2nthrt (problem 3.4.6)

Percentage Accurate: 53.7% → 82.9%

Time: 27.8s

Alternatives: 15

Speedup: 1.8×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% 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: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-198)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-12)
     (/ (- (log (/ x (+ 1.0 x)))) n)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-198) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = -log((x / (1.0 + x))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-198) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = -Math.log((x / (1.0 + x))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-198:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = -math.log((x / (1.0 + x))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-198)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-198], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -4.9999999999999999e-198

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf 87.8%

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

      1. mul-1-neg 87.8%

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

      2. log-rec 87.8%

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

      3. mul-1-neg 87.8%

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

      4. distribute-neg-frac 87.8%

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

      5. mul-1-neg 87.8%

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

      6. remove-double-neg 87.8%

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

      7. *-commutative 87.8%

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

   5. Simplified 87.8%

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

   if -4.9999999999999999e-198 < (/.f64 1 n) < 1.99999999999999996e-12

   1. Initial program 37.6%

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

   3. Taylor expanded in n around inf 88.1%

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

      1. log1p-def 88.1%

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

   5. Simplified 88.1%

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

      1. log1p-udef 88.1%

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

      2. diff-log 88.2%

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

      3. +-commutative 88.2%

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

   7. Applied egg-rr 88.2%

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

      1. clear-num 88.2%

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

      2. log-rec 88.2%

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

   9. Applied egg-rr 88.2%

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

   if 1.99999999999999996e-12 < (/.f64 1 n)

   1. Initial program 56.8%

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

   3. Taylor expanded in n around 0 56.8%

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

      1. log1p-def 97.2%

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

   5. Simplified 97.2%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+184}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (- (log (/ x (+ 1.0 x)))) n)))
   (if (<= (/ 1.0 n) -5e+184)
     (- 1.0 t_0)
     (if (<= (/ 1.0 n) -2e-20)
       t_1
       (if (<= (/ 1.0 n) -5e-198)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 0.0002)
           t_1
           (if (<= (/ 1.0 n) 1e+166)
             (- (+ 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 t_1 = -log((x / (1.0 + x))) / n;
	double tmp;
	if ((1.0 / n) <= -5e+184) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= -2e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-198) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 0.0002) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+166) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = -log((x / (1.0d0 + x))) / n
    if ((1.0d0 / n) <= (-5d+184)) then
        tmp = 1.0d0 - t_0
    else if ((1.0d0 / n) <= (-2d-20)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-198)) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 0.0002d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+166) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = sqrt(((n * x) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = -Math.log((x / (1.0 + x))) / n;
	double tmp;
	if ((1.0 / n) <= -5e+184) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= -2e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-198) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 0.0002) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+166) {
		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))
	t_1 = -math.log((x / (1.0 + x))) / n
	tmp = 0
	if (1.0 / n) <= -5e+184:
		tmp = 1.0 - t_0
	elif (1.0 / n) <= -2e-20:
		tmp = t_1
	elif (1.0 / n) <= -5e-198:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 0.0002:
		tmp = t_1
	elif (1.0 / n) <= 1e+166:
		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)
	t_1 = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+184)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= -2e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-198)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 0.0002)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+166)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = -log((x / (1.0 + x))) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e+184)
		tmp = 1.0 - t_0;
	elseif ((1.0 / n) <= -2e-20)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-198)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 0.0002)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+166)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = sqrt(((n * x) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+184], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-198], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0002], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+166], 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) < -4.9999999999999999e184

   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 58.0%

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

   if -4.9999999999999999e184 < (/.f64 1 n) < -1.99999999999999989e-20 or -4.9999999999999999e-198 < (/.f64 1 n) < 2.0000000000000001e-4

   1. Initial program 54.4%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. log1p-def 80.0%

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

   5. Simplified 80.0%

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

      1. log1p-udef 80.0%

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

      2. diff-log 80.1%

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

      3. +-commutative 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. clear-num 80.1%

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

      2. log-rec 80.1%

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

   9. Applied egg-rr 80.1%

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

   if -1.99999999999999989e-20 < (/.f64 1 n) < -4.9999999999999999e-198

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. log1p-def 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around inf 76.5%

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

      1. *-commutative 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in x around 0 76.5%

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

       1. associate-/r* 78.3%

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

   11. Simplified 78.3%

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

   if 2.0000000000000001e-4 < (/.f64 1 n) < 9.9999999999999994e165

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0 86.0%

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

   if 9.9999999999999994e165 < (/.f64 1 n)

   1. Initial program 12.1%

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

   3. Taylor expanded in n around inf 8.6%

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

      1. log1p-def 8.6%

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

   5. Simplified 8.6%

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

   6. Taylor expanded in x around inf 61.8%

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

      1. *-commutative 61.8%

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

   8. Simplified 61.8%

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

      1. add-sqr-sqrt 61.8%

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

      2. sqrt-unprod 91.9%

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

      3. inv-pow 91.9%

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

      4. inv-pow 91.9%

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

      5. pow-prod-up 91.9%

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

      6. metadata-eval 91.9%

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

   10. Applied egg-rr 91.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+184}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-20}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+184}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e+184)
     (- 1.0 t_0)
     (if (<= (/ 1.0 n) -2e-20)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) -5e-198)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 0.0002)
           (/ (- (log (/ x (+ 1.0 x)))) n)
           (if (<= (/ 1.0 n) 1e+166)
             (- (+ 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) <= -5e+184) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= -2e-20) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= -5e-198) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 0.0002) {
		tmp = -log((x / (1.0 + x))) / n;
	} else if ((1.0 / n) <= 1e+166) {
		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) <= -5e+184) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= -2e-20) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= -5e-198) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 0.0002) {
		tmp = -Math.log((x / (1.0 + x))) / n;
	} else if ((1.0 / n) <= 1e+166) {
		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) <= -5e+184:
		tmp = 1.0 - t_0
	elif (1.0 / n) <= -2e-20:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= -5e-198:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 0.0002:
		tmp = -math.log((x / (1.0 + x))) / n
	elif (1.0 / n) <= 1e+166:
		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) <= -5e+184)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= -2e-20)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= -5e-198)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 0.0002)
		tmp = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n);
	elseif (Float64(1.0 / n) <= 1e+166)
		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], -5e+184], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-20], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-198], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0002], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+166], 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 6 regimes

