2nthrt (problem 3.4.6)

Percentage Accurate: 53.9% → 98.3%

Time: 29.3s

Alternatives: 19

Speedup: 1.9×

• could not determine a ground truth

  1. x = -4.540582180596144e-150
  2. n = -1.9123067983566522e+179

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    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: 53.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    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: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{{n}^{-1}}{4}\right)}\\ \mathbf{if}\;n \leq -3600000 \lor \neg \left(n \leq 5600000\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\sqrt{{x}^{\left({n}^{-1}\right)}}\right) \cdot t\_0, t\_0, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ (pow n -1.0) 4.0))))
   (if (or (<= n -3600000.0) (not (<= n 5600000.0)))
     (/ (log1p (pow x -1.0)) n)
     (fma (* (- (sqrt (pow x (pow n -1.0)))) t_0) t_0 (exp (/ (log1p x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (pow(n, -1.0) / 4.0));
	double tmp;
	if ((n <= -3600000.0) || !(n <= 5600000.0)) {
		tmp = log1p(pow(x, -1.0)) / n;
	} else {
		tmp = fma((-sqrt(pow(x, pow(n, -1.0))) * t_0), t_0, exp((log1p(x) / n)));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64((n ^ -1.0) / 4.0)
	tmp = 0.0
	if ((n <= -3600000.0) || !(n <= 5600000.0))
		tmp = Float64(log1p((x ^ -1.0)) / n);
	else
		tmp = fma(Float64(Float64(-sqrt((x ^ (n ^ -1.0)))) * t_0), t_0, exp(Float64(log1p(x) / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(N[Power[n, -1.0], $MachinePrecision] / 4.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[n, -3600000.0], N[Not[LessEqual[n, 5600000.0]], $MachinePrecision]], N[(N[Log[1 + N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[((-N[Sqrt[N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]) * t$95$0), $MachinePrecision] * t$95$0 + N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.6e6 or 5.6e6 < n

   1. Initial program 29.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 99.3%

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

   if -3.6e6 < n < 5.6e6

   1. Initial program 84.8%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. sqr-pow N/A

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

      7. lift-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. sqr-pow N/A

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

      11. distribute-lft-neg-in N/A

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

      12. sqr-pow N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right) \cdot {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}, {x}^{\left(\frac{\frac{\frac{1}{n}}{2}}{2}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)} \]

   4. Applied rewrites 95.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3600000 \lor \neg \left(n \leq 5600000\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\sqrt{{x}^{\left({n}^{-1}\right)}}\right) \cdot {x}^{\left(\frac{{n}^{-1}}{4}\right)}, {x}^{\left(\frac{{n}^{-1}}{4}\right)}, e^{\frac{\mathsf{log1p}\left(x\right)}{n}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -20.0)
   (/ 0.3333333333333333 (* (pow x 3.0) n))
   (if (<= (pow n -1.0) 5e-12)
     (/ (log1p (pow x -1.0)) n)
     (-
      (fma (fma (/ (+ -0.5 (/ 0.5 n)) n) x (pow n -1.0)) x 1.0)
      (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -20.0) {
		tmp = 0.3333333333333333 / (pow(x, 3.0) * n);
	} else if (pow(n, -1.0) <= 5e-12) {
		tmp = log1p(pow(x, -1.0)) / n;
	} else {
		tmp = fma(fma(((-0.5 + (0.5 / n)) / n), x, pow(n, -1.0)), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -20.0)
		tmp = Float64(0.3333333333333333 / Float64((x ^ 3.0) * n));
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = Float64(log1p((x ^ -1.0)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(-0.5 + Float64(0.5 / n)) / n), x, (n ^ -1.0)), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -20.0], N[(0.3333333333333333 / N[(N[Power[x, 3.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-12], N[(N[Log[1 + N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -20

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{3}}{n \cdot \color{blue}{{x}^{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.3%

