2nthrt (problem 3.4.6)

Percentage Accurate: 53.2% → 85.0%

Time: 38.1s

Alternatives: 13

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 0.0) (/ (log (/ x (+ x 1.0))) (- n)) (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = log((x / (x + 1.0))) / -n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = Math.log((x / (x + 1.0))) / -n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 1.0 - t_0
	elif t_1 <= 0.0:
		tmp = math.log((x / (x + 1.0))) / -n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n));
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 0.0)
		tmp = log((x / (x + 1.0))) / -n;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n))) < 0.0

   1. Initial program 40.3%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

      1. log1p-undefine 79.4%

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

      2. diff-log 79.6%

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

   7. Applied egg-rr 79.6%

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

      1. clear-num 79.6%

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

      2. log-rec 79.6%

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

   9. Applied egg-rr 79.6%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x 1) (/.f64 1 n)) (pow.f64 x (/.f64 1 n)))

   1. Initial program 51.3%

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

   3. Taylor expanded in n around 0 51.3%

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

      1. log1p-define 97.4%

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

      2. *-rgt-identity 97.4%

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

      3. associate-/l* 97.4%

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

      4. exp-to-pow 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around 0 97.4%

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

4. Final simplification 84.6%

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

Alternative 2: 73.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ t_2 := \frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+37}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq -500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+199}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n))))
        (t_1 (/ (log (/ (+ x 1.0) x)) n))
        (t_2 (/ 0.3333333333333333 (* n (pow x 3.0)))))
   (if (<= (/ 1.0 n) -5e+37)
     t_2
     (if (<= (/ 1.0 n) -500000000.0)
       t_0
       (if (<= (/ 1.0 n) -5e-59)
         t_1
         (if (<= (/ 1.0 n) -1e-119)
           (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)
           (if (<= (/ 1.0 n) 0.001)
             t_1
             (if (<= (/ 1.0 n) 1e+199) t_0 t_2))))))))

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 t_2 = 0.3333333333333333 / (n * pow(x, 3.0));
	double tmp;
	if ((1.0 / n) <= -5e+37) {
		tmp = t_2;
	} else if ((1.0 / n) <= -500000000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= -5e-59) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-119) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else if ((1.0 / n) <= 0.001) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+199) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = log(((x + 1.0d0) / x)) / n
    t_2 = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    if ((1.0d0 / n) <= (-5d+37)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-500000000.0d0)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-5d-59)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-1d-119)) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else if ((1.0d0 / n) <= 0.001d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+199) then
        tmp = t_0
    else
        tmp = t_2
    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 t_2 = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	double tmp;
	if ((1.0 / n) <= -5e+37) {
		tmp = t_2;
	} else if ((1.0 / n) <= -500000000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= -5e-59) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-119) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else if ((1.0 / n) <= 0.001) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+199) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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
	t_2 = 0.3333333333333333 / (n * math.pow(x, 3.0))
	tmp = 0
	if (1.0 / n) <= -5e+37:
		tmp = t_2
	elif (1.0 / n) <= -500000000.0:
		tmp = t_0
	elif (1.0 / n) <= -5e-59:
		tmp = t_1
	elif (1.0 / n) <= -1e-119:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	elif (1.0 / n) <= 0.001:
		tmp = t_1
	elif (1.0 / n) <= 1e+199:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	t_2 = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+37)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -500000000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -5e-59)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -1e-119)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 0.001)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+199)
		tmp = t_0;
	else
		tmp = t_2;
	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;
	t_2 = 0.3333333333333333 / (n * (x ^ 3.0));
	tmp = 0.0;
	if ((1.0 / n) <= -5e+37)
		tmp = t_2;
	elseif ((1.0 / n) <= -500000000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= -5e-59)
		tmp = t_1;
	elseif ((1.0 / n) <= -1e-119)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	elseif ((1.0 / n) <= 0.001)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+199)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+37], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-59], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-119], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.001], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+199], t$95$0, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.99999999999999989e37 or 1.0000000000000001e199 < (/.f64 1 n)

   1. Initial program 83.0%

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

   3. Taylor expanded in n around inf 48.6%

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

      1. log1p-define 48.6%

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

   5. Simplified 48.6%

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

   6. Taylor expanded in x around inf 27.9%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   7. Step-by-step derivation