2. if (/.f64 1 n) < -4.9999999999999999e184

   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 58.0%

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

   if -4.9999999999999999e184 < (/.f64 1 n) < -1.99999999999999989e-20

   1. Initial program 94.5%

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

   3. Taylor expanded in n around inf 62.7%

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

      1. log1p-def 62.7%

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

   5. Simplified 62.7%

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

   if -1.99999999999999989e-20 < (/.f64 1 n) < -4.9999999999999999e-198

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. log1p-def 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around inf 76.5%

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

      1. *-commutative 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in x around 0 76.5%

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

       1. associate-/r* 78.3%

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

   11. Simplified 78.3%

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

   if -4.9999999999999999e-198 < (/.f64 1 n) < 2.0000000000000001e-4

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 87.4%

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

      1. log1p-def 87.4%

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

   5. Simplified 87.4%

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

      1. log1p-udef 87.4%

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

      2. diff-log 87.5%

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

      3. +-commutative 87.5%

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

   7. Applied egg-rr 87.5%

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

      1. clear-num 87.5%

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

      2. log-rec 87.5%

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

   9. Applied egg-rr 87.5%

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

   if 2.0000000000000001e-4 < (/.f64 1 n) < 9.9999999999999994e165

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0 86.0%

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

   if 9.9999999999999994e165 < (/.f64 1 n)

   1. Initial program 12.1%

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

   3. Taylor expanded in n around inf 8.6%

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

      1. log1p-def 8.6%

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

   5. Simplified 8.6%

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

   6. Taylor expanded in x around inf 61.8%

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

      1. *-commutative 61.8%

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

   8. Simplified 61.8%

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

      1. add-sqr-sqrt 61.8%

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

      2. sqrt-unprod 91.9%

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

      3. inv-pow 91.9%

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

      4. inv-pow 91.9%

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

      5. pow-prod-up 91.9%

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

      6. metadata-eval 91.9%

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

   10. Applied egg-rr 91.9%

       \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+184}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-198)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 0.0002)
     (/ (- (log (/ x (+ 1.0 x)))) n)
     (if (<= (/ 1.0 n) 1e+166)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (sqrt (pow (* n x) -2.0))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-198) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 0.0002) {
		tmp = -log((x / (1.0 + x))) / n;
	} else if ((1.0 / n) <= 1e+166) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-5d-198)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 0.0002d0) then
        tmp = -log((x / (1.0d0 + x))) / n
    else if ((1.0d0 / n) <= 1d+166) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else
        tmp = sqrt(((n * x) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-198) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 0.0002) {
		tmp = -Math.log((x / (1.0 + x))) / n;
	} else if ((1.0 / n) <= 1e+166) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-198:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 0.0002:
		tmp = -math.log((x / (1.0 + x))) / n
	elif (1.0 / n) <= 1e+166:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-198)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.0002)
		tmp = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n);
	elseif (Float64(1.0 / n) <= 1e+166)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5e-198)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 0.0002)
		tmp = -log((x / (1.0 + x))) / n;
	elseif ((1.0 / n) <= 1e+166)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	else
		tmp = sqrt(((n * x) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-198], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0002], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+166], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.9999999999999999e-198

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf 87.8%

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

      1. mul-1-neg 87.8%

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

      2. log-rec 87.8%

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

      3. mul-1-neg 87.8%

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

      4. distribute-neg-frac 87.8%

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

      5. mul-1-neg 87.8%

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

      6. remove-double-neg 87.8%

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

      7. *-commutative 87.8%

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

   5. Simplified 87.8%

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

   if -4.9999999999999999e-198 < (/.f64 1 n) < 2.0000000000000001e-4

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 87.4%

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

      1. log1p-def 87.4%

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

   5. Simplified 87.4%

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

      1. log1p-udef 87.4%

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

      2. diff-log 87.5%

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

      3. +-commutative 87.5%

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

   7. Applied egg-rr 87.5%

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

      1. clear-num 87.5%

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

      2. log-rec 87.5%

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

   9. Applied egg-rr 87.5%

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

   if 2.0000000000000001e-4 < (/.f64 1 n) < 9.9999999999999994e165

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0 86.0%

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

   if 9.9999999999999994e165 < (/.f64 1 n)

   1. Initial program 12.1%

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

   3. Taylor expanded in n around inf 8.6%

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

      1. log1p-def 8.6%

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

   5. Simplified 8.6%

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

   6. Taylor expanded in x around inf 61.8%

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

      1. *-commutative 61.8%

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

   8. Simplified 61.8%

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

      1. add-sqr-sqrt 61.8%

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

      2. sqrt-unprod 91.9%

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

      3. inv-pow 91.9%

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

      4. inv-pow 91.9%

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

      5. pow-prod-up 91.9%

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

      6. metadata-eval 91.9%

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

   10. Applied egg-rr 91.9%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-198)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 0.0002)
     (/ (- (log (/ x (+ 1.0 x)))) n)
     (- (exp (/ x n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-198) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 0.0002) {
		tmp = -log((x / (1.0 + x))) / n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-5d-198)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 0.0002d0) then
        tmp = -log((x / (1.0d0 + x))) / n
    else
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-198) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 0.0002) {
		tmp = -Math.log((x / (1.0 + x))) / n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-198:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 0.0002:
		tmp = -math.log((x / (1.0 + x))) / n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-198)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.0002)
		tmp = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5e-198)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 0.0002)
		tmp = -log((x / (1.0 + x))) / n;
	else
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-198], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0002], N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -4.9999999999999999e-198

