d+154) then
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 + ((-0.25d0) / x)) / x)) / x)) / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.8e-234) {
		tmp = Math.log(x) / -n;
	} else if (x <= 3.5e-196) {
		tmp = 1.0 / (x * n);
	} else if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.1e+154) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.8e-234:
		tmp = math.log(x) / -n
	elif x <= 3.5e-196:
		tmp = 1.0 / (x * n)
	elif x <= 0.9:
		tmp = (x - math.log(x)) / n
	elif x <= 2.1e+154:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.8e-234)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 3.5e-196)
		tmp = Float64(1.0 / Float64(x * n));
	elseif (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.1e+154)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x)) / x)) / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.8e-234)
		tmp = log(x) / -n;
	elseif (x <= 3.5e-196)
		tmp = 1.0 / (x * n);
	elseif (x <= 0.9)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.1e+154)
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.8e-234], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 3.5e-196], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.1e+154], 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], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.7999999999999999e-234

   1. Initial program 37.1%

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

   3. Taylor expanded in n around inf 65.1%

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

      1. log1p-define 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in x around 0 65.1%

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

      1. neg-mul-1 65.1%

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

   8. Simplified 65.1%

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

   if 1.7999999999999999e-234 < x < 3.50000000000000004e-196

   1. Initial program 56.1%

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

   3. Taylor expanded in n around inf 33.3%

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

      1. log1p-define 33.3%

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

   5. Simplified 33.3%

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

   6. Taylor expanded in x around inf 54.2%

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

   if 3.50000000000000004e-196 < x < 0.900000000000000022

   1. Initial program 39.7%

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

   3. Taylor expanded in n around inf 56.5%

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

      1. log1p-define 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in x around 0 55.4%

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

   if 0.900000000000000022 < x < 2.09999999999999994e154

   1. Initial program 52.7%

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

   3. Taylor expanded in n around inf 50.5%

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

      1. log1p-define 50.5%

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

   5. Simplified 50.5%

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

      1. log1p-undefine 50.5%

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

      2. diff-log 50.8%

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

   7. Applied egg-rr 50.8%

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

      1. +-commutative 50.8%

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

   9. Simplified 50.8%

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

       1. clear-num 50.8%

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

       2. log-div 50.9%

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

       3. metadata-eval 50.9%

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

   11. Applied egg-rr 50.9%

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

       1. neg-sub0 50.9%

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

   13. Simplified 50.9%

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

   14. Taylor expanded in x around -inf 76.4%

       \[\leadsto \frac{-\color{blue}{-1 \cdot \frac{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x}}{x}}{x}}}{n} \]
   15. Step-by-step derivation

       1. mul-1-neg 76.4%

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

       2. distribute-neg-frac2 76.4%

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

   16. Simplified 76.4%

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

   if 2.09999999999999994e154 < x

   1. Initial program 89.0%

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

   3. Taylor expanded in n around inf 89.0%

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

      1. log1p-define 89.0%

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

   5. Simplified 89.0%

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

      1. log1p-undefine 89.0%

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

      2. diff-log 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. log-div 89.0%

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

       3. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. neg-sub0 89.0%

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

   13. Simplified 89.0%

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

   14. Taylor expanded in x around inf 89.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-234}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 1.8 \cdot 10^{-234}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-197}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 1.8e-234)
     t_0
     (if (<= x 8e-197)
       (/ 1.0 (* x n))
       (if (<= x 0.7)
         t_0
         (if (<= x 9e+153)
           (/
            (/
             (- 1.0 (/ (- 0.5 (/ (+ 0.3333333333333333 (/ -0.25 x)) x)) x))
             x)
            n)
           (/ 0.0 n)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 1.8e-234) {
		tmp = t_0;
	} else if (x <= 8e-197) {
		tmp = 1.0 / (x * n);
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 9e+153) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 1.8d-234) then
        tmp = t_0
    else if (x <= 8d-197) then
        tmp = 1.0d0 / (x * n)
    else if (x <= 0.7d0) then
        tmp = t_0
    else if (x <= 9d+153) then
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 + ((-0.25d0) / x)) / x)) / x)) / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 1.8e-234) {
		tmp = t_0;
	} else if (x <= 8e-197) {
		tmp = 1.0 / (x * n);
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 9e+153) {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 1.8e-234:
		tmp = t_0
	elif x <= 8e-197:
		tmp = 1.0 / (x * n)
	elif x <= 0.7:
		tmp = t_0
	elif x <= 9e+153:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 1.8e-234)
		tmp = t_0;
	elseif (x <= 8e-197)
		tmp = Float64(1.0 / Float64(x * n));
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 9e+153)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x)) / x)) / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 1.8e-234)
		tmp = t_0;
	elseif (x <= 8e-197)
		tmp = 1.0 / (x * n);
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 9e+153)
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 1.8e-234], t$95$0, If[LessEqual[x, 8e-197], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.7], t$95$0, If[LessEqual[x, 9e+153], 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], N[(0.0 / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.7999999999999999e-234 or 7.9999999999999999e-197 < x < 0.69999999999999996