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

   if -20 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 30.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 98.0%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 71.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{-n}\right) - 1}{-n}, x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))))
   (if (<= (pow n -1.0) -1e-5)
     (- (pow (+ x 1.0) (pow n -1.0)) t_0)
     (if (<= (pow n -1.0) 5e-12)
       (/ (log1p (pow x -1.0)) n)
       (-
        (fma
         (/
          (-
           (fma
            (fma -0.3333333333333333 x 0.5)
            x
            (/
             (fma (/ (* x x) n) 0.16666666666666666 (* (fma -0.5 x 0.5) x))
             (- n)))
           1.0)
          (- n))
         x
         1.0)
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double tmp;
	if (pow(n, -1.0) <= -1e-5) {
		tmp = pow((x + 1.0), pow(n, -1.0)) - t_0;
	} else if (pow(n, -1.0) <= 5e-12) {
		tmp = log1p(pow(x, -1.0)) / n;
	} else {
		tmp = fma(((fma(fma(-0.3333333333333333, x, 0.5), x, (fma(((x * x) / n), 0.16666666666666666, (fma(-0.5, x, 0.5) * x)) / -n)) - 1.0) / -n), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	tmp = 0.0
	if ((n ^ -1.0) <= -1e-5)
		tmp = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0);
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = Float64(log1p((x ^ -1.0)) / n);
	else
		tmp = Float64(fma(Float64(Float64(fma(fma(-0.3333333333333333, x, 0.5), x, Float64(fma(Float64(Float64(x * x) / n), 0.16666666666666666, Float64(fma(-0.5, x, 0.5) * x)) / Float64(-n))) - 1.0) / Float64(-n)), x, 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1e-5], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-12], N[(N[Log[1 + N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] * x + N[(N[(N[(N[(x * x), $MachinePrecision] / n), $MachinePrecision] * 0.16666666666666666 + N[(N[(-0.5 * x + 0.5), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / (-n)), $MachinePrecision] * x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000008e-5

   1. Initial program 99.1%

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

   if -1.00000000000000008e-5 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 29.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 99.3%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around -inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{2}}{n} + x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n} + x \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot x\right)\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.2%

      \[\leadsto \mathsf{fma}\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -\frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{n}\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1 \cdot 10^{-5}:\\ \;\;\;\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{-n}\right) - 1}{-n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{-n}\right) - 1}{-n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -1.6e-5)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (pow n -1.0) 5e-12)
     (/ (log1p (pow x -1.0)) n)
     (-
      (fma
       (/
        (-
         (fma
          (fma -0.3333333333333333 x 0.5)
          x
          (/
           (fma (/ (* x x) n) 0.16666666666666666 (* (fma -0.5 x 0.5) x))
           (- n)))
         1.0)
        (- n))
       x
       1.0)
      (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -1.6e-5) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if (pow(n, -1.0) <= 5e-12) {
		tmp = log1p(pow(x, -1.0)) / n;
	} else {
		tmp = fma(((fma(fma(-0.3333333333333333, x, 0.5), x, (fma(((x * x) / n), 0.16666666666666666, (fma(-0.5, x, 0.5) * x)) / -n)) - 1.0) / -n), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1.6e-5)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = Float64(log1p((x ^ -1.0)) / n);
	else
		tmp = Float64(fma(Float64(Float64(fma(fma(-0.3333333333333333, x, 0.5), x, Float64(fma(Float64(Float64(x * x) / n), 0.16666666666666666, Float64(fma(-0.5, x, 0.5) * x)) / Float64(-n))) - 1.0) / Float64(-n)), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1.6e-5], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-12], N[(N[Log[1 + N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] * x + N[(N[(N[(N[(x * x), $MachinePrecision] / n), $MachinePrecision] * 0.16666666666666666 + N[(N[(-0.5 * x + 0.5), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / (-n)), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.59999999999999993e-5

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1.59999999999999993e-5 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 98.4%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around -inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{2}}{n} + x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n} + x \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot x\right)\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.2%

      \[\leadsto \mathsf{fma}\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -\frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{n}\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{-n}\right) - 1}{-n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{-n}\right) - 1}{-n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -20.0)
   (/ 0.3333333333333333 (* (pow x 3.0) n))
   (if (<= (pow n -1.0) 5e-12)
     (/ (log1p (pow x -1.0)) n)
     (-
      (fma
       (/
        (-
         (fma
          (fma -0.3333333333333333 x 0.5)
          x
          (/
           (fma (/ (* x x) n) 0.16666666666666666 (* (fma -0.5 x 0.5) x))
           (- n)))
         1.0)
        (- n))
       x
       1.0)
      (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -20.0) {
		tmp = 0.3333333333333333 / (pow(x, 3.0) * n);
	} else if (pow(n, -1.0) <= 5e-12) {
		tmp = log1p(pow(x, -1.0)) / n;
	} else {
		tmp = fma(((fma(fma(-0.3333333333333333, x, 0.5), x, (fma(((x * x) / n), 0.16666666666666666, (fma(-0.5, x, 0.5) * x)) / -n)) - 1.0) / -n), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -20.0)
		tmp = Float64(0.3333333333333333 / Float64((x ^ 3.0) * n));
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = Float64(log1p((x ^ -1.0)) / n);
	else
		tmp = Float64(fma(Float64(Float64(fma(fma(-0.3333333333333333, x, 0.5), x, Float64(fma(Float64(Float64(x * x) / n), 0.16666666666666666, Float64(fma(-0.5, x, 0.5) * x)) / Float64(-n))) - 1.0) / Float64(-n)), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -20.0], N[(0.3333333333333333 / N[(N[Power[x, 3.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-12], N[(N[Log[1 + N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] * x + N[(N[(N[(N[(x * x), $MachinePrecision] / n), $MachinePrecision] * 0.16666666666666666 + N[(N[(-0.5 * x + 0.5), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / (-n)), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -20