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf 87.8%

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

      1. mul-1-neg 87.8%

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

      2. log-rec 87.8%

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

      3. mul-1-neg 87.8%

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

      4. distribute-neg-frac 87.8%

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

      5. mul-1-neg 87.8%

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

      6. remove-double-neg 87.8%

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

      7. *-commutative 87.8%

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

   5. Simplified 87.8%

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

   if -4.9999999999999999e-198 < (/.f64 1 n) < 2.0000000000000001e-4

   1. Initial program 37.3%

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

   3. Taylor expanded in n around inf 87.4%

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

      1. log1p-def 87.4%

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

   5. Simplified 87.4%

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

      1. log1p-udef 87.4%

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

      2. diff-log 87.5%

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

      3. +-commutative 87.5%

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

   7. Applied egg-rr 87.5%

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

      1. clear-num 87.5%

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

      2. log-rec 87.5%

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

   9. Applied egg-rr 87.5%

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

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

   1. Initial program 58.2%

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

   3. Taylor expanded in n around 0 58.2%

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

      1. log1p-def 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+184}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (- (log (/ x (+ 1.0 x)))) n)))
   (if (<= (/ 1.0 n) -5e+184)
     (- 1.0 t_0)
     (if (<= (/ 1.0 n) -2e-20)
       t_1
       (if (<= (/ 1.0 n) -5e-198)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 0.0002)
           t_1
           (if (<= (/ 1.0 n) 1e+166)
             (- (+ 1.0 (/ x n)) t_0)
             (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = -log((x / (1.0 + x))) / n;
	double tmp;
	if ((1.0 / n) <= -5e+184) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= -2e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-198) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 0.0002) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+166) {
		tmp = (1.0 + (x / n)) - t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = -log((x / (1.0d0 + x))) / n
    if ((1.0d0 / n) <= (-5d+184)) then
        tmp = 1.0d0 - t_0
    else if ((1.0d0 / n) <= (-2d-20)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-198)) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 0.0002d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+166) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = -Math.log((x / (1.0 + x))) / n;
	double tmp;
	if ((1.0 / n) <= -5e+184) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= -2e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-198) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 0.0002) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+166) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = -math.log((x / (1.0 + x))) / n
	tmp = 0
	if (1.0 / n) <= -5e+184:
		tmp = 1.0 - t_0
	elif (1.0 / n) <= -2e-20:
		tmp = t_1
	elif (1.0 / n) <= -5e-198:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 0.0002:
		tmp = t_1
	elif (1.0 / n) <= 1e+166:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+184)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= -2e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-198)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 0.0002)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+166)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = -log((x / (1.0 + x))) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e+184)
		tmp = 1.0 - t_0;
	elseif ((1.0 / n) <= -2e-20)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-198)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 0.0002)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+166)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+184], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-198], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0002], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+166], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.9999999999999999e184

   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 58.0%

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

   if -4.9999999999999999e184 < (/.f64 1 n) < -1.99999999999999989e-20 or -4.9999999999999999e-198 < (/.f64 1 n) < 2.0000000000000001e-4

   1. Initial program 54.4%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. log1p-def 80.0%

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

   5. Simplified 80.0%

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

      1. log1p-udef 80.0%

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

      2. diff-log 80.1%

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

      3. +-commutative 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. clear-num 80.1%

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

      2. log-rec 80.1%

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

   9. Applied egg-rr 80.1%

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

   if -1.99999999999999989e-20 < (/.f64 1 n) < -4.9999999999999999e-198

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. log1p-def 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around inf 76.5%

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

      1. *-commutative 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in x around 0 76.5%

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

       1. associate-/r* 78.3%

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

   11. Simplified 78.3%

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

   if 2.0000000000000001e-4 < (/.f64 1 n) < 9.9999999999999994e165

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0 86.0%

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

   if 9.9999999999999994e165 < (/.f64 1 n)

   1. Initial program 12.1%

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

   3. Taylor expanded in n around inf 8.6%

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

      1. log1p-def 8.6%

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

   5. Simplified 8.6%

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

   6. Taylor expanded in x around inf 61.8%

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

      1. *-commutative 61.8%

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

   8. Simplified 61.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+184}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-20}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+184}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e+184)
     t_0
     (if (<= (/ 1.0 n) -2e-20)
       t_1
       (if (<= (/ 1.0 n) -5e-198)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 0.0002)
           t_1
           (if (<= (/ 1.0 n) 1e+166) t_0 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+184) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-198) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 0.0002) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+166) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d+184)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-2d-20)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-198)) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 0.0002d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+166) then
        tmp = t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+184) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-198) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 0.0002) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+166) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e+184:
		tmp = t_0
	elif (1.0 / n) <= -2e-20:
		tmp = t_1
	elif (1.0 / n) <= -5e-198:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 0.0002:
		tmp = t_1
	elif (1.0 / n) <= 1e+166:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+184)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -2e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-198)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 0.0002)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+166)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e+184)
		tmp = t_0;
	elseif ((1.0 / n) <= -2e-20)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-198)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 0.0002)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+166)
		tmp = t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+184], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-198], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0002], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+166], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.9999999999999999e184 or 2.0000000000000001e-4 < (/.f64 1 n) < 9.9999999999999994e165

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0 69.2%

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

   if -4.9999999999999999e184 < (/.f64 1 n) < -1.99999999999999989e-20 or -4.9999999999999999e-198 < (/.f64 1 n) < 2.0000000000000001e-4

   1. Initial program 54.4%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. log1p-def 80.0%

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

   5. Simplified 80.0%

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

      1. log1p-udef 80.0%

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

      2. diff-log 80.1%

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

      3. +-commutative 80.1%

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

   7. Applied egg-rr 80.1%

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

   if -1.99999999999999989e-20 < (/.f64 1 n) < -4.9999999999999999e-198

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. log1p-def 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around inf 76.5%

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

      1. *-commutative 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in x around 0 76.5%

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

       1. associate-/r* 78.3%

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

   11. Simplified 78.3%

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

   if 9.9999999999999994e165 < (/.f64 1 n)

