2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 86.1%

Time: 25.3s

Alternatives: 13

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-22)
     (/ (/ (exp (/ (log x) n)) n) x)
     (if (<= (/ 1.0 n) 4e-52)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1000.0)
         (/ t_0 (* n x))
         (pow (sqrt (- (exp (/ (log1p x) n)) t_0)) 2.0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-22) {
		tmp = (exp((log(x) / n)) / n) / x;
	} else if ((1.0 / n) <= 4e-52) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = pow(sqrt((exp((log1p(x) / n)) - t_0)), 2.0);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-22) {
		tmp = (Math.exp((Math.log(x) / n)) / n) / x;
	} else if ((1.0 / n) <= 4e-52) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = Math.pow(Math.sqrt((Math.exp((Math.log1p(x) / n)) - t_0)), 2.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-22:
		tmp = (math.exp((math.log(x) / n)) / n) / x
	elif (1.0 / n) <= 4e-52:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1000.0:
		tmp = t_0 / (n * x)
	else:
		tmp = math.pow(math.sqrt((math.exp((math.log1p(x) / n)) - t_0)), 2.0)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-22)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	elseif (Float64(1.0 / n) <= 4e-52)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1000.0)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = sqrt(Float64(exp(Float64(log1p(x) / n)) - t_0)) ^ 2.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-22], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-52], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2.0000000000000001e-22

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf 97.1%

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

      1. log-rec 97.1%

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

      2. mul-1-neg 97.1%

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

      3. associate-*r/ 97.1%

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

      4. neg-mul-1 97.1%

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

      5. mul-1-neg 97.1%

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

      6. remove-double-neg 97.1%

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

      7. *-commutative 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0 97.1%

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

      1. associate-/r* 97.2%

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

      2. *-lft-identity 97.2%

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

      3. associate-*r/ 97.2%

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

      4. associate-*l/ 97.2%

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

      5. *-commutative 97.2%

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

      6. exp-to-pow 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in x around 0 97.2%

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

   if -2.0000000000000001e-22 < (/.f64 1 n) < 4e-52

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

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

   if 4e-52 < (/.f64 1 n) < 1e3

   1. Initial program 5.0%

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

   3. Taylor expanded in x around inf 82.7%

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

      1. log-rec 82.7%

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

      2. mul-1-neg 82.7%

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

      3. associate-*r/ 82.7%

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

      4. neg-mul-1 82.7%

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

      5. mul-1-neg 82.7%

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

      6. remove-double-neg 82.7%

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

      7. *-commutative 82.7%

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

   5. Simplified 82.7%

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

   6. Taylor expanded in x around 0 82.7%

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

      1. associate-/r* 82.4%

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

      2. *-lft-identity 82.4%

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

      3. associate-*r/ 82.4%

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

      4. associate-*l/ 82.4%

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

      5. *-commutative 82.4%

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

      6. exp-to-pow 82.4%

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

   8. Simplified 82.4%

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

      1. add0 82.4%

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

      2. associate-/l/ 82.7%

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

   10. Applied egg-rr 82.7%

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

       1. add0 82.7%

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

   12. Simplified 82.7%

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

   if 1e3 < (/.f64 1 n)

   1. Initial program 67.2%

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

      1. add-sqr-sqrt 67.2%

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

      2. pow2 67.2%

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

      3. pow-to-exp 67.2%

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

      4. un-div-inv 67.2%

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

      5. +-commutative 67.2%

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

      6. log1p-define 99.8%

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{{\left(\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ t_1 := \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ t_2 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.7 \cdot 10^{-142}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-134}:\\ \;\;\;\;1 - t\_2\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-74}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_2\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_2}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n))
        (t_1 (log1p (expm1 (/ 1.0 (* n x)))))
        (t_2 (pow x (/ 1.0 n))))
   (if (<= x 1.7e-142)
     t_0
     (if (<= x 8.4e-134)
       (- 1.0 t_2)
       (if (<= x 1.55e-116)
         t_0
         (if (<= x 7.2e-96)
           t_1
           (if (<= x 4.1e-84)
             t_0
             (if (<= x 1.75e-74)
               (- (+ 1.0 (/ x n)) t_2)
               (if (<= x 2.85e-68)
                 t_1
                 (if (<= x 2.55e-11)
                   (/ (- x (log x)) n)
                   (/ (/ t_2 n) x)))))))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double t_1 = log1p(expm1((1.0 / (n * x))));
	double t_2 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.7e-142) {
		tmp = t_0;
	} else if (x <= 8.4e-134) {
		tmp = 1.0 - t_2;
	} else if (x <= 1.55e-116) {
		tmp = t_0;
	} else if (x <= 7.2e-96) {
		tmp = t_1;
	} else if (x <= 4.1e-84) {
		tmp = t_0;
	} else if (x <= 1.75e-74) {
		tmp = (1.0 + (x / n)) - t_2;
	} else if (x <= 2.85e-68) {
		tmp = t_1;
	} else if (x <= 2.55e-11) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (t_2 / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double t_1 = Math.log1p(Math.expm1((1.0 / (n * x))));
	double t_2 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.7e-142) {
		tmp = t_0;
	} else if (x <= 8.4e-134) {
		tmp = 1.0 - t_2;
	} else if (x <= 1.55e-116) {
		tmp = t_0;
	} else if (x <= 7.2e-96) {
		tmp = t_1;
	} else if (x <= 4.1e-84) {
		tmp = t_0;
	} else if (x <= 1.75e-74) {
		tmp = (1.0 + (x / n)) - t_2;
	} else if (x <= 2.85e-68) {
		tmp = t_1;
	} else if (x <= 2.55e-11) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (t_2 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	t_1 = math.log1p(math.expm1((1.0 / (n * x))))
	t_2 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.7e-142:
		tmp = t_0
	elif x <= 8.4e-134:
		tmp = 1.0 - t_2
	elif x <= 1.55e-116:
		tmp = t_0
	elif x <= 7.2e-96:
		tmp = t_1
	elif x <= 4.1e-84:
		tmp = t_0
	elif x <= 1.75e-74:
		tmp = (1.0 + (x / n)) - t_2
	elif x <= 2.85e-68:
		tmp = t_1
	elif x <= 2.55e-11:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (t_2 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	t_1 = log1p(expm1(Float64(1.0 / Float64(n * x))))
	t_2 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.7e-142)
		tmp = t_0;
	elseif (x <= 8.4e-134)
		tmp = Float64(1.0 - t_2);
	elseif (x <= 1.55e-116)
		tmp = t_0;
	elseif (x <= 7.2e-96)
		tmp = t_1;
	elseif (x <= 4.1e-84)
		tmp = t_0;
	elseif (x <= 1.75e-74)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_2);
	elseif (x <= 2.85e-68)
		tmp = t_1;
	elseif (x <= 2.55e-11)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(t_2 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, Block[{t$95$1 = N[Log[1 + N[(Exp[N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.7e-142], t$95$0, If[LessEqual[x, 8.4e-134], N[(1.0 - t$95$2), $MachinePrecision], If[LessEqual[x, 1.55e-116], t$95$0, If[LessEqual[x, 7.2e-96], t$95$1, If[LessEqual[x, 4.1e-84], t$95$0, If[LessEqual[x, 1.75e-74], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[x, 2.85e-68], t$95$1, If[LessEqual[x, 2.55e-11], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(t$95$2 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 1.70000000000000014e-142 or 8.3999999999999996e-134 < x < 1.55000000000000009e-116 or 7.20000000000000016e-96 < x < 4.10000000000000005e-84