      1. associate--l+ 27.9%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. associate-*r/ 27.9%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. metadata-eval 27.9%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      4. associate-*r/ 29.7%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{n} \]

      5. metadata-eval 29.7%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{n} \]

   8. Simplified 29.7%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}}{n} \]

   9. Taylor expanded in x around 0 73.1%

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

   if -4.99999999999999989e37 < (/.f64 1 n) < -5e8 or 1e-3 < (/.f64 1 n) < 1.0000000000000001e199

   1. Initial program 74.8%

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

   3. Taylor expanded in x around 0 69.3%

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

      1. *-rgt-identity 69.3%

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

      2. associate-/l* 69.3%

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

      3. exp-to-pow 69.3%

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

   5. Simplified 69.3%

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

   if -5e8 < (/.f64 1 n) < -5.0000000000000001e-59 or -1.00000000000000001e-119 < (/.f64 1 n) < 1e-3

   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 79.5%

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

      1. log1p-define 79.5%

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

   5. Simplified 79.5%

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

      1. log1p-undefine 79.5%

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

      2. diff-log 79.6%

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

   7. Applied egg-rr 79.6%

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

   if -5.0000000000000001e-59 < (/.f64 1 n) < -1.00000000000000001e-119

   1. Initial program 12.8%

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

   3. Taylor expanded in n around inf 42.4%

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

      1. log1p-define 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in x around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. metadata-eval 71.4%

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

   8. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{{x}^{2}}}}{n} \]
   9. Step-by-step derivation

      1. unpow2 71.4%

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

   10. Applied egg-rr 71.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq -500000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+199}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 1.03 \cdot 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ x (+ x 1.0))) (- n))))
   (if (<= (/ 1.0 n) -1e-10)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-59)
       t_1
       (if (<= (/ 1.0 n) -1e-119)
         (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)
         (if (<= (/ 1.0 n) 0.001)
           t_1
           (if (<= (/ 1.0 n) 1.03e+214)
             (- (+ 1.0 (/ x n)) t_0)
             (/ 0.3333333333333333 (* n (pow x 3.0))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log((x / (x + 1.0))) / -n;
	double tmp;
	if ((1.0 / n) <= -1e-10) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-59) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-119) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else if ((1.0 / n) <= 0.001) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1.03e+214) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	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 / (x + 1.0d0))) / -n
    if ((1.0d0 / n) <= (-1d-10)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= (-5d-59)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-1d-119)) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else if ((1.0d0 / n) <= 0.001d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1.03d+214) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    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 / (x + 1.0))) / -n;
	double tmp;
	if ((1.0 / n) <= -1e-10) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-59) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-119) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else if ((1.0 / n) <= 0.001) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1.03e+214) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log((x / (x + 1.0))) / -n
	tmp = 0
	if (1.0 / n) <= -1e-10:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-59:
		tmp = t_1
	elif (1.0 / n) <= -1e-119:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	elif (1.0 / n) <= 0.001:
		tmp = t_1
	elif (1.0 / n) <= 1.03e+214:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-10)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-59)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -1e-119)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 0.001)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1.03e+214)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log((x / (x + 1.0))) / -n;
	tmp = 0.0;
	if ((1.0 / n) <= -1e-10)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= -5e-59)
		tmp = t_1;
	elseif ((1.0 / n) <= -1e-119)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	elseif ((1.0 / n) <= 0.001)
		tmp = t_1;
	elseif ((1.0 / n) <= 1.03e+214)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	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[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-10], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-59], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-119], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.001], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.03e+214], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-10

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. exp-to-pow 100.0%

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

      9. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -1.00000000000000004e-10 < (/.f64 1 n) < -5.0000000000000001e-59 or -1.00000000000000001e-119 < (/.f64 1 n) < 1e-3

   1. Initial program 31.4%

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

   3. Taylor expanded in n around inf 79.9%

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

      1. log1p-define 79.9%

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

   5. Simplified 79.9%

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

      1. log1p-undefine 79.9%

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

      2. diff-log 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. clear-num 80.1%

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

      2. log-rec 80.1%

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

   9. Applied egg-rr 80.1%

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

   if -5.0000000000000001e-59 < (/.f64 1 n) < -1.00000000000000001e-119

   1. Initial program 12.8%

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

   3. Taylor expanded in n around inf 42.4%

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

      1. log1p-define 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in x around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. metadata-eval 71.4%