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{3}}{n \cdot \color{blue}{{x}^{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.3%

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

   if -20 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 30.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 98.0%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around -inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{2}}{n} + x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n} + x \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot x\right)\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.2%

      \[\leadsto \mathsf{fma}\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -\frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{n}\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{-n}\right) - 1}{-n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{-0.16666666666666666}{n} \cdot \frac{x \cdot x}{n}\right) - 1}{-n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -20.0)
   (/ 0.3333333333333333 (* (pow x 3.0) n))
   (if (<= (pow n -1.0) 5e-12)
     (/ (log1p (pow x -1.0)) n)
     (-
      (fma
       (/
        (-
         (fma
          (fma -0.3333333333333333 x 0.5)
          x
          (* (/ -0.16666666666666666 n) (/ (* x x) n)))
         1.0)
        (- n))
       x
       1.0)
      (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -20.0) {
		tmp = 0.3333333333333333 / (pow(x, 3.0) * n);
	} else if (pow(n, -1.0) <= 5e-12) {
		tmp = log1p(pow(x, -1.0)) / n;
	} else {
		tmp = fma(((fma(fma(-0.3333333333333333, x, 0.5), x, ((-0.16666666666666666 / n) * ((x * x) / n))) - 1.0) / -n), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -20.0)
		tmp = Float64(0.3333333333333333 / Float64((x ^ 3.0) * n));
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = Float64(log1p((x ^ -1.0)) / n);
	else
		tmp = Float64(fma(Float64(Float64(fma(fma(-0.3333333333333333, x, 0.5), x, Float64(Float64(-0.16666666666666666 / n) * Float64(Float64(x * x) / n))) - 1.0) / Float64(-n)), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -20.0], N[(0.3333333333333333 / N[(N[Power[x, 3.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-12], N[(N[Log[1 + N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * x + 0.5), $MachinePrecision] * x + N[(N[(-0.16666666666666666 / n), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / (-n)), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -20

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{3}}{n \cdot \color{blue}{{x}^{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.3%

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

   if -20 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 30.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 98.0%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}, x, \frac{-0.5 + \frac{0.5}{n}}{n}\right), x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around -inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{2}}{n} + x \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)}{n} + x \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot x\right)\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.2%

      \[\leadsto \mathsf{fma}\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, -\frac{\mathsf{fma}\left(\frac{x \cdot x}{n}, 0.16666666666666666, \mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x\right)}{n}\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   9. Taylor expanded in n around 0

      \[\leadsto \mathsf{fma}\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, x, \frac{1}{2}\right), x, \frac{-1}{6} \cdot \frac{{x}^{2}}{{n}^{2}}\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.7%

       \[\leadsto \mathsf{fma}\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{-0.16666666666666666}{n} \cdot \frac{x \cdot x}{n}\right) - 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 0.5\right), x, \frac{-0.16666666666666666}{n} \cdot \frac{x \cdot x}{n}\right) - 1}{-n}, x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -20.0)
   (/ 0.3333333333333333 (* (pow x 3.0) n))
   (if (<= (pow n -1.0) 5e-12)
     (/ (log1p (pow x -1.0)) n)
     (- (+ (/ x n) 1.0) (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -20.0) {
		tmp = 0.3333333333333333 / (pow(x, 3.0) * n);
	} else if (pow(n, -1.0) <= 5e-12) {
		tmp = log1p(pow(x, -1.0)) / n;
	} else {
		tmp = ((x / n) + 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -20.0) {
		tmp = 0.3333333333333333 / (Math.pow(x, 3.0) * n);
	} else if (Math.pow(n, -1.0) <= 5e-12) {
		tmp = Math.log1p(Math.pow(x, -1.0)) / n;
	} else {
		tmp = ((x / n) + 1.0) - Math.pow(x, Math.pow(n, -1.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -20.0:
		tmp = 0.3333333333333333 / (math.pow(x, 3.0) * n)
	elif math.pow(n, -1.0) <= 5e-12:
		tmp = math.log1p(math.pow(x, -1.0)) / n
	else:
		tmp = ((x / n) + 1.0) - math.pow(x, math.pow(n, -1.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -20.0)
		tmp = Float64(0.3333333333333333 / Float64((x ^ 3.0) * n));
	elseif ((n ^ -1.0) <= 5e-12)
		tmp = Float64(log1p((x ^ -1.0)) / n);
	else
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -20.0], N[(0.3333333333333333 / N[(N[Power[x, 3.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 5e-12], N[(N[Log[1 + N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -20