   1. Initial program 34.4%

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

   3. Taylor expanded in x around 0 34.4%

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

   4. Taylor expanded in n around inf 64.5%

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

      1. associate-*r/ 64.5%

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

      2. mul-1-neg 64.5%

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

   6. Simplified 64.5%

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

   if 1.70000000000000014e-142 < x < 8.3999999999999996e-134

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 1.55000000000000009e-116 < x < 7.20000000000000016e-96 or 1.75000000000000007e-74 < x < 2.8500000000000001e-68

   1. Initial program 51.9%

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

   3. Taylor expanded in x around inf 51.4%

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

      1. log-rec 51.4%

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

      2. mul-1-neg 51.4%

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

      3. associate-*r/ 51.4%

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

      4. neg-mul-1 51.4%

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

      5. mul-1-neg 51.4%

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

      6. remove-double-neg 51.4%

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

      7. *-commutative 51.4%

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

   5. Simplified 51.4%

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

   6. Taylor expanded in n around inf 34.3%

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

      1. log1p-expm1-u 81.2%

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

   8. Applied egg-rr 81.2%

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

   if 4.10000000000000005e-84 < x < 1.75000000000000007e-74

   1. Initial program 81.7%

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

   3. Taylor expanded in x around 0 82.2%

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

   if 2.8500000000000001e-68 < x < 2.54999999999999992e-11

   1. Initial program 35.9%

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

   3. Taylor expanded in x around 0 32.2%

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

   4. Taylor expanded in n around inf 65.9%

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

   if 2.54999999999999992e-11 < x

   1. Initial program 67.1%

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

   3. Taylor expanded in x around inf 95.2%

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

      1. log-rec 95.2%

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

      2. mul-1-neg 95.2%

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

      3. associate-*r/ 95.2%

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

      4. neg-mul-1 95.2%

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

      5. mul-1-neg 95.2%

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

      6. remove-double-neg 95.2%

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

      7. *-commutative 95.2%

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

   5. Simplified 95.2%

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

   6. Taylor expanded in x around 0 95.2%

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

      1. associate-/r* 96.8%

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

      2. *-lft-identity 96.8%

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

      3. associate-*r/ 96.8%

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

      4. associate-*l/ 96.8%

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

      5. *-commutative 96.8%

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

      6. exp-to-pow 96.7%

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

   8. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-134}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-116}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-74}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-22)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 4e-52)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1000.0)
         (/ t_0 (* n x))
         (if (<= (/ 1.0 n) 2e+179)
           (- (+ 1.0 (/ x n)) t_0)
           (sqrt (pow (* n x) -2.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-22) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-52) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e+179) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-22) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-52) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e+179) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-22:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 4e-52:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1000.0:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e+179:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-22)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-52)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1000.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e+179)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-22], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-52], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+179], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2.0000000000000001e-22

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf 97.1%

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

      1. log-rec 97.1%

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

      2. mul-1-neg 97.1%

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

      3. associate-*r/ 97.1%

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

      4. neg-mul-1 97.1%

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

      5. mul-1-neg 97.1%

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

      6. remove-double-neg 97.1%

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

      7. *-commutative 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0 97.1%

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

      1. associate-/r* 97.2%

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

      2. *-lft-identity 97.2%

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

      3. associate-*r/ 97.2%

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

      4. associate-*l/ 97.2%

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

      5. *-commutative 97.2%

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

      6. exp-to-pow 97.1%

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

   8. Simplified 97.1%

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

   if -2.0000000000000001e-22 < (/.f64 1 n) < 4e-52

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

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

   if 4e-52 < (/.f64 1 n) < 1e3

   1. Initial program 5.0%

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

   3. Taylor expanded in x around inf 82.7%

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

      1. log-rec 82.7%

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

      2. mul-1-neg 82.7%

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

      3. associate-*r/ 82.7%

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

      4. neg-mul-1 82.7%

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

      5. mul-1-neg 82.7%

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

      6. remove-double-neg 82.7%

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

      7. *-commutative 82.7%

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

   5. Simplified 82.7%

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

   6. Taylor expanded in x around 0 82.7%

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

      1. associate-/r* 82.4%

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

      2. *-lft-identity 82.4%

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

      3. associate-*r/ 82.4%

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

      4. associate-*l/ 82.4%

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

      5. *-commutative 82.4%

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

      6. exp-to-pow 82.4%

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

   8. Simplified 82.4%

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

      1. add0 82.4%

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

      2. associate-/l/ 82.7%

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

   10. Applied egg-rr 82.7%

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

       1. add0 82.7%

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

   12. Simplified 82.7%

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

   if 1e3 < (/.f64 1 n) < 1.99999999999999996e179

   1. Initial program 95.9%

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

   3. Taylor expanded in x around 0 86.5%

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

   if 1.99999999999999996e179 < (/.f64 1 n)

   1. Initial program 15.2%

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

   3. Taylor expanded in x around inf 0.4%

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

      1. log-rec 0.4%

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

      2. mul-1-neg 0.4%

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

      3. associate-*r/ 0.4%

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

      4. neg-mul-1 0.4%

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

      5. mul-1-neg 0.4%

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

      6. remove-double-neg 0.4%

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

      7. *-commutative 0.4%

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

   5. Simplified 0.4%

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

   6. Taylor expanded in n around inf 57.2%

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

      1. add-sqr-sqrt 57.2%

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

      2. sqrt-unprod 82.5%

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

      3. inv-pow 82.5%

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

      4. inv-pow 82.5%

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

      5. pow-prod-up 82.5%

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

      6. metadata-eval 82.5%

         \[\leadsto \sqrt{{\left(x \cdot n\right)}^{\color{blue}{-2}}} \]