   1. Initial program 12.1%

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

   3. Taylor expanded in n around inf 8.6%

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

      1. log1p-def 8.6%

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

   5. Simplified 8.6%

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

   6. Taylor expanded in x around inf 61.8%

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

      1. *-commutative 61.8%

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

   8. Simplified 61.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+184}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+184}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (- (log (/ x (+ 1.0 x)))) n)))
   (if (<= (/ 1.0 n) -5e+184)
     t_0
     (if (<= (/ 1.0 n) -2e-20)
       t_1
       (if (<= (/ 1.0 n) -5e-198)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 0.0002)
           t_1
           (if (<= (/ 1.0 n) 1e+166) t_0 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = -log((x / (1.0 + x))) / n;
	double tmp;
	if ((1.0 / n) <= -5e+184) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-198) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 0.0002) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+166) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = -log((x / (1.0d0 + x))) / n
    if ((1.0d0 / n) <= (-5d+184)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-2d-20)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-198)) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 0.0002d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+166) then
        tmp = t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = -Math.log((x / (1.0 + x))) / n;
	double tmp;
	if ((1.0 / n) <= -5e+184) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-198) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 0.0002) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+166) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = -math.log((x / (1.0 + x))) / n
	tmp = 0
	if (1.0 / n) <= -5e+184:
		tmp = t_0
	elif (1.0 / n) <= -2e-20:
		tmp = t_1
	elif (1.0 / n) <= -5e-198:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 0.0002:
		tmp = t_1
	elif (1.0 / n) <= 1e+166:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-log(Float64(x / Float64(1.0 + x)))) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+184)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -2e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-198)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 0.0002)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+166)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = -log((x / (1.0 + x))) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e+184)
		tmp = t_0;
	elseif ((1.0 / n) <= -2e-20)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-198)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 0.0002)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+166)
		tmp = t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+184], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-198], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0002], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+166], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.9999999999999999e184 or 2.0000000000000001e-4 < (/.f64 1 n) < 9.9999999999999994e165

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0 69.2%

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

   if -4.9999999999999999e184 < (/.f64 1 n) < -1.99999999999999989e-20 or -4.9999999999999999e-198 < (/.f64 1 n) < 2.0000000000000001e-4

   1. Initial program 54.4%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. log1p-def 80.0%

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

   5. Simplified 80.0%

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

      1. log1p-udef 80.0%

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

      2. diff-log 80.1%

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

      3. +-commutative 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. clear-num 80.1%

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

      2. log-rec 80.1%

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

   9. Applied egg-rr 80.1%

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

   if -1.99999999999999989e-20 < (/.f64 1 n) < -4.9999999999999999e-198

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. log1p-def 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around inf 76.5%

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

      1. *-commutative 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in x around 0 76.5%

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

       1. associate-/r* 78.3%

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

   11. Simplified 78.3%

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

   if 9.9999999999999994e165 < (/.f64 1 n)

   1. Initial program 12.1%

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

   3. Taylor expanded in n around inf 8.6%

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

      1. log1p-def 8.6%

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

   5. Simplified 8.6%

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

   6. Taylor expanded in x around inf 61.8%

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

      1. *-commutative 61.8%

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

   8. Simplified 61.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+184}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-20}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0002:\\ \;\;\;\;\frac{-\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+166}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 4.5 \cdot 10^{-268}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-240}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-229}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-208}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-205}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+198} \lor \neg \left(x \leq 1.35 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 4.5e-268)
     (/ (- (log x)) n)
     (if (<= x 1.06e-240)
       t_0
       (if (<= x 3.9e-229)
         (* (log x) (/ -1.0 n))
         (if (<= x 7e-208)
           (/ 1.0 (* n x))
           (if (<= x 1.1e-205)
             t_0
             (if (<= x 1.0)
               (/ (- x (log x)) n)
               (if (<= x 7.4e+98)
                 (/ (/ 1.0 n) x)
                 (if (or (<= x 2e+198) (not (<= x 1.35e+216)))
                   (/ 0.0 n)
                   (/ (/ 1.0 x) n)))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 4.5e-268) {
		tmp = -log(x) / n;
	} else if (x <= 1.06e-240) {
		tmp = t_0;
	} else if (x <= 3.9e-229) {
		tmp = log(x) * (-1.0 / n);
	} else if (x <= 7e-208) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.1e-205) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 7.4e+98) {
		tmp = (1.0 / n) / x;
	} else if ((x <= 2e+198) || !(x <= 1.35e+216)) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 4.5d-268) then
        tmp = -log(x) / n
    else if (x <= 1.06d-240) then
        tmp = t_0
    else if (x <= 3.9d-229) then
        tmp = log(x) * ((-1.0d0) / n)
    else if (x <= 7d-208) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 1.1d-205) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 7.4d+98) then
        tmp = (1.0d0 / n) / x
    else if ((x <= 2d+198) .or. (.not. (x <= 1.35d+216))) then
        tmp = 0.0d0 / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 4.5e-268) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.06e-240) {
		tmp = t_0;
	} else if (x <= 3.9e-229) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if (x <= 7e-208) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.1e-205) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 7.4e+98) {
		tmp = (1.0 / n) / x;
	} else if ((x <= 2e+198) || !(x <= 1.35e+216)) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 4.5e-268:
		tmp = -math.log(x) / n
	elif x <= 1.06e-240:
		tmp = t_0
	elif x <= 3.9e-229:
		tmp = math.log(x) * (-1.0 / n)
	elif x <= 7e-208:
		tmp = 1.0 / (n * x)
	elif x <= 1.1e-205:
		tmp = t_0
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 7.4e+98:
		tmp = (1.0 / n) / x
	elif (x <= 2e+198) or not (x <= 1.35e+216):
		tmp = 0.0 / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 4.5e-268)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.06e-240)
		tmp = t_0;
	elseif (x <= 3.9e-229)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (x <= 7e-208)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 1.1e-205)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 7.4e+98)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif ((x <= 2e+198) || !(x <= 1.35e+216))
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 4.5e-268)
		tmp = -log(x) / n;
	elseif (x <= 1.06e-240)
		tmp = t_0;
	elseif (x <= 3.9e-229)
		tmp = log(x) * (-1.0 / n);
	elseif (x <= 7e-208)
		tmp = 1.0 / (n * x);
	elseif (x <= 1.1e-205)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 7.4e+98)
		tmp = (1.0 / n) / x;
	elseif ((x <= 2e+198) || ~((x <= 1.35e+216)))
		tmp = 0.0 / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.5e-268], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.06e-240], t$95$0, If[LessEqual[x, 3.9e-229], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-208], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e-205], t$95$0, If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 7.4e+98], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[x, 2e+198], N[Not[LessEqual[x, 1.35e+216]], $MachinePrecision]], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if x < 4.5000000000000001e-268