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{3}}{n \cdot \color{blue}{{x}^{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.3%

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

   if -20 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 30.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 98.0%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 53.1

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

   5. Applied rewrites 53.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;{n}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -240000000 \lor \neg \left(n \leq 7000000\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -240000000.0) (not (<= n 7000000.0)))
   (/ (log1p (pow x -1.0)) n)
   (- (exp (/ (log1p x) n)) (pow x (pow n -1.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -240000000.0) || !(n <= 7000000.0)) {
		tmp = log1p(pow(x, -1.0)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -240000000.0) || !(n <= 7000000.0)) {
		tmp = Math.log1p(Math.pow(x, -1.0)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, Math.pow(n, -1.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -240000000.0) or not (n <= 7000000.0):
		tmp = math.log1p(math.pow(x, -1.0)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, math.pow(n, -1.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -240000000.0) || !(n <= 7000000.0))
		tmp = Float64(log1p((x ^ -1.0)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -240000000.0], N[Not[LessEqual[n, 7000000.0]], $MachinePrecision]], N[(N[Log[1 + N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.4e8 or 7e6 < n

   1. Initial program 29.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 99.3%

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

   if -2.4e8 < n < 7e6

   1. Initial program 84.8%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 95.7

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

   4. Applied rewrites 95.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -240000000 \lor \neg \left(n \leq 7000000\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{if}\;n \leq -10:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-156}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;n \leq 1800000:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log1p (pow x -1.0)) n)))
   (if (<= n -10.0)
     t_0
     (if (<= n 7e-156)
       (/ 0.3333333333333333 (* (pow x 3.0) n))
       (if (<= n 1800000.0) (- 1.0 (pow x (pow n -1.0))) t_0)))))

C

double code(double x, double n) {
	double t_0 = log1p(pow(x, -1.0)) / n;
	double tmp;
	if (n <= -10.0) {
		tmp = t_0;
	} else if (n <= 7e-156) {
		tmp = 0.3333333333333333 / (pow(x, 3.0) * n);
	} else if (n <= 1800000.0) {
		tmp = 1.0 - pow(x, pow(n, -1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p(Math.pow(x, -1.0)) / n;
	double tmp;
	if (n <= -10.0) {
		tmp = t_0;
	} else if (n <= 7e-156) {
		tmp = 0.3333333333333333 / (Math.pow(x, 3.0) * n);
	} else if (n <= 1800000.0) {
		tmp = 1.0 - Math.pow(x, Math.pow(n, -1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p(math.pow(x, -1.0)) / n
	tmp = 0
	if n <= -10.0:
		tmp = t_0
	elif n <= 7e-156:
		tmp = 0.3333333333333333 / (math.pow(x, 3.0) * n)
	elif n <= 1800000.0:
		tmp = 1.0 - math.pow(x, math.pow(n, -1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log1p((x ^ -1.0)) / n)
	tmp = 0.0
	if (n <= -10.0)
		tmp = t_0;
	elseif (n <= 7e-156)
		tmp = Float64(0.3333333333333333 / Float64((x ^ 3.0) * n));
	elseif (n <= 1800000.0)
		tmp = Float64(1.0 - (x ^ (n ^ -1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[1 + N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -10.0], t$95$0, If[LessEqual[n, 7e-156], N[(0.3333333333333333 / N[(N[Power[x, 3.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1800000.0], N[(1.0 - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -10 or 1.8e6 < n

   1. Initial program 30.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 98.0%

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

   if -10 < n < 6.9999999999999999e-156

   1. Initial program 89.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 19.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{3}}{n \cdot \color{blue}{{x}^{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.4%