   8. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-22)
     (/ (exp (/ (log x) n)) (* n x))
     (if (<= (/ 1.0 n) 4e-52)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1000.0)
         (/ t_0 (* n x))
         (if (<= (/ 1.0 n) 2e+179)
           (- (+ 1.0 (/ x n)) t_0)
           (sqrt (pow (* n x) -2.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-22) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 4e-52) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e+179) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-22) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 4e-52) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e+179) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-22:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 4e-52:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1000.0:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e+179:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-22)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 4e-52)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1000.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e+179)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-22], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-52], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+179], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2.0000000000000001e-22

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf 97.1%

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

      1. log-rec 97.1%

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

      2. mul-1-neg 97.1%

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

      3. associate-*r/ 97.1%

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

      4. neg-mul-1 97.1%

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

      5. mul-1-neg 97.1%

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

      6. remove-double-neg 97.1%

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

      7. *-commutative 97.1%

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

   5. Simplified 97.1%

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

   if -2.0000000000000001e-22 < (/.f64 1 n) < 4e-52

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

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

   if 4e-52 < (/.f64 1 n) < 1e3

   1. Initial program 5.0%

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

   3. Taylor expanded in x around inf 82.7%

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

      1. log-rec 82.7%

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

      2. mul-1-neg 82.7%

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

      3. associate-*r/ 82.7%

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

      4. neg-mul-1 82.7%

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

      5. mul-1-neg 82.7%

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

      6. remove-double-neg 82.7%

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

      7. *-commutative 82.7%

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

   5. Simplified 82.7%

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

   6. Taylor expanded in x around 0 82.7%

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

      1. associate-/r* 82.4%

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

      2. *-lft-identity 82.4%

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

      3. associate-*r/ 82.4%

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

      4. associate-*l/ 82.4%

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

      5. *-commutative 82.4%

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

      6. exp-to-pow 82.4%

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

   8. Simplified 82.4%

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

      1. add0 82.4%

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

      2. associate-/l/ 82.7%

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

   10. Applied egg-rr 82.7%

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

       1. add0 82.7%

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

   12. Simplified 82.7%

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

   if 1e3 < (/.f64 1 n) < 1.99999999999999996e179

   1. Initial program 95.9%

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

   3. Taylor expanded in x around 0 86.5%

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

   if 1.99999999999999996e179 < (/.f64 1 n)

   1. Initial program 15.2%

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

   3. Taylor expanded in x around inf 0.4%

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

      1. log-rec 0.4%

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

      2. mul-1-neg 0.4%

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

      3. associate-*r/ 0.4%

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

      4. neg-mul-1 0.4%

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

      5. mul-1-neg 0.4%

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

      6. remove-double-neg 0.4%

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

      7. *-commutative 0.4%

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

   5. Simplified 0.4%

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

   6. Taylor expanded in n around inf 57.2%

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

      1. add-sqr-sqrt 57.2%

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

      2. sqrt-unprod 82.5%

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

      3. inv-pow 82.5%

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

      4. inv-pow 82.5%

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

      5. pow-prod-up 82.5%

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

      6. metadata-eval 82.5%

         \[\leadsto \sqrt{{\left(x \cdot n\right)}^{\color{blue}{-2}}} \]

   8. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-22)
     (/ (/ (exp (/ (log x) n)) n) x)
     (if (<= (/ 1.0 n) 4e-52)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1000.0)
         (/ t_0 (* n x))
         (if (<= (/ 1.0 n) 2e+179)
           (- (+ 1.0 (/ x n)) t_0)
           (sqrt (pow (* n x) -2.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-22) {
		tmp = (exp((log(x) / n)) / n) / x;
	} else if ((1.0 / n) <= 4e-52) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e+179) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-22) {
		tmp = (Math.exp((Math.log(x) / n)) / n) / x;
	} else if ((1.0 / n) <= 4e-52) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1000.0) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e+179) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-22:
		tmp = (math.exp((math.log(x) / n)) / n) / x
	elif (1.0 / n) <= 4e-52:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1000.0:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e+179:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-22)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	elseif (Float64(1.0 / n) <= 4e-52)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1000.0)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e+179)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-22], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-52], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+179], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2.0000000000000001e-22

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf 97.1%

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

      1. log-rec 97.1%

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

      2. mul-1-neg 97.1%

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

      3. associate-*r/ 97.1%

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

      4. neg-mul-1 97.1%

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

      5. mul-1-neg 97.1%

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

      6. remove-double-neg 97.1%

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

      7. *-commutative 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0 97.1%

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

      1. associate-/r* 97.2%

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

      2. *-lft-identity 97.2%

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

      3. associate-*r/ 97.2%

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

      4. associate-*l/ 97.2%

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

      5. *-commutative 97.2%

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

      6. exp-to-pow 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in x around 0 97.2%

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

   if -2.0000000000000001e-22 < (/.f64 1 n) < 4e-52

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

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

   if 4e-52 < (/.f64 1 n) < 1e3

   1. Initial program 5.0%

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

   3. Taylor expanded in x around inf 82.7%

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

      1. log-rec 82.7%

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

      2. mul-1-neg 82.7%

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

      3. associate-*r/ 82.7%

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

      4. neg-mul-1 82.7%

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

      5. mul-1-neg 82.7%

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

      6. remove-double-neg 82.7%

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

      7. *-commutative 82.7%

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

   5. Simplified 82.7%

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

   6. Taylor expanded in x around 0 82.7%

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

      1. associate-/r* 82.4%

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

      2. *-lft-identity 82.4%

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

      3. associate-*r/ 82.4%

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

      4. associate-*l/ 82.4%

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

      5. *-commutative 82.4%

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

      6. exp-to-pow 82.4%

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

   8. Simplified 82.4%

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

      1. add0 82.4%

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

      2. associate-/l/ 82.7%

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

   10. Applied egg-rr 82.7%

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

       1. add0 82.7%

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

   12. Simplified 82.7%

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

   if 1e3 < (/.f64 1 n) < 1.99999999999999996e179

   1. Initial program 95.9%

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

   3. Taylor expanded in x around 0 86.5%

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

   if 1.99999999999999996e179 < (/.f64 1 n)