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

   8. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{{x}^{2}}}}{n} \]
   9. Step-by-step derivation

      1. unpow2 71.4%

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

   10. Applied egg-rr 71.4%

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

   if 1e-3 < (/.f64 1 n) < 1.03e214

   1. Initial program 63.8%

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

   3. Taylor expanded in x around 0 61.9%

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

   if 1.03e214 < (/.f64 1 n)

   1. Initial program 20.7%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

   6. Taylor expanded in x around inf 46.4%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   7. Step-by-step derivation

      1. associate--l+ 46.4%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. associate-*r/ 46.4%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. metadata-eval 46.4%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      4. associate-*r/ 46.4%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{n} \]

      5. metadata-eval 46.4%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{n} \]

   8. Simplified 46.4%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}}{n} \]

   9. Taylor expanded in x around 0 74.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.03 \cdot 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{\log t\_0}{-n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_1}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\frac{1}{n} \leq 1.03 \cdot 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(t\_0 + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0)))
        (t_1 (pow x (/ 1.0 n)))
        (t_2 (/ (log t_0) (- n))))
   (if (<= (/ 1.0 n) -1e-10)
     (/ t_1 (* x n))
     (if (<= (/ 1.0 n) -5e-59)
       t_2
       (if (<= (/ 1.0 n) -1e-119)
         (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)
         (if (<= (/ 1.0 n) 0.001)
           t_2
           (if (<= (/ 1.0 n) 1.03e+214)
             (- (+ 1.0 (/ x n)) t_1)
             (/ (log1p (+ t_0 -1.0)) (- n)))))))))

C

double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = pow(x, (1.0 / n));
	double t_2 = log(t_0) / -n;
	double tmp;
	if ((1.0 / n) <= -1e-10) {
		tmp = t_1 / (x * n);
	} else if ((1.0 / n) <= -5e-59) {
		tmp = t_2;
	} else if ((1.0 / n) <= -1e-119) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else if ((1.0 / n) <= 0.001) {
		tmp = t_2;
	} else if ((1.0 / n) <= 1.03e+214) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = log1p((t_0 + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = Math.log(t_0) / -n;
	double tmp;
	if ((1.0 / n) <= -1e-10) {
		tmp = t_1 / (x * n);
	} else if ((1.0 / n) <= -5e-59) {
		tmp = t_2;
	} else if ((1.0 / n) <= -1e-119) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else if ((1.0 / n) <= 0.001) {
		tmp = t_2;
	} else if ((1.0 / n) <= 1.03e+214) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = Math.log1p((t_0 + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / (x + 1.0)
	t_1 = math.pow(x, (1.0 / n))
	t_2 = math.log(t_0) / -n
	tmp = 0
	if (1.0 / n) <= -1e-10:
		tmp = t_1 / (x * n)
	elif (1.0 / n) <= -5e-59:
		tmp = t_2
	elif (1.0 / n) <= -1e-119:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	elif (1.0 / n) <= 0.001:
		tmp = t_2
	elif (1.0 / n) <= 1.03e+214:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = math.log1p((t_0 + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(log(t_0) / Float64(-n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-10)
		tmp = Float64(t_1 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-59)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -1e-119)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 0.001)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 1.03e+214)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = Float64(log1p(Float64(t_0 + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t$95$0], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-10], N[(t$95$1 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-59], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-119], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.001], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.03e+214], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[Log[1 + N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-10

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. exp-to-pow 100.0%

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

      9. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -1.00000000000000004e-10 < (/.f64 1 n) < -5.0000000000000001e-59 or -1.00000000000000001e-119 < (/.f64 1 n) < 1e-3

   1. Initial program 31.4%

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

   3. Taylor expanded in n around inf 79.9%

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

      1. log1p-define 79.9%

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

   5. Simplified 79.9%

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

      1. log1p-undefine 79.9%

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

      2. diff-log 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. clear-num 80.1%

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

      2. log-rec 80.1%

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

   9. Applied egg-rr 80.1%

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

   if -5.0000000000000001e-59 < (/.f64 1 n) < -1.00000000000000001e-119

   1. Initial program 12.8%

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

   3. Taylor expanded in n around inf 42.4%

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

      1. log1p-define 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in x around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. metadata-eval 71.4%