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

   if 6.9999999999999999e-156 < n < 1.8e6

   1. Initial program 68.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -10:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-156}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \mathbf{elif}\;n \leq 1800000:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left({x}^{-1}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -20.0)
   (/ (/ (/ (fma -0.5 x 0.3333333333333333) n) (* x x)) x)
   (/ (pow n -1.0) x)))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -20.0) {
		tmp = ((fma(-0.5, x, 0.3333333333333333) / n) / (x * x)) / x;
	} else {
		tmp = pow(n, -1.0) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -20.0)
		tmp = Float64(Float64(Float64(fma(-0.5, x, 0.3333333333333333) / n) / Float64(x * x)) / x);
	else
		tmp = Float64((n ^ -1.0) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -20.0], N[(N[(N[(N[(-0.5 * x + 0.3333333333333333), $MachinePrecision] / n), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -20

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2} \cdot \frac{x}{n} + \frac{1}{3} \cdot \frac{1}{n}}{{x}^{2}}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 44.8%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x} \]

   if -20 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.3%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -20.0)
   (/ (/ (/ (fma -0.5 x 0.3333333333333333) n) (* x x)) x)
   (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -20.0) {
		tmp = ((fma(-0.5, x, 0.3333333333333333) / n) / (x * x)) / x;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -20.0)
		tmp = Float64(Float64(Float64(fma(-0.5, x, 0.3333333333333333) / n) / Float64(x * x)) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -20.0], N[(N[(N[(N[(-0.5 * x + 0.3333333333333333), $MachinePrecision] / n), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -20

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2} \cdot \frac{x}{n} + \frac{1}{3} \cdot \frac{1}{n}}{{x}^{2}}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 44.8%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x} \]

   if -20 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}}{x} + \frac{-0.5 + \frac{0.5}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.6%

      \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} - -1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -20.0)
   (/ (/ (/ (fma -0.5 x 0.3333333333333333) n) (* x x)) x)
   (/ (/ (- (/ (- (/ 0.3333333333333333 x) 0.5) x) -1.0) x) n)))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -20.0) {
		tmp = ((fma(-0.5, x, 0.3333333333333333) / n) / (x * x)) / x;
	} else {
		tmp = (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -20.0)
		tmp = Float64(Float64(Float64(fma(-0.5, x, 0.3333333333333333) / n) / Float64(x * x)) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -20.0], N[(N[(N[(N[(-0.5 * x + 0.3333333333333333), $MachinePrecision] / n), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -20

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2} \cdot \frac{x}{n} + \frac{1}{3} \cdot \frac{1}{n}}{{x}^{2}}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 44.8%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x} \]

   if -20 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

      \[\leadsto \frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} - -1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -20.0)
   (/ (/ (/ (fma -0.5 x 0.3333333333333333) n) (* x x)) x)
   (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) (* n x))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -20.0) {
		tmp = ((fma(-0.5, x, 0.3333333333333333) / n) / (x * x)) / x;
	} else {
		tmp = (((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -20.0)
		tmp = Float64(Float64(Float64(fma(-0.5, x, 0.3333333333333333) / n) / Float64(x * x)) / x);
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -20.0], N[(N[(N[(N[(-0.5 * x + 0.3333333333333333), $MachinePrecision] / n), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -20

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{-1}{2} \cdot \frac{x}{n} + \frac{1}{3} \cdot \frac{1}{n}}{{x}^{2}}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 44.8%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x} \]

   if -20 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}}{x} + \frac{-0.5 + \frac{0.5}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 46.1%

      \[\leadsto \frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{\color{blue}{n \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -20:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-0.5, x, 0.3333333333333333\right)}{n}}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+204}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \left(\frac{0.5}{\left(x \cdot x\right) \cdot n} + \frac{0.5}{x}\right)}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{3} \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.35e-5)
   (/ (- x (log x)) n)
   (if (<= x 3.1e+204)
     (/
      (/
       (-
        (+ (/ 0.3333333333333333 (* x x)) 1.0)
        (+ (/ 0.5 (* (* x x) n)) (/ 0.5 x)))
       n)
      x)
     (/ 0.3333333333333333 (* (pow x 3.0) n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.35e-5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 3.1e+204) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / ((x * x) * n)) + (0.5 / x))) / n) / x;
	} else {
		tmp = 0.3333333333333333 / (pow(x, 3.0) * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.35d-5) then
        tmp = (x - log(x)) / n
    else if (x <= 3.1d+204) then
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - ((0.5d0 / ((x * x) * n)) + (0.5d0 / x))) / n) / x