   1. Initial program 15.2%

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

   3. Taylor expanded in x around inf 0.4%

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

      1. log-rec 0.4%

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

      2. mul-1-neg 0.4%

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

      3. associate-*r/ 0.4%

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

      4. neg-mul-1 0.4%

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

      5. mul-1-neg 0.4%

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

      6. remove-double-neg 0.4%

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

      7. *-commutative 0.4%

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

   5. Simplified 0.4%

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

   6. Taylor expanded in n around inf 57.2%

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

      1. add-sqr-sqrt 57.2%

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

      2. sqrt-unprod 82.5%

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

      3. inv-pow 82.5%

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

      4. inv-pow 82.5%

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

      5. pow-prod-up 82.5%

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

      6. metadata-eval 82.5%

         \[\leadsto \sqrt{{\left(x \cdot n\right)}^{\color{blue}{-2}}} \]

   8. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-22)
     (/ (/ (exp (/ (log x) n)) n) x)
     (if (<= (/ 1.0 n) 4e-52)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1000.0) (/ t_0 (* n x)) (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-22) {
		tmp = (exp((log(x) / n)) / n) / x;
	} else if ((1.0 / n) <= 4e-52) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-22) {
		tmp = (Math.exp((Math.log(x) / n)) / n) / x;
	} else if ((1.0 / n) <= 4e-52) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1000.0) {
		tmp = t_0 / (n * x);
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-22:
		tmp = (math.exp((math.log(x) / n)) / n) / x
	elif (1.0 / n) <= 4e-52:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1000.0:
		tmp = t_0 / (n * x)
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-22)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	elseif (Float64(1.0 / n) <= 4e-52)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1000.0)
		tmp = Float64(t_0 / Float64(n * x));
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-22], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-52], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000.0], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2.0000000000000001e-22

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf 97.1%

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

      1. log-rec 97.1%

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

      2. mul-1-neg 97.1%

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

      3. associate-*r/ 97.1%

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

      4. neg-mul-1 97.1%

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

      5. mul-1-neg 97.1%

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

      6. remove-double-neg 97.1%

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

      7. *-commutative 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in x around 0 97.1%

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

      1. associate-/r* 97.2%

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

      2. *-lft-identity 97.2%

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

      3. associate-*r/ 97.2%

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

      4. associate-*l/ 97.2%

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

      5. *-commutative 97.2%

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

      6. exp-to-pow 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in x around 0 97.2%

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

   if -2.0000000000000001e-22 < (/.f64 1 n) < 4e-52

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

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

   if 4e-52 < (/.f64 1 n) < 1e3

   1. Initial program 5.0%

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

   3. Taylor expanded in x around inf 82.7%

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

      1. log-rec 82.7%

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

      2. mul-1-neg 82.7%

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

      3. associate-*r/ 82.7%

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

      4. neg-mul-1 82.7%

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

      5. mul-1-neg 82.7%

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

      6. remove-double-neg 82.7%

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

      7. *-commutative 82.7%

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

   5. Simplified 82.7%

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

   6. Taylor expanded in x around 0 82.7%

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

      1. associate-/r* 82.4%

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

      2. *-lft-identity 82.4%

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

      3. associate-*r/ 82.4%

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

      4. associate-*l/ 82.4%

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

      5. *-commutative 82.4%

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

      6. exp-to-pow 82.4%

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

   8. Simplified 82.4%

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

      1. add0 82.4%

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

      2. associate-/l/ 82.7%

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

   10. Applied egg-rr 82.7%

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

       1. add0 82.7%

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

   12. Simplified 82.7%

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

   if 1e3 < (/.f64 1 n)

   1. Initial program 67.2%

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

   3. Taylor expanded in n around 0 67.2%

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

      1. log1p-define 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 99.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.7 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-134}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 1.7e-142)
     (/ (- (log x)) n)
     (if (<= x 7.5e-134)
       (- 1.0 t_0)
       (if (<= x 2.55e-11) (/ (- x (log x)) n) (/ (/ t_0 n) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.7e-142) {
		tmp = -log(x) / n;
	} else if (x <= 7.5e-134) {
		tmp = 1.0 - t_0;
	} else if (x <= 2.55e-11) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 1.7d-142) then
        tmp = -log(x) / n
    else if (x <= 7.5d-134) then
        tmp = 1.0d0 - t_0
    else if (x <= 2.55d-11) then
        tmp = (x - log(x)) / n
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.7e-142) {
		tmp = -Math.log(x) / n;
	} else if (x <= 7.5e-134) {
		tmp = 1.0 - t_0;
	} else if (x <= 2.55e-11) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.7e-142:
		tmp = -math.log(x) / n
	elif x <= 7.5e-134:
		tmp = 1.0 - t_0
	elif x <= 2.55e-11:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.7e-142)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 7.5e-134)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 2.55e-11)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 1.7e-142)
		tmp = -log(x) / n;
	elseif (x <= 7.5e-134)
		tmp = 1.0 - t_0;
	elseif (x <= 2.55e-11)
		tmp = (x - log(x)) / n;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.7e-142], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 7.5e-134], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 2.55e-11], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.70000000000000014e-142

   1. Initial program 37.4%

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

   3. Taylor expanded in x around 0 37.4%

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

   4. Taylor expanded in n around inf 60.7%

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

      1. associate-*r/ 60.7%

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

      2. mul-1-neg 60.7%

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

   6. Simplified 60.7%

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

   if 1.70000000000000014e-142 < x < 7.50000000000000048e-134

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 7.50000000000000048e-134 < x < 2.54999999999999992e-11