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

   8. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{{x}^{2}}}}{n} \]
   9. Step-by-step derivation

      1. unpow2 71.4%

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

   10. Applied egg-rr 71.4%

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

   if 1e-3 < (/.f64 1 n) < 1.03e214

   1. Initial program 63.8%

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

   3. Taylor expanded in x around 0 61.9%

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

   if 1.03e214 < (/.f64 1 n)

   1. Initial program 20.7%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

      1. log1p-undefine 6.6%

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

      2. diff-log 6.6%

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

   7. Applied egg-rr 6.6%

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

      1. clear-num 6.6%

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

      2. log-rec 6.6%

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

   9. Applied egg-rr 6.6%

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

       1. log1p-expm1-u 82.4%

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

       2. expm1-undefine 82.4%

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

       3. add-exp-log 82.4%

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

       4. +-commutative 82.4%

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

   11. Applied egg-rr 82.4%

       \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\frac{x}{x + 1} - 1\right)}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1.03 \cdot 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{x}{x + 1} + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+199}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ x (+ x 1.0))) (- n))))
   (if (<= (/ 1.0 n) -1e-10)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-59)
       t_1
       (if (<= (/ 1.0 n) -1e-119)
         (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)
         (if (<= (/ 1.0 n) 0.001)
           t_1
           (if (<= (/ 1.0 n) 1e+199)
             (- 1.0 t_0)
             (/ 0.3333333333333333 (* n (pow x 3.0))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log((x / (x + 1.0))) / -n;
	double tmp;
	if ((1.0 / n) <= -1e-10) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-59) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-119) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else if ((1.0 / n) <= 0.001) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+199) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	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 / (x + 1.0d0))) / -n
    if ((1.0d0 / n) <= (-1d-10)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= (-5d-59)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-1d-119)) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else if ((1.0d0 / n) <= 0.001d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+199) then
        tmp = 1.0d0 - t_0
    else
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    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 / (x + 1.0))) / -n;
	double tmp;
	if ((1.0 / n) <= -1e-10) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-59) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-119) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else if ((1.0 / n) <= 0.001) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+199) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log((x / (x + 1.0))) / -n
	tmp = 0
	if (1.0 / n) <= -1e-10:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-59:
		tmp = t_1
	elif (1.0 / n) <= -1e-119:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	elif (1.0 / n) <= 0.001:
		tmp = t_1
	elif (1.0 / n) <= 1e+199:
		tmp = 1.0 - t_0
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-10)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-59)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -1e-119)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	elseif (Float64(1.0 / n) <= 0.001)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+199)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log((x / (x + 1.0))) / -n;
	tmp = 0.0;
	if ((1.0 / n) <= -1e-10)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= -5e-59)
		tmp = t_1;
	elseif ((1.0 / n) <= -1e-119)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	elseif ((1.0 / n) <= 0.001)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+199)
		tmp = 1.0 - t_0;
	else
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	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[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-10], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-59], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-119], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.001], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+199], N[(1.0 - t$95$0), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-10

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. exp-to-pow 100.0%

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

      9. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -1.00000000000000004e-10 < (/.f64 1 n) < -5.0000000000000001e-59 or -1.00000000000000001e-119 < (/.f64 1 n) < 1e-3

   1. Initial program 31.4%

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

   3. Taylor expanded in n around inf 79.9%

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

      1. log1p-define 79.9%

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

   5. Simplified 79.9%

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

      1. log1p-undefine 79.9%

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

      2. diff-log 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. clear-num 80.1%

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

      2. log-rec 80.1%

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

   9. Applied egg-rr 80.1%

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

   if -5.0000000000000001e-59 < (/.f64 1 n) < -1.00000000000000001e-119

   1. Initial program 12.8%

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

   3. Taylor expanded in n around inf 42.4%

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

      1. log1p-define 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in x around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. metadata-eval 71.4%

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

   8. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{{x}^{2}}}}{n} \]
   9. Step-by-step derivation

      1. unpow2 71.4%

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

   10. Applied egg-rr 71.4%

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

   if 1e-3 < (/.f64 1 n) < 1.0000000000000001e199

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0 60.9%

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

      1. *-rgt-identity 60.9%

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

      2. associate-/l* 60.9%

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

      3. exp-to-pow 60.9%

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

   5. Simplified 60.9%

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

   if 1.0000000000000001e199 < (/.f64 1 n)