   1. Initial program 34.6%

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

   3. Taylor expanded in n around inf 67.0%

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

      1. log1p-def 67.0%

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

   5. Simplified 67.0%

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

   6. Taylor expanded in x around 0 67.0%

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

      1. neg-mul-1 67.0%

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

   8. Simplified 67.0%

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

   if 4.5000000000000001e-268 < x < 1.06e-240 or 6.99999999999999982e-208 < x < 1.10000000000000005e-205

   1. Initial program 68.4%

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

   3. Taylor expanded in x around 0 68.4%

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

   if 1.06e-240 < x < 3.89999999999999985e-229

   1. Initial program 32.9%

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

   3. Taylor expanded in n around inf 72.7%

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

      1. log1p-def 72.7%

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

   5. Simplified 72.7%

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

      1. div-inv 73.2%

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

   7. Applied egg-rr 73.2%

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

   8. Taylor expanded in x around 0 73.2%

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

      1. neg-mul-1 72.7%

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

   10. Simplified 73.2%

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

   if 3.89999999999999985e-229 < x < 6.99999999999999982e-208

   1. Initial program 51.5%

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

   3. Taylor expanded in n around inf 22.4%

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

      1. log1p-def 22.4%

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

   5. Simplified 22.4%

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

   6. Taylor expanded in x around inf 68.3%

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

      1. *-commutative 68.3%

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

   8. Simplified 68.3%

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

   if 1.10000000000000005e-205 < x < 1

   1. Initial program 39.0%

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

   3. Taylor expanded in n around inf 58.9%

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

      1. log1p-def 58.9%

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

   5. Simplified 58.9%

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

      1. log1p-udef 58.9%

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

      2. diff-log 58.9%

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

      3. +-commutative 58.9%

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

   7. Applied egg-rr 58.9%

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

   8. Taylor expanded in x around 0 58.9%

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

      1. neg-mul-1 58.9%

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

      2. sub-neg 58.9%

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

   10. Simplified 58.9%

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

   if 1 < x < 7.3999999999999997e98

   1. Initial program 41.0%

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

   3. Taylor expanded in n around inf 38.2%

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

      1. log1p-def 38.2%

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

   5. Simplified 38.2%

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

   6. Taylor expanded in x around inf 64.7%

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

      1. *-commutative 64.7%

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

   8. Simplified 64.7%

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

   9. Taylor expanded in x around 0 64.7%

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

       1. associate-/r* 65.9%

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

   11. Simplified 65.9%

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

   if 7.3999999999999997e98 < x < 2.00000000000000004e198 or 1.3500000000000001e216 < x