   1. Initial program 39.3%

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

   3. Taylor expanded in x around 0 37.5%

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

   4. Taylor expanded in n around inf 57.0%

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

   if 2.54999999999999992e-11 < x

   1. Initial program 67.1%

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

   3. Taylor expanded in x around inf 95.2%

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

      1. log-rec 95.2%

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

      2. mul-1-neg 95.2%

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

      3. associate-*r/ 95.2%

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

      4. neg-mul-1 95.2%

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

      5. mul-1-neg 95.2%

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

      6. remove-double-neg 95.2%

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

      7. *-commutative 95.2%

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

   5. Simplified 95.2%

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

   6. Taylor expanded in x around 0 95.2%

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

      1. associate-/r* 96.8%

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

      2. *-lft-identity 96.8%

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

      3. associate-*r/ 96.8%

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

      4. associate-*l/ 96.8%

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

      5. *-commutative 96.8%

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

      6. exp-to-pow 96.7%

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

   8. Simplified 96.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-134}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-133}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.000114:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.42 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.7e-142)
   (/ (- (log x)) n)
   (if (<= x 2.1e-133)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.000114)
       (/ (- x (log x)) n)
       (if (<= x 2.42e+123) (/ (/ 1.0 n) x) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.7e-142) {
		tmp = -log(x) / n;
	} else if (x <= 2.1e-133) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.000114) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.42e+123) {
		tmp = (1.0 / n) / 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 <= 1.7d-142) then
        tmp = -log(x) / n
    else if (x <= 2.1d-133) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.000114d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.42d+123) then
        tmp = (1.0d0 / n) / 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 <= 1.7e-142) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.1e-133) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.000114) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.42e+123) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.7e-142:
		tmp = -math.log(x) / n
	elif x <= 2.1e-133:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.000114:
		tmp = (x - math.log(x)) / n
	elif x <= 2.42e+123:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.7e-142)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.1e-133)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.000114)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.42e+123)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.7e-142)
		tmp = -log(x) / n;
	elseif (x <= 2.1e-133)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.000114)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.42e+123)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.7e-142], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.1e-133], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000114], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.42e+123], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.70000000000000014e-142

   1. Initial program 37.4%

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

   3. Taylor expanded in x around 0 37.4%

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

   4. Taylor expanded in n around inf 60.7%

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

      1. associate-*r/ 60.7%

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

      2. mul-1-neg 60.7%

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

   6. Simplified 60.7%

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

   if 1.70000000000000014e-142 < x < 2.1000000000000001e-133

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 2.1000000000000001e-133 < x < 1.1400000000000001e-4

   1. Initial program 42.0%

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

   3. Taylor expanded in x around 0 38.6%

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

   4. Taylor expanded in n around inf 54.7%

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

   if 1.1400000000000001e-4 < x < 2.4200000000000001e123

   1. Initial program 42.5%

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

   3. Taylor expanded in x around inf 94.1%

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

      1. log-rec 94.1%

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

      2. mul-1-neg 94.1%

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

      3. associate-*r/ 94.1%

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

      4. neg-mul-1 94.1%

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

      5. mul-1-neg 94.1%

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

      6. remove-double-neg 94.1%

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

      7. *-commutative 94.1%

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

   5. Simplified 94.1%

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

   6. Taylor expanded in x around 0 94.1%

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

      1. associate-/r* 95.2%

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

      2. *-lft-identity 95.2%

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

      3. associate-*r/ 95.2%

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

      4. associate-*l/ 95.2%

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

      5. *-commutative 95.2%

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

      6. exp-to-pow 95.2%

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

   8. Simplified 95.2%

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

   9. Taylor expanded in n around inf 69.7%

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

   if 2.4200000000000001e123 < x

   1. Initial program 81.7%

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

      1. flip3-- 50.9%

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

      2. div-inv 50.9%

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

      3. pow-to-exp 50.9%

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

      4. un-div-inv 50.9%

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

      5. +-commutative 50.9%

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

      6. log1p-define 50.9%

         \[\leadsto \left({\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

      7. pow-pow 50.8%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - \color{blue}{{x}^{\left(\frac{1}{n} \cdot 3\right)}}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

   4. Applied egg-rr 50.9%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 50.9%

         \[\leadsto \color{blue}{\frac{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot 1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      2. *-rgt-identity 50.9%

         \[\leadsto \frac{\color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. associate-*l/ 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      4. metadata-eval 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      5. +-commutative 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}}} \]

      6. +-commutative 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}} \]

      7. associate-+l+ 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{{\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

   6. Simplified 50.9%

      \[\leadsto \color{blue}{\frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{{\left(x + {x}^{2}\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

   7. Taylor expanded in n around -inf 81.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n}} \]
   8. Step-by-step derivation

      1. *-commutative 81.7%

         \[\leadsto \color{blue}{\frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n} \cdot -0.3333333333333333} \]

      2. cancel-sign-sub-inv 81.7%

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) + \left(--3\right) \cdot \log x}}{n} \cdot -0.3333333333333333 \]

      3. distribute-rgt-out 81.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) \cdot \left(-2 + -1\right)} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      4. log1p-define 81.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot \left(-2 + -1\right) + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      5. metadata-eval 81.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot \color{blue}{-3} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{3} \cdot \log x}{n} \cdot -0.3333333333333333 \]

      7. *-commutative 81.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{\log x \cdot 3}}{n} \cdot -0.3333333333333333 \]

   9. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot -3 + \log x \cdot 3}{n} \cdot -0.3333333333333333} \]

   10. Taylor expanded in x around inf 81.7%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)}{n}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.7%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)\right)}{n}} \]

       2. distribute-rgt-out 81.7%

          \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(\log \left(\frac{1}{x}\right) \cdot \left(-3 + 3\right)\right)}}{n} \]

       3. metadata-eval 81.7%

          \[\leadsto \frac{-0.3333333333333333 \cdot \left(\log \left(\frac{1}{x}\right) \cdot \color{blue}{0}\right)}{n} \]

       4. mul0-rgt 81.7%

          \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{0}}{n} \]

       5. metadata-eval 81.7%

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

   12. Simplified 81.7%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-133}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.000114:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.42 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000114:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.000114)
   (/ (- x (log x)) n)
   (if (<= x 2.95e+122) (/ (/ 1.0 n) x) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.000114) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.95e+122) {
		tmp = (1.0 / n) / 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 <= 0.000114d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.95d+122) then
        tmp = (1.0d0 / n) / 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 <= 0.000114) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.95e+122) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.000114:
		tmp = (x - math.log(x)) / n
	elif x <= 2.95e+122:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.000114)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.95e+122)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.000114)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.95e+122)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.000114], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.95e+122], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.1400000000000001e-4