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf 6.3%

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

      1. log1p-define 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in x around inf 47.0%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   7. Step-by-step derivation

      1. associate--l+ 47.0%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. associate-*r/ 47.0%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. metadata-eval 47.0%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      4. associate-*r/ 47.0%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{n} \]

      5. metadata-eval 47.0%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{n} \]

   8. Simplified 47.0%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}}{n} \]

   9. Taylor expanded in x around 0 70.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.001:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+199}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -1.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -3.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;n \leq -3.8 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 9.7 \cdot 10^{-215}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 1400:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ x (+ x 1.0))) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= n -1.7e+124)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= n -3.7e+58)
       (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)
       (if (<= n -3.8e-9)
         t_0
         (if (<= n -2e-38)
           t_1
           (if (<= n 9.7e-215)
             (/ 0.3333333333333333 (* n (pow x 3.0)))
             (if (<= n 1400.0) t_1 t_0))))))))

C

double code(double x, double n) {
	double t_0 = log((x / (x + 1.0))) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (n <= -1.7e+124) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if (n <= -3.7e+58) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else if (n <= -3.8e-9) {
		tmp = t_0;
	} else if (n <= -2e-38) {
		tmp = t_1;
	} else if (n <= 9.7e-215) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 1400.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log((x / (x + 1.0d0))) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (n <= (-1.7d+124)) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if (n <= (-3.7d+58)) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else if (n <= (-3.8d-9)) then
        tmp = t_0
    else if (n <= (-2d-38)) then
        tmp = t_1
    else if (n <= 9.7d-215) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 1400.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log((x / (x + 1.0))) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (n <= -1.7e+124) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if (n <= -3.7e+58) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else if (n <= -3.8e-9) {
		tmp = t_0;
	} else if (n <= -2e-38) {
		tmp = t_1;
	} else if (n <= 9.7e-215) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 1400.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((x / (x + 1.0))) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if n <= -1.7e+124:
		tmp = math.log(((x + 1.0) / x)) / n
	elif n <= -3.7e+58:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	elif n <= -3.8e-9:
		tmp = t_0
	elif n <= -2e-38:
		tmp = t_1
	elif n <= 9.7e-215:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 1400.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (n <= -1.7e+124)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (n <= -3.7e+58)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	elseif (n <= -3.8e-9)
		tmp = t_0;
	elseif (n <= -2e-38)
		tmp = t_1;
	elseif (n <= 9.7e-215)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 1400.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log((x / (x + 1.0))) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (n <= -1.7e+124)
		tmp = log(((x + 1.0) / x)) / n;
	elseif (n <= -3.7e+58)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	elseif (n <= -3.8e-9)
		tmp = t_0;
	elseif (n <= -2e-38)
		tmp = t_1;
	elseif (n <= 9.7e-215)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 1400.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.7e+124], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -3.7e+58], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -3.8e-9], t$95$0, If[LessEqual[n, -2e-38], t$95$1, If[LessEqual[n, 9.7e-215], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1400.0], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.7e124

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf 89.1%

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

      1. log1p-define 89.1%

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

   5. Simplified 89.1%

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

      1. log1p-undefine 89.1%

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

      2. diff-log 89.4%

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

   7. Applied egg-rr 89.4%

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

   if -1.7e124 < n < -3.7000000000000002e58

   1. Initial program 23.7%

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

   3. Taylor expanded in n around inf 49.6%

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

      1. log1p-define 49.6%

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

   5. Simplified 49.6%

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

   6. Taylor expanded in x around inf 75.0%

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

      1. associate-*r/ 75.0%

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

      2. metadata-eval 75.0%

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

   8. Simplified 75.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{{x}^{2}}}}{n} \]
   9. Step-by-step derivation

      1. unpow2 75.0%

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

   10. Applied egg-rr 75.0%

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

   if -3.7000000000000002e58 < n < -3.80000000000000011e-9 or 1400 < n

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf 74.3%

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

      1. log1p-define 74.3%

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

   5. Simplified 74.3%

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

      1. log1p-undefine 74.3%

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

      2. diff-log 74.4%

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

   7. Applied egg-rr 74.4%

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

      1. clear-num 74.4%

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

      2. log-rec 74.4%

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

   9. Applied egg-rr 74.4%

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

   if -3.80000000000000011e-9 < n < -1.9999999999999999e-38 or 9.7000000000000001e-215 < n < 1400