   1. Initial program 82.9%

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

      1. flip3-- 46.6%

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

      2. div-inv 46.6%

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

      3. add-exp-log 46.6%

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

      4. log-pow 46.6%

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

      5. +-commutative 46.6%

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

      6. log1p-udef 46.6%

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

      7. *-commutative 46.6%

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

      8. un-div-inv 46.6%

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

      9. pow-pow 46.5%

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

   4. Applied egg-rr 46.6%

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

      1. associate-*r/ 46.6%

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

      2. *-rgt-identity 46.6%

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

      3. associate-*l/ 46.6%

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

      4. metadata-eval 46.6%

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

      5. *-commutative 46.6%

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

      6. +-commutative 46.6%

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

   6. Simplified 46.6%

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

   7. Taylor expanded in n around -inf 82.9%

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

      1. *-commutative 82.9%

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

      2. cancel-sign-sub-inv 82.9%

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

      3. log1p-def 82.9%

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

      4. log1p-def 82.9%

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

      5. distribute-rgt-out 82.9%

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

      6. metadata-eval 82.9%

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

      7. metadata-eval 82.9%

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

      8. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in x around inf 82.9%

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

       1. associate-*r/ 82.9%

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

       2. distribute-rgt-out 82.9%

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

       3. metadata-eval 82.9%

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

       4. mul0-rgt 82.9%

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

       5. metadata-eval 82.9%

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

   12. Simplified 82.9%

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

   if 2.00000000000000004e198 < x < 1.3500000000000001e216

   1. Initial program 53.8%

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

   3. Taylor expanded in n around inf 53.8%

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

      1. log1p-def 53.8%

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

   5. Simplified 53.8%

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

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 8 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-268}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-240}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-229}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-208}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-205}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+198} \lor \neg \left(x \leq 1.35 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-229}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-208}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+202} \lor \neg \left(x \leq 1.35 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.1e-229)
   (/ (- (log x)) n)
   (if (<= x 7e-208)
     (/ 1.0 (* n x))
     (if (<= x 1.0)
       (/ (- x (log x)) n)
       (if (<= x 6.2e+98)
         (/ (/ 1.0 n) x)
         (if (or (<= x 1.55e+202) (not (<= x 1.35e+216)))
           (/ 0.0 n)
           (/ (/ 1.0 x) n)))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.1e-229) {
		tmp = -log(x) / n;
	} else if (x <= 7e-208) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 6.2e+98) {
		tmp = (1.0 / n) / x;
	} else if ((x <= 1.55e+202) || !(x <= 1.35e+216)) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.1d-229) then
        tmp = -log(x) / n
    else if (x <= 7d-208) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 6.2d+98) then
        tmp = (1.0d0 / n) / x
    else if ((x <= 1.55d+202) .or. (.not. (x <= 1.35d+216))) then
        tmp = 0.0d0 / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.1e-229) {
		tmp = -Math.log(x) / n;
	} else if (x <= 7e-208) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 6.2e+98) {
		tmp = (1.0 / n) / x;
	} else if ((x <= 1.55e+202) || !(x <= 1.35e+216)) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.1e-229:
		tmp = -math.log(x) / n
	elif x <= 7e-208:
		tmp = 1.0 / (n * x)
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 6.2e+98:
		tmp = (1.0 / n) / x
	elif (x <= 1.55e+202) or not (x <= 1.35e+216):
		tmp = 0.0 / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.1e-229)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 7e-208)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 6.2e+98)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif ((x <= 1.55e+202) || !(x <= 1.35e+216))
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.1e-229)
		tmp = -log(x) / n;
	elseif (x <= 7e-208)
		tmp = 1.0 / (n * x);
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 6.2e+98)
		tmp = (1.0 / n) / x;
	elseif ((x <= 1.55e+202) || ~((x <= 1.35e+216)))
		tmp = 0.0 / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.1e-229], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 7e-208], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 6.2e+98], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[x, 1.55e+202], N[Not[LessEqual[x, 1.35e+216]], $MachinePrecision]], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 3.1000000000000001e-229

   1. Initial program 44.0%

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

   3. Taylor expanded in n around inf 55.5%

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

      1. log1p-def 55.5%

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

   5. Simplified 55.5%

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

   6. Taylor expanded in x around 0 55.5%

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

      1. neg-mul-1 55.5%

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

   8. Simplified 55.5%

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

   if 3.1000000000000001e-229 < x < 6.99999999999999982e-208

   1. Initial program 51.5%

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

   3. Taylor expanded in n around inf 22.4%

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

      1. log1p-def 22.4%

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

   5. Simplified 22.4%

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

   6. Taylor expanded in x around inf 68.3%

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

      1. *-commutative 68.3%

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

   8. Simplified 68.3%

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

   if 6.99999999999999982e-208 < x < 1

   1. Initial program 40.1%

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

   3. Taylor expanded in n around inf 58.1%

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

      1. log1p-def 58.1%

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

   5. Simplified 58.1%

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

      1. log1p-udef 58.1%

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

      2. diff-log 58.1%

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

      3. +-commutative 58.1%

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

   7. Applied egg-rr 58.1%

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

   8. Taylor expanded in x around 0 58.1%

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

      1. neg-mul-1 58.1%

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

      2. sub-neg 58.1%

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

   10. Simplified 58.1%

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

   if 1 < x < 6.20000000000000038e98

   1. Initial program 41.0%

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

   3. Taylor expanded in n around inf 38.2%

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

      1. log1p-def 38.2%

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

   5. Simplified 38.2%

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

   6. Taylor expanded in x around inf 64.7%

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

      1. *-commutative 64.7%

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

   8. Simplified 64.7%

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

   9. Taylor expanded in x around 0 64.7%

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

       1. associate-/r* 65.9%

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

   11. Simplified 65.9%

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

   if 6.20000000000000038e98 < x < 1.54999999999999996e202 or 1.3500000000000001e216 < x