   1. Initial program 42.2%

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

   3. Taylor expanded in x around 0 40.9%

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

   4. Taylor expanded in n around inf 55.7%

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

   if 1.1400000000000001e-4 < x < 2.95000000000000016e122

   1. Initial program 42.5%

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

   3. Taylor expanded in x around inf 94.1%

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

      1. log-rec 94.1%

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

      2. mul-1-neg 94.1%

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

      3. associate-*r/ 94.1%

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

      4. neg-mul-1 94.1%

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

      5. mul-1-neg 94.1%

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

      6. remove-double-neg 94.1%

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

      7. *-commutative 94.1%

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

   5. Simplified 94.1%

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

   6. Taylor expanded in x around 0 94.1%

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

      1. associate-/r* 95.2%

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

      2. *-lft-identity 95.2%

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

      3. associate-*r/ 95.2%

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

      4. associate-*l/ 95.2%

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

      5. *-commutative 95.2%

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

      6. exp-to-pow 95.2%

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

   8. Simplified 95.2%

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

   9. Taylor expanded in n around inf 69.7%

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

   if 2.95000000000000016e122 < x

   1. Initial program 81.7%

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

      1. flip3-- 50.9%

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

      2. div-inv 50.9%

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

      3. pow-to-exp 50.9%

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

      4. un-div-inv 50.9%

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

      5. +-commutative 50.9%

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

      6. log1p-define 50.9%

         \[\leadsto \left({\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

      7. pow-pow 50.8%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - \color{blue}{{x}^{\left(\frac{1}{n} \cdot 3\right)}}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

   4. Applied egg-rr 50.9%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 50.9%

         \[\leadsto \color{blue}{\frac{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot 1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      2. *-rgt-identity 50.9%

         \[\leadsto \frac{\color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. associate-*l/ 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      4. metadata-eval 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      5. +-commutative 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}}} \]

      6. +-commutative 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}} \]

      7. associate-+l+ 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{{\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

   6. Simplified 50.9%

      \[\leadsto \color{blue}{\frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{{\left(x + {x}^{2}\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

   7. Taylor expanded in n around -inf 81.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n}} \]
   8. Step-by-step derivation

      1. *-commutative 81.7%

         \[\leadsto \color{blue}{\frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n} \cdot -0.3333333333333333} \]

      2. cancel-sign-sub-inv 81.7%

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) + \left(--3\right) \cdot \log x}}{n} \cdot -0.3333333333333333 \]

      3. distribute-rgt-out 81.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) \cdot \left(-2 + -1\right)} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      4. log1p-define 81.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot \left(-2 + -1\right) + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      5. metadata-eval 81.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot \color{blue}{-3} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{3} \cdot \log x}{n} \cdot -0.3333333333333333 \]

      7. *-commutative 81.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{\log x \cdot 3}}{n} \cdot -0.3333333333333333 \]

   9. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot -3 + \log x \cdot 3}{n} \cdot -0.3333333333333333} \]

   10. Taylor expanded in x around inf 81.7%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)}{n}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.7%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)\right)}{n}} \]

       2. distribute-rgt-out 81.7%

          \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(\log \left(\frac{1}{x}\right) \cdot \left(-3 + 3\right)\right)}}{n} \]

       3. metadata-eval 81.7%

          \[\leadsto \frac{-0.3333333333333333 \cdot \left(\log \left(\frac{1}{x}\right) \cdot \color{blue}{0}\right)}{n} \]

       4. mul0-rgt 81.7%

          \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{0}}{n} \]

       5. metadata-eval 81.7%

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

   12. Simplified 81.7%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000114:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000114:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.000114)
   (/ (- (log x)) n)
   (if (<= x 1.55e+123) (/ (/ 1.0 n) x) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.000114) {
		tmp = -log(x) / n;
	} else if (x <= 1.55e+123) {
		tmp = (1.0 / n) / 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 <= 0.000114d0) then
        tmp = -log(x) / n
    else if (x <= 1.55d+123) then
        tmp = (1.0d0 / n) / 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 <= 0.000114) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.55e+123) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.000114:
		tmp = -math.log(x) / n
	elif x <= 1.55e+123:
		tmp = (1.0 / n) / x
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.000114)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.55e+123)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.000114)
		tmp = -log(x) / n;
	elseif (x <= 1.55e+123)
		tmp = (1.0 / n) / x;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.000114], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.55e+123], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.1400000000000001e-4

   1. Initial program 42.2%

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

   3. Taylor expanded in x around 0 40.7%

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

   4. Taylor expanded in n around inf 55.3%

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

      1. associate-*r/ 55.3%

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

      2. mul-1-neg 55.3%

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

   6. Simplified 55.3%

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

   if 1.1400000000000001e-4 < x < 1.55000000000000003e123

   1. Initial program 42.5%

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

   3. Taylor expanded in x around inf 94.1%

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

      1. log-rec 94.1%

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

      2. mul-1-neg 94.1%

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

      3. associate-*r/ 94.1%

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

      4. neg-mul-1 94.1%

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

      5. mul-1-neg 94.1%

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

      6. remove-double-neg 94.1%

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

      7. *-commutative 94.1%

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

   5. Simplified 94.1%

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

   6. Taylor expanded in x around 0 94.1%

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

      1. associate-/r* 95.2%

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

      2. *-lft-identity 95.2%

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

      3. associate-*r/ 95.2%

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

      4. associate-*l/ 95.2%

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

      5. *-commutative 95.2%

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

      6. exp-to-pow 95.2%

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

   8. Simplified 95.2%

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

   9. Taylor expanded in n around inf 69.7%

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

   if 1.55000000000000003e123 < x

   1. Initial program 81.7%

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

      1. flip3-- 50.9%

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

      2. div-inv 50.9%

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

      3. pow-to-exp 50.9%

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

      4. un-div-inv 50.9%

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

      5. +-commutative 50.9%

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

      6. log1p-define 50.9%

         \[\leadsto \left({\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

      7. pow-pow 50.8%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - \color{blue}{{x}^{\left(\frac{1}{n} \cdot 3\right)}}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

   4. Applied egg-rr 50.9%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 50.9%

         \[\leadsto \color{blue}{\frac{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot 1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      2. *-rgt-identity 50.9%

         \[\leadsto \frac{\color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. associate-*l/ 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      4. metadata-eval 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      5. +-commutative 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}}} \]

      6. +-commutative 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}} \]

      7. associate-+l+ 50.9%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{{\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

   6. Simplified 50.9%

      \[\leadsto \color{blue}{\frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{{\left(x + {x}^{2}\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

   7. Taylor expanded in n around -inf 81.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n}} \]
   8. Step-by-step derivation

      1. *-commutative 81.7%

         \[\leadsto \color{blue}{\frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n} \cdot -0.3333333333333333} \]

      2. cancel-sign-sub-inv 81.7%

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) + \left(--3\right) \cdot \log x}}{n} \cdot -0.3333333333333333 \]