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0 68.3%

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

      1. *-rgt-identity 68.3%

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

      2. associate-/l* 68.3%

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

      3. exp-to-pow 68.3%

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

   5. Simplified 68.3%

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

   if -1.9999999999999999e-38 < n < 9.7000000000000001e-215

   1. Initial program 84.1%

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

   3. Taylor expanded in n around inf 50.2%

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

      1. log1p-define 50.2%

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

   5. Simplified 50.2%

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

   6. Taylor expanded in x around inf 27.1%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   7. Step-by-step derivation

      1. associate--l+ 27.1%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. associate-*r/ 27.1%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. metadata-eval 27.1%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      4. associate-*r/ 28.9%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{n} \]

      5. metadata-eval 28.9%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{n} \]

   8. Simplified 28.9%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}}{n} \]

   9. Taylor expanded in x around 0 73.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq -3.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{elif}\;n \leq -3.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-38}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 9.7 \cdot 10^{-215}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 1400:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-242}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-202}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7.2e-242)
   (/ (log x) (- n))
   (if (<= x 1.45e-202)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.96)
       (/ (- x (log x)) n)
       (if (<= x 1.3e+185) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7.2e-242) {
		tmp = log(x) / -n;
	} else if (x <= 1.45e-202) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.96) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.3e+185) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 7.2d-242) then
        tmp = log(x) / -n
    else if (x <= 1.45d-202) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.96d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.3d+185) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 7.2e-242) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.45e-202) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.96) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.3e+185) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7.2e-242:
		tmp = math.log(x) / -n
	elif x <= 1.45e-202:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.96:
		tmp = (x - math.log(x)) / n
	elif x <= 1.3e+185:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7.2e-242)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.45e-202)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.96)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.3e+185)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 7.2e-242)
		tmp = log(x) / -n;
	elseif (x <= 1.45e-202)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.96)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.3e+185)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7.2e-242], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.45e-202], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.96], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.3e+185], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 7.20000000000000028e-242

   1. Initial program 35.9%

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

   3. Taylor expanded in n around inf 69.0%

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

      1. log1p-define 69.0%

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

   5. Simplified 69.0%

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

   6. Taylor expanded in x around 0 69.0%

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

      1. neg-mul-1 69.0%

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

   8. Simplified 69.0%

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

   if 7.20000000000000028e-242 < x < 1.44999999999999994e-202

   1. Initial program 61.3%

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

   3. Taylor expanded in x around 0 61.3%

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

      1. *-rgt-identity 61.3%

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

      2. associate-/l* 61.3%

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

      3. exp-to-pow 61.3%

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

   5. Simplified 61.3%

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

   if 1.44999999999999994e-202 < x < 0.95999999999999996

   1. Initial program 32.5%

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

   3. Taylor expanded in n around inf 57.4%

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

      1. log1p-define 57.4%

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

   5. Simplified 57.4%

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

   6. Taylor expanded in x around 0 56.8%

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

      1. neg-mul-1 56.8%

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

      2. sub-neg 56.8%

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

   8. Simplified 56.8%

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

   if 0.95999999999999996 < x < 1.3e185

   1. Initial program 46.6%

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

   3. Taylor expanded in n around inf 51.6%

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

      1. log1p-define 51.6%

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

   5. Simplified 51.6%

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

   6. Taylor expanded in x around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. metadata-eval 71.4%

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

   8. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{{x}^{2}}}}{n} \]
   9. Step-by-step derivation

      1. unpow2 71.4%

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

   10. Applied egg-rr 71.4%

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

   if 1.3e185 < x

   1. Initial program 86.1%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-define 86.1%

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

   5. Simplified 86.1%

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

      1. log1p-undefine 86.1%

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

      2. diff-log 86.1%

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

   7. Applied egg-rr 86.1%

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

   8. Taylor expanded in x around inf 86.1%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-242}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-202}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.96)
   (/ (- x (log x)) n)
   (if (<= x 1.9e+185) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.96) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.9e+185) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.96d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.9d+185) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.96) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.9e+185) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.96:
		tmp = (x - math.log(x)) / n
	elif x <= 1.9e+185:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.96)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.9e+185)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.96)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.9e+185)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.96], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.9e+185], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.95999999999999996