   1. Initial program 82.9%

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

      1. flip3-- 46.6%

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

      2. div-inv 46.6%

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

      3. add-exp-log 46.6%

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

      4. log-pow 46.6%

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

      5. +-commutative 46.6%

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

      6. log1p-udef 46.6%

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

      7. *-commutative 46.6%

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

      8. un-div-inv 46.6%

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

      9. pow-pow 46.5%

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

   4. Applied egg-rr 46.6%

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

      1. associate-*r/ 46.6%

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

      2. *-rgt-identity 46.6%

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

      3. associate-*l/ 46.6%

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

      4. metadata-eval 46.6%

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

      5. *-commutative 46.6%

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

      6. +-commutative 46.6%

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

   6. Simplified 46.6%

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

   7. Taylor expanded in n around -inf 82.9%

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

      1. *-commutative 82.9%

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

      2. cancel-sign-sub-inv 82.9%

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

      3. log1p-def 82.9%

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

      4. log1p-def 82.9%

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

      5. distribute-rgt-out 82.9%

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

      6. metadata-eval 82.9%

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

      7. metadata-eval 82.9%

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

      8. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in x around inf 82.9%

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

       1. associate-*r/ 82.9%

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

       2. distribute-rgt-out 82.9%

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

       3. metadata-eval 82.9%

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

       4. mul0-rgt 82.9%

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

       5. metadata-eval 82.9%

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

   12. Simplified 82.9%

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

   if 1.54999999999999996e202 < x < 1.3500000000000001e216

   1. Initial program 53.8%

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

   3. Taylor expanded in n around inf 53.8%

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

      1. log1p-def 53.8%

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

   5. Simplified 53.8%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-229}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-208}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+202} \lor \neg \left(x \leq 1.35 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 3.9 \cdot 10^{-229}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-208}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+198} \lor \neg \left(x \leq 1.35 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 3.9e-229)
     t_0
     (if (<= x 7e-208)
       (/ 1.0 (* n x))
       (if (<= x 0.55)
         t_0
         (if (<= x 2.45e+97)
           (/ (/ 1.0 n) x)
           (if (or (<= x 4.7e+198) (not (<= x 1.35e+216)))
             (/ 0.0 n)
             (/ (/ 1.0 x) n))))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 3.9e-229) {
		tmp = t_0;
	} else if (x <= 7e-208) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.55) {
		tmp = t_0;
	} else if (x <= 2.45e+97) {
		tmp = (1.0 / n) / x;
	} else if ((x <= 4.7e+198) || !(x <= 1.35e+216)) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 3.9d-229) then
        tmp = t_0
    else if (x <= 7d-208) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.55d0) then
        tmp = t_0
    else if (x <= 2.45d+97) then
        tmp = (1.0d0 / n) / x
    else if ((x <= 4.7d+198) .or. (.not. (x <= 1.35d+216))) then
        tmp = 0.0d0 / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 3.9e-229) {
		tmp = t_0;
	} else if (x <= 7e-208) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.55) {
		tmp = t_0;
	} else if (x <= 2.45e+97) {
		tmp = (1.0 / n) / x;
	} else if ((x <= 4.7e+198) || !(x <= 1.35e+216)) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 3.9e-229:
		tmp = t_0
	elif x <= 7e-208:
		tmp = 1.0 / (n * x)
	elif x <= 0.55:
		tmp = t_0
	elif x <= 2.45e+97:
		tmp = (1.0 / n) / x
	elif (x <= 4.7e+198) or not (x <= 1.35e+216):
		tmp = 0.0 / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 3.9e-229)
		tmp = t_0;
	elseif (x <= 7e-208)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.55)
		tmp = t_0;
	elseif (x <= 2.45e+97)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif ((x <= 4.7e+198) || !(x <= 1.35e+216))
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 3.9e-229)
		tmp = t_0;
	elseif (x <= 7e-208)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.55)
		tmp = t_0;
	elseif (x <= 2.45e+97)
		tmp = (1.0 / n) / x;
	elseif ((x <= 4.7e+198) || ~((x <= 1.35e+216)))
		tmp = 0.0 / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 3.9e-229], t$95$0, If[LessEqual[x, 7e-208], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.55], t$95$0, If[LessEqual[x, 2.45e+97], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[x, 4.7e+198], N[Not[LessEqual[x, 1.35e+216]], $MachinePrecision]], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 3.89999999999999985e-229 or 6.99999999999999982e-208 < x < 0.55000000000000004

   1. Initial program 41.4%

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

   3. Taylor expanded in n around inf 57.2%

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

      1. log1p-def 57.2%

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

   5. Simplified 57.2%

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

   6. Taylor expanded in x around 0 57.0%

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

      1. neg-mul-1 57.0%

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

   8. Simplified 57.0%

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

   if 3.89999999999999985e-229 < x < 6.99999999999999982e-208

   1. Initial program 51.5%

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

   3. Taylor expanded in n around inf 22.4%

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

      1. log1p-def 22.4%

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

   5. Simplified 22.4%

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

   6. Taylor expanded in x around inf 68.3%

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

      1. *-commutative 68.3%

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

   8. Simplified 68.3%

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

   if 0.55000000000000004 < x < 2.44999999999999982e97

   1. Initial program 41.0%

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

   3. Taylor expanded in n around inf 38.2%

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

      1. log1p-def 38.2%

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

   5. Simplified 38.2%

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

   6. Taylor expanded in x around inf 64.7%

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

      1. *-commutative 64.7%

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

   8. Simplified 64.7%

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

   9. Taylor expanded in x around 0 64.7%

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

       1. associate-/r* 65.9%

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

   11. Simplified 65.9%

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

   if 2.44999999999999982e97 < x < 4.7000000000000002e198 or 1.3500000000000001e216 < x

   1. Initial program 82.9%

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

      1. flip3-- 46.6%

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

      2. div-inv 46.6%

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

      3. add-exp-log 46.6%

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

      4. log-pow 46.6%

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

      5. +-commutative 46.6%

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

      6. log1p-udef 46.6%

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

      7. *-commutative 46.6%

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

      8. un-div-inv 46.6%

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

      9. pow-pow 46.5%

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

   4. Applied egg-rr 46.6%

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

      1. associate-*r/ 46.6%

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

      2. *-rgt-identity 46.6%

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

      3. associate-*l/ 46.6%

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

      4. metadata-eval 46.6%

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

      5. *-commutative 46.6%

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

      6. +-commutative 46.6%

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

   6. Simplified 46.6%

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

   7. Taylor expanded in n around -inf 82.9%

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

      1. *-commutative 82.9%

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

      2. cancel-sign-sub-inv 82.9%

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

      3. log1p-def 82.9%

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

      4. log1p-def 82.9%

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

      5. distribute-rgt-out 82.9%

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

      6. metadata-eval 82.9%

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

      7. metadata-eval 82.9%

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

      8. *-commutative 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in x around inf 82.9%

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

       1. associate-*r/ 82.9%

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

       2. distribute-rgt-out 82.9%

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

       3. metadata-eval 82.9%

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

       4. mul0-rgt 82.9%

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

       5. metadata-eval 82.9%

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

   12. Simplified 82.9%

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

   if 4.7000000000000002e198 < x < 1.3500000000000001e216

   1. Initial program 53.8%

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

   3. Taylor expanded in n around inf 53.8%

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

      1. log1p-def 53.8%

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

   5. Simplified 53.8%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{-229}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-208}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+198} \lor \neg \left(x \leq 1.35 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.7% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.017 \lor \neg \left(n \leq -3.3 \cdot 10^{-183}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -0.017) (not (<= n -3.3e-183))) (/ 1.0 (* n x)) (/ 0.0 n)))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -0.017) || !(n <= -3.3e-183)) {
		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 <= (-0.017d0)) .or. (.not. (n <= (-3.3d-183)))) 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 <= -0.017) || !(n <= -3.3e-183)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -0.017) or not (n <= -3.3e-183):
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -0.017) || !(n <= -3.3e-183))
		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 <= -0.017) || ~((n <= -3.3e-183)))
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -0.017], N[Not[LessEqual[n, -3.3e-183]], $MachinePrecision]], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -0.017000000000000001 or -3.3e-183 < n