      3. distribute-rgt-out 81.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) \cdot \left(-2 + -1\right)} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      4. log1p-define 81.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot \left(-2 + -1\right) + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      5. metadata-eval 81.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot \color{blue}{-3} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{3} \cdot \log x}{n} \cdot -0.3333333333333333 \]

      7. *-commutative 81.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{\log x \cdot 3}}{n} \cdot -0.3333333333333333 \]

   9. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot -3 + \log x \cdot 3}{n} \cdot -0.3333333333333333} \]

   10. Taylor expanded in x around inf 81.7%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)}{n}} \]
   11. Step-by-step derivation

       1. associate-*r/ 81.7%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)\right)}{n}} \]

       2. distribute-rgt-out 81.7%

          \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(\log \left(\frac{1}{x}\right) \cdot \left(-3 + 3\right)\right)}}{n} \]

       3. metadata-eval 81.7%

          \[\leadsto \frac{-0.3333333333333333 \cdot \left(\log \left(\frac{1}{x}\right) \cdot \color{blue}{0}\right)}{n} \]

       4. mul0-rgt 81.7%

          \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{0}}{n} \]

       5. metadata-eval 81.7%

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

   12. Simplified 81.7%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000114:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -22 \lor \neg \left(n \leq -3.7 \cdot 10^{-291}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -22.0) (not (<= n -3.7e-291))) (/ 1.0 (* n x)) (/ 0.0 n)))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -22.0) || !(n <= -3.7e-291)) {
		tmp = 1.0 / (n * 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 ((n <= (-22.0d0)) .or. (.not. (n <= (-3.7d-291)))) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -22.0) || !(n <= -3.7e-291)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -22.0) or not (n <= -3.7e-291):
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -22.0) || !(n <= -3.7e-291))
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -22.0) || ~((n <= -3.7e-291)))
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -22.0], N[Not[LessEqual[n, -3.7e-291]], $MachinePrecision]], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -22 or -3.7000000000000001e-291 < n

   1. Initial program 39.8%

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

   3. Taylor expanded in x around inf 48.1%

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

      1. log-rec 48.1%

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

      2. mul-1-neg 48.1%

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

      3. associate-*r/ 48.1%

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

      4. neg-mul-1 48.1%

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

      5. mul-1-neg 48.1%

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

      6. remove-double-neg 48.1%

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

      7. *-commutative 48.1%

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

   5. Simplified 48.1%

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

   6. Taylor expanded in n around inf 49.2%

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

   if -22 < n < -3.7000000000000001e-291

   1. Initial program 98.6%

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

      1. flip3-- 0.0%

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

      2. div-inv 0.0%

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

      3. pow-to-exp 0.0%

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

      4. un-div-inv 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-define 0.0%

         \[\leadsto \left({\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

      7. pow-pow 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - \color{blue}{{x}^{\left(\frac{1}{n} \cdot 3\right)}}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot 1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      2. *-rgt-identity 0.0%

         \[\leadsto \frac{\color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. associate-*l/ 0.0%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      4. metadata-eval 0.0%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      5. +-commutative 0.0%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}}} \]

      6. +-commutative 0.0%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}} \]

      7. associate-+l+ 0.0%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{{\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

   6. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{{\left(x + {x}^{2}\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

   7. Taylor expanded in n around -inf 51.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n}} \]
   8. Step-by-step derivation

      1. *-commutative 51.7%

         \[\leadsto \color{blue}{\frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n} \cdot -0.3333333333333333} \]

      2. cancel-sign-sub-inv 51.7%

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) + \left(--3\right) \cdot \log x}}{n} \cdot -0.3333333333333333 \]

      3. distribute-rgt-out 51.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) \cdot \left(-2 + -1\right)} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      4. log1p-define 51.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot \left(-2 + -1\right) + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      5. metadata-eval 51.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot \color{blue}{-3} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      6. metadata-eval 51.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{3} \cdot \log x}{n} \cdot -0.3333333333333333 \]

      7. *-commutative 51.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{\log x \cdot 3}}{n} \cdot -0.3333333333333333 \]

   9. Simplified 51.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot -3 + \log x \cdot 3}{n} \cdot -0.3333333333333333} \]

   10. Taylor expanded in x around inf 54.8%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)}{n}} \]
   11. Step-by-step derivation

       1. associate-*r/ 54.8%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)\right)}{n}} \]

       2. distribute-rgt-out 54.8%

          \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(\log \left(\frac{1}{x}\right) \cdot \left(-3 + 3\right)\right)}}{n} \]

       3. metadata-eval 54.8%

          \[\leadsto \frac{-0.3333333333333333 \cdot \left(\log \left(\frac{1}{x}\right) \cdot \color{blue}{0}\right)}{n} \]

       4. mul0-rgt 54.8%

          \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{0}}{n} \]

       5. metadata-eval 54.8%

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

   12. Simplified 54.8%

       \[\leadsto \color{blue}{\frac{0}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -22 \lor \neg \left(n \leq -3.7 \cdot 10^{-291}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2.0) (/ 0.0 n) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2.0d0)) then
        tmp = 0.0d0 / n
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2.0:
		tmp = 0.0 / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.0)
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2.0)
		tmp = 0.0 / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.0], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -2

   1. Initial program 98.6%

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

      1. flip3-- 0.0%

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

      2. div-inv 0.0%

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

      3. pow-to-exp 0.0%

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

      4. un-div-inv 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-define 0.0%

         \[\leadsto \left({\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

      7. pow-pow 0.0%

         \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - \color{blue}{{x}^{\left(\frac{1}{n} \cdot 3\right)}}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot 1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

      2. *-rgt-identity 0.0%

         \[\leadsto \frac{\color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      3. associate-*l/ 0.0%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      4. metadata-eval 0.0%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

      5. +-commutative 0.0%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}}} \]

      6. +-commutative 0.0%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}} \]

      7. associate-+l+ 0.0%

         \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{{\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

   6. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{{\left(x + {x}^{2}\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

   7. Taylor expanded in n around -inf 51.2%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n}} \]
   8. Step-by-step derivation

      1. *-commutative 51.2%

         \[\leadsto \color{blue}{\frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n} \cdot -0.3333333333333333} \]

      2. cancel-sign-sub-inv 51.2%

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) + \left(--3\right) \cdot \log x}}{n} \cdot -0.3333333333333333 \]