   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 57.0%

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

      1. log1p-define 57.0%

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

   5. Simplified 57.0%

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

   6. Taylor expanded in x around 0 56.5%

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

      1. neg-mul-1 56.5%

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

      2. sub-neg 56.5%

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

   8. Simplified 56.5%

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

   if 0.95999999999999996 < x < 1.8999999999999999e185

   1. Initial program 46.6%

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

   3. Taylor expanded in n around inf 51.6%

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

      1. log1p-define 51.6%

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

   5. Simplified 51.6%

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

   6. Taylor expanded in x around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. metadata-eval 71.4%

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

   8. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{{x}^{2}}}}{n} \]
   9. Step-by-step derivation

      1. unpow2 71.4%

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

   10. Applied egg-rr 71.4%

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

   if 1.8999999999999999e185 < x

   1. Initial program 86.1%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-define 86.1%

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

   5. Simplified 86.1%

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

      1. log1p-undefine 86.1%

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

      2. diff-log 86.1%

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

   7. Applied egg-rr 86.1%

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

   8. Taylor expanded in x around inf 86.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.68)
   (/ (log x) (- n))
   (if (<= x 1.05e+185) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = log(x) / -n;
	} else if (x <= 1.05e+185) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = log(x) / -n
    else if (x <= 1.05d+185) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.05e+185) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.68:
		tmp = math.log(x) / -n
	elif x <= 1.05e+185:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.05e+185)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = log(x) / -n;
	elseif (x <= 1.05e+185)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.68], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.05e+185], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.680000000000000049

   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 57.0%

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

      1. log1p-define 57.0%

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

   5. Simplified 57.0%

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

   6. Taylor expanded in x around 0 55.4%

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

      1. neg-mul-1 55.4%

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

   8. Simplified 55.4%

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

   if 0.680000000000000049 < x < 1.05e185

   1. Initial program 46.6%

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

   3. Taylor expanded in n around inf 51.6%

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

      1. log1p-define 51.6%

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

   5. Simplified 51.6%

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

   6. Taylor expanded in x around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. metadata-eval 71.4%

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

   8. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{{x}^{2}}}}{n} \]
   9. Step-by-step derivation

      1. unpow2 71.4%

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

   10. Applied egg-rr 71.4%

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

   if 1.05e185 < x

   1. Initial program 86.1%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-define 86.1%

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

   5. Simplified 86.1%

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

      1. log1p-undefine 86.1%

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

      2. diff-log 86.1%

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

   7. Applied egg-rr 86.1%

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

   8. Taylor expanded in x around inf 86.1%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.3% accurate, 17.6× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5.0) (/ 0.0 n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5.0) {
		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) <= (-5.0d0)) 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) <= -5.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5.0:
		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) <= -5.0)
		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) <= -5.0)
		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], -5.0], 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) < -5

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 55.5%

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

      1. log1p-define 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. Step-by-step derivation

      1. log1p-undefine 55.5%

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

      2. diff-log 55.5%

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

   7. Applied egg-rr 55.5%

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

   8. Taylor expanded in x around inf 50.8%

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

   if -5 < (/.f64 1 n)

   1. Initial program 33.6%

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

   3. Taylor expanded in n around inf 62.4%

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

      1. log1p-define 62.4%

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

   5. Simplified 62.4%

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

   6. Taylor expanded in x around inf 46.3%

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

4. Final simplification 47.3%

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

Alternative 11: 41.1% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in n around inf 60.8%

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

   1. log1p-define 60.8%

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

5. Simplified 60.8%

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

6. Taylor expanded in x around inf 41.2%

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

   1. *-commutative 41.2%

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

8. Simplified 41.2%

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

9. Final simplification 41.2%

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

Alternative 12: 41.7% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in n around inf 60.8%

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

   1. log1p-define 60.8%

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

5. Simplified 60.8%

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

6. Taylor expanded in x around inf 41.7%

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

7. Final simplification 41.7%

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

Alternative 13: 4.6% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.7%

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

3. Taylor expanded in x around 0 29.3%

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

4. Taylor expanded in x around inf 4.8%

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

5. Final simplification 4.8%

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

Reproduce

herbie shell --seed 2024046 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))