   1. Initial program 39.0%

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

   3. Taylor expanded in n around inf 59.0%

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

      1. log1p-define 59.0%

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

   5. Simplified 59.0%

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

   6. Taylor expanded in x around 0 57.6%

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

      1. neg-mul-1 57.6%

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

   8. Simplified 57.6%

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

   if 1.7999999999999999e-234 < x < 7.9999999999999999e-197

   1. Initial program 56.1%

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

   3. Taylor expanded in n around inf 33.3%

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

      1. log1p-define 33.3%

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

   5. Simplified 33.3%

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

   6. Taylor expanded in x around inf 54.2%

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

   if 0.69999999999999996 < x < 9.0000000000000002e153

   1. Initial program 52.7%

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

   3. Taylor expanded in n around inf 50.5%

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

      1. log1p-define 50.5%

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

   5. Simplified 50.5%

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

      1. log1p-undefine 50.5%

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

      2. diff-log 50.8%

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

   7. Applied egg-rr 50.8%

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

      1. +-commutative 50.8%

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

   9. Simplified 50.8%

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

       1. clear-num 50.8%

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

       2. log-div 50.9%

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

       3. metadata-eval 50.9%

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

   11. Applied egg-rr 50.9%

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

       1. neg-sub0 50.9%

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

   13. Simplified 50.9%

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

   14. Taylor expanded in x around -inf 76.4%

       \[\leadsto \frac{-\color{blue}{-1 \cdot \frac{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{x}}{x}}{x}}{x}}}{n} \]
   15. Step-by-step derivation

       1. mul-1-neg 76.4%

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

       2. distribute-neg-frac2 76.4%

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

   16. Simplified 76.4%

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

   if 9.0000000000000002e153 < x

   1. Initial program 89.0%

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

   3. Taylor expanded in n around inf 89.0%

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

      1. log1p-define 89.0%

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

   5. Simplified 89.0%

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

      1. log1p-undefine 89.0%

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

      2. diff-log 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. log-div 89.0%

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

       3. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. neg-sub0 89.0%

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

   13. Simplified 89.0%

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

   14. Taylor expanded in x around inf 89.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-234}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-197}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.5% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.5e+154)
   (/ (+ (/ 1.0 n) (/ (/ (+ -0.5 (/ 0.3333333333333333 x)) n) x)) x)
   (/ 0.0 n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.5e+154) {
		tmp = ((1.0 / n) + (((-0.5 + (0.3333333333333333 / x)) / n) / x)) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.5d+154) then
        tmp = ((1.0d0 / n) + ((((-0.5d0) + (0.3333333333333333d0 / x)) / n) / x)) / x
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.5e+154) {
		tmp = ((1.0 / n) + (((-0.5 + (0.3333333333333333 / x)) / n) / x)) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.5e+154:
		tmp = ((1.0 / n) + (((-0.5 + (0.3333333333333333 / x)) / n) / x)) / x
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.5e+154)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / n) / x)) / x);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.5e+154)
		tmp = ((1.0 / n) + (((-0.5 + (0.3333333333333333 / x)) / n) / x)) / x;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.5e+154], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.50000000000000013e154