      3. distribute-rgt-out 51.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) \cdot \left(-2 + -1\right)} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      4. log1p-define 51.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot \left(-2 + -1\right) + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      5. metadata-eval 51.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot \color{blue}{-3} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

      6. metadata-eval 51.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{3} \cdot \log x}{n} \cdot -0.3333333333333333 \]

      7. *-commutative 51.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{\log x \cdot 3}}{n} \cdot -0.3333333333333333 \]

   9. Simplified 51.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot -3 + \log x \cdot 3}{n} \cdot -0.3333333333333333} \]

   10. Taylor expanded in x around inf 53.9%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)}{n}} \]
   11. Step-by-step derivation

       1. associate-*r/ 53.9%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)\right)}{n}} \]

       2. distribute-rgt-out 53.9%

          \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(\log \left(\frac{1}{x}\right) \cdot \left(-3 + 3\right)\right)}}{n} \]

       3. metadata-eval 53.9%

          \[\leadsto \frac{-0.3333333333333333 \cdot \left(\log \left(\frac{1}{x}\right) \cdot \color{blue}{0}\right)}{n} \]

       4. mul0-rgt 53.9%

          \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{0}}{n} \]

       5. metadata-eval 53.9%

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

   12. Simplified 53.9%

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

   if -2 < (/.f64 1 n)

   1. Initial program 38.9%

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

   3. Taylor expanded in x around inf 47.3%

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

      1. log-rec 47.3%

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

      2. mul-1-neg 47.3%

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

      3. associate-*r/ 47.3%

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

      4. neg-mul-1 47.3%

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

      5. mul-1-neg 47.3%

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

      6. remove-double-neg 47.3%

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

      7. *-commutative 47.3%

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

   5. Simplified 47.3%

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

   6. Taylor expanded in x around 0 47.3%

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

      1. associate-/r* 48.5%

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

      2. *-lft-identity 48.5%

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

      3. associate-*r/ 48.5%

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

      4. associate-*l/ 48.5%

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

      5. *-commutative 48.5%

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

      6. exp-to-pow 48.5%

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

   8. Simplified 48.5%

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

   9. Taylor expanded in n around inf 50.0%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.6% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ \frac{0}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 0.0 n))

C

double code(double x, double n) {
	return 0.0 / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0 / n
end function

Java

public static double code(double x, double n) {
	return 0.0 / n;
}

Python

def code(x, n):
	return 0.0 / n

Julia

function code(x, n)
	return Float64(0.0 / n)
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0 / n;
end

Wolfram

code[x_, n_] := N[(0.0 / n), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. flip3-- 27.7%

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

   2. div-inv 27.7%

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

   3. pow-to-exp 27.7%

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

   4. un-div-inv 27.7%

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

   5. +-commutative 27.7%

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

   6. log1p-define 30.7%

      \[\leadsto \left({\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

   7. pow-pow 30.7%

      \[\leadsto \left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - \color{blue}{{x}^{\left(\frac{1}{n} \cdot 3\right)}}\right) \cdot \frac{1}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)} \]

4. Applied egg-rr 28.2%

   \[\leadsto \color{blue}{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot \frac{1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]
5. Step-by-step derivation

   1. associate-*r/ 28.2%

      \[\leadsto \color{blue}{\frac{\left({\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}\right) \cdot 1}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)}} \]

   2. *-rgt-identity 28.2%

      \[\leadsto \frac{\color{blue}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{1}{n} \cdot 3\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

   3. associate-*l/ 28.2%

      \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\color{blue}{\left(\frac{1 \cdot 3}{n}\right)}}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

   4. metadata-eval 28.2%

      \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{\color{blue}{3}}{n}\right)}}{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right)} \]

   5. +-commutative 28.2%

      \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}}} \]

   6. +-commutative 28.2%

      \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{\left({\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + {\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}} \]

   7. associate-+l+ 28.2%

      \[\leadsto \frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{\color{blue}{{\left(\left(x + 1\right) \cdot x\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

6. Simplified 28.2%

   \[\leadsto \color{blue}{\frac{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{3} - {x}^{\left(\frac{3}{n}\right)}}{{\left(x + {x}^{2}\right)}^{\left(\frac{1}{n}\right)} + \left({\left({x}^{2}\right)}^{\left(\frac{1}{n}\right)} + {\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)}^{2}\right)}} \]

7. Taylor expanded in n around -inf 60.1%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n}} \]
8. Step-by-step derivation

   1. *-commutative 60.1%

      \[\leadsto \color{blue}{\frac{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) - -3 \cdot \log x}{n} \cdot -0.3333333333333333} \]

   2. cancel-sign-sub-inv 60.1%

      \[\leadsto \frac{\color{blue}{\left(-2 \cdot \log \left(1 + x\right) + -1 \cdot \log \left(1 + x\right)\right) + \left(--3\right) \cdot \log x}}{n} \cdot -0.3333333333333333 \]

   3. distribute-rgt-out 60.1%

      \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) \cdot \left(-2 + -1\right)} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

   4. log1p-define 60.1%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot \left(-2 + -1\right) + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

   5. metadata-eval 60.1%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot \color{blue}{-3} + \left(--3\right) \cdot \log x}{n} \cdot -0.3333333333333333 \]

   6. metadata-eval 60.1%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{3} \cdot \log x}{n} \cdot -0.3333333333333333 \]

   7. *-commutative 60.1%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) \cdot -3 + \color{blue}{\log x \cdot 3}}{n} \cdot -0.3333333333333333 \]

9. Simplified 60.1%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) \cdot -3 + \log x \cdot 3}{n} \cdot -0.3333333333333333} \]

10. Taylor expanded in x around inf 34.4%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)}{n}} \]
11. Step-by-step derivation

    1. associate-*r/ 34.4%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(-3 \cdot \log \left(\frac{1}{x}\right) + 3 \cdot \log \left(\frac{1}{x}\right)\right)}{n}} \]

    2. distribute-rgt-out 34.4%

       \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(\log \left(\frac{1}{x}\right) \cdot \left(-3 + 3\right)\right)}}{n} \]

    3. metadata-eval 34.4%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\log \left(\frac{1}{x}\right) \cdot \color{blue}{0}\right)}{n} \]

    4. mul0-rgt 34.4%

       \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{0}}{n} \]

    5. metadata-eval 34.4%

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

12. Simplified 34.4%

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

13. Final simplification 34.4%

    \[\leadsto \frac{0}{n} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))