   1. Initial program 47.0%

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

   3. Taylor expanded in n around inf 60.7%

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

      1. log1p-def 60.7%

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

   5. Simplified 60.7%

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

   6. Taylor expanded in x around inf 48.1%

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

      1. *-commutative 48.1%

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

   8. Simplified 48.1%

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

   if -0.017000000000000001 < n < -3.3e-183

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. add-exp-log 0.0%

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

      4. log-pow 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-udef 0.0%

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

      7. *-commutative 0.0%

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

      8. un-div-inv 0.0%

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

      9. 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%

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

      6. +-commutative 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in n around -inf 64.0%

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

      1. *-commutative 64.0%

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

      2. cancel-sign-sub-inv 64.0%

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

      3. log1p-def 64.0%

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

      4. log1p-def 64.0%

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

      5. distribute-rgt-out 64.0%

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

      6. metadata-eval 64.0%

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

      7. metadata-eval 64.0%

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

      8. *-commutative 64.0%

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

   9. Simplified 64.0%

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

   10. Taylor expanded in x around inf 65.5%

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

       1. associate-*r/ 65.5%

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

       2. distribute-rgt-out 65.5%

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

       3. metadata-eval 65.5%

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

       4. mul0-rgt 65.5%

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

       5. metadata-eval 65.5%

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

   12. Simplified 65.5%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.017 \lor \neg \left(n \leq -3.3 \cdot 10^{-183}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.9% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -20000000.0) (/ 0.0 n) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		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) <= (-20000000.0d0)) 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) <= -20000000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -20000000.0:
		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) <= -20000000.0)
		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) <= -20000000.0)
		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], -20000000.0], 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) < -2e7

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. add-exp-log 0.0%

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

      4. log-pow 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-udef 0.0%

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

      7. *-commutative 0.0%

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

      8. un-div-inv 0.0%

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

      9. 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%

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

      6. +-commutative 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in n around -inf 55.3%

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

      1. *-commutative 55.3%

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

      2. cancel-sign-sub-inv 55.3%

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

      3. log1p-def 55.3%

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

      4. log1p-def 55.3%

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

      5. distribute-rgt-out 55.3%

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

      6. metadata-eval 55.3%

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

      7. metadata-eval 55.3%

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

      8. *-commutative 55.3%

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

   9. Simplified 55.3%

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

   10. Taylor expanded in x around inf 56.4%

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

       1. associate-*r/ 56.4%

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

       2. distribute-rgt-out 56.4%

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

       3. metadata-eval 56.4%

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

       4. mul0-rgt 56.4%

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

       5. metadata-eval 56.4%

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

   12. Simplified 56.4%

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

   if -2e7 < (/.f64 1 n)

   1. Initial program 38.4%

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

   3. Taylor expanded in n around inf 63.4%

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

      1. log1p-def 63.4%

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

   5. Simplified 63.4%

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

   6. Taylor expanded in x around inf 48.5%

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

      1. *-commutative 48.5%

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

   8. Simplified 48.5%

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

   9. Taylor expanded in x around 0 48.5%

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

       1. associate-/r* 49.6%

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

   11. Simplified 49.6%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.9% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -20000000.0) (/ 0.0 n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-20000000.0d0)) then
        tmp = 0.0d0 / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -20000000.0:
		tmp = 0.0 / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -20000000.0)
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -20000000.0)
		tmp = 0.0 / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000.0], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -2e7

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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. add-exp-log 0.0%

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

      4. log-pow 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-udef 0.0%

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

      7. *-commutative 0.0%

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

      8. un-div-inv 0.0%

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

      9. 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%

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

      6. +-commutative 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in n around -inf 55.3%

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

      1. *-commutative 55.3%

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

      2. cancel-sign-sub-inv 55.3%

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

      3. log1p-def 55.3%

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

      4. log1p-def 55.3%

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

      5. distribute-rgt-out 55.3%

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

      6. metadata-eval 55.3%

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

      7. metadata-eval 55.3%

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

      8. *-commutative 55.3%

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

   9. Simplified 55.3%

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

   10. Taylor expanded in x around inf 56.4%

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

       1. associate-*r/ 56.4%

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

       2. distribute-rgt-out 56.4%

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

       3. metadata-eval 56.4%

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

       4. mul0-rgt 56.4%

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

       5. metadata-eval 56.4%

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

   12. Simplified 56.4%

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

   if -2e7 < (/.f64 1 n)

   1. Initial program 38.4%

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

   3. Taylor expanded in n around inf 63.4%

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

      1. log1p-def 63.4%

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

   5. Simplified 63.4%

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

   6. Taylor expanded in x around inf 49.6%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.5% 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 55.3%

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

   1. flip3-- 27.2%

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

   2. div-inv 27.2%

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

   3. add-exp-log 27.2%

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

   4. log-pow 27.2%

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

   5. +-commutative 27.2%

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

   6. log1p-udef 31.7%

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

   7. *-commutative 31.7%

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

   8. un-div-inv 31.7%

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

   9. pow-pow 31.7%

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

4. Applied egg-rr 27.2%

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

   1. associate-*r/ 27.2%

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

   2. *-rgt-identity 27.2%

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

   3. associate-*l/ 27.2%

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

   4. metadata-eval 27.2%

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

   5. *-commutative 27.2%

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

   6. +-commutative 27.2%

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

6. Simplified 27.2%

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

7. Taylor expanded in n around -inf 61.4%

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

   1. *-commutative 61.4%

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

   2. cancel-sign-sub-inv 61.4%

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

   3. log1p-def 61.4%

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

   4. log1p-def 61.4%

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

   5. distribute-rgt-out 61.4%

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

   6. metadata-eval 61.4%

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

   7. metadata-eval 61.4%

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

   8. *-commutative 61.4%

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

9. Simplified 61.4%

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

10. Taylor expanded in x around inf 35.1%

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

    1. associate-*r/ 35.1%

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

    2. distribute-rgt-out 35.1%

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

    3. metadata-eval 35.1%

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

    4. mul0-rgt 35.1%

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

    5. metadata-eval 35.1%

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

12. Simplified 35.1%

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

13. Final simplification 35.1%

    \[\leadsto \frac{0}{n} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024029 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))