   1. Initial program 44.5%

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

   3. Taylor expanded in n around inf 54.0%

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

      1. log1p-define 54.0%

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

   5. Simplified 54.0%

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

      1. log1p-undefine 54.0%

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

      2. diff-log 54.1%

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

   7. Applied egg-rr 54.1%

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

      1. +-commutative 54.1%

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

   9. Simplified 54.1%

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

   10. Taylor expanded in x around inf 25.9%

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

   12. Simplified 41.6%

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

   13. Taylor expanded in x around inf 41.6%

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

       1. associate-*r/ 41.6%

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

       2. metadata-eval 41.6%

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

       3. associate-/l/ 41.6%

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

       4. metadata-eval 41.6%

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

       5. associate-*r/ 41.6%

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

       6. associate-*r/ 41.6%

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

       7. metadata-eval 41.6%

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

       8. div-sub 41.6%

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

       9. sub-neg 41.6%

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

       10. associate-*r/ 41.6%

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

       11. metadata-eval 41.6%

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

       12. metadata-eval 41.6%

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

   15. Simplified 41.6%

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

   if 1.50000000000000013e154 < x

   1. Initial program 89.0%

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

   3. Taylor expanded in n around inf 89.0%

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

      1. log1p-define 89.0%

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

   5. Simplified 89.0%

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

      1. log1p-undefine 89.0%

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

      2. diff-log 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. +-commutative 89.0%

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

   9. Simplified 89.0%

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

       1. clear-num 89.0%

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

       2. log-div 89.0%

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

       3. metadata-eval 89.0%

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

   11. Applied egg-rr 89.0%

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

       1. neg-sub0 89.0%

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

   13. Simplified 89.0%

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

   14. Taylor expanded in x around inf 89.0%

       \[\leadsto \frac{-\log \color{blue}{1}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.4% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in n around inf 61.9%

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

   1. log1p-define 61.9%

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

5. Simplified 61.9%

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

   1. log1p-undefine 61.9%

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

   2. diff-log 62.0%

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

7. Applied egg-rr 62.0%

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

   1. +-commutative 62.0%

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

9. Simplified 62.0%

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

10. Taylor expanded in x around inf 34.5%

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

12. Simplified 46.7%

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

13. Taylor expanded in x around inf 46.7%

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

    1. associate-*r/ 46.7%

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

    2. metadata-eval 46.7%

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

    3. associate-/l/ 46.7%

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

    4. metadata-eval 46.7%

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

    5. associate-*r/ 46.7%

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

    6. associate-*r/ 46.7%

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

    7. metadata-eval 46.7%

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

    8. div-sub 46.7%

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

    9. sub-neg 46.7%

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

    10. associate-*r/ 46.7%

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

    11. metadata-eval 46.7%

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

    12. metadata-eval 46.7%

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

15. Simplified 46.7%

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

16. Final simplification 46.7%

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

Alternative 14: 45.8% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in n around inf 61.9%

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

   1. log1p-define 61.9%

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

5. Simplified 61.9%

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

6. Taylor expanded in x around -inf 46.6%

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

7. Taylor expanded in n around 0 46.4%

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

9. Simplified 46.4%

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

10. Final simplification 46.4%

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

Alternative 15: 46.4% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in n around inf 61.9%

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

   1. log1p-define 61.9%

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

5. Simplified 61.9%

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

   1. log1p-undefine 61.9%

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

   2. diff-log 62.0%

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

7. Applied egg-rr 62.0%

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

   1. +-commutative 62.0%

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

9. Simplified 62.0%

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

10. Taylor expanded in x around inf 46.6%

    \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

12. Simplified 46.6%

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

13. Final simplification 46.6%

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

Alternative 16: 39.9% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in n around inf 61.9%

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

   1. log1p-define 61.9%

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

5. Simplified 61.9%

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

6. Taylor expanded in x around inf 44.1%

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

7. Final simplification 44.1%

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

Alternative 17: 40.5% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in n around inf 61.9%

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

   1. log1p-define 61.9%

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

5. Simplified 61.9%

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

6. Taylor expanded in x around inf 44.1%

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

   1. associate-/r* 44.3%

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

8. Simplified 44.3%

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

9. Final simplification 44.3%

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

Alternative 18: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0 32.4%

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

4. Taylor expanded in x around inf 4.7%

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

5. Final simplification 4.7%

   \[\leadsto \frac{x}{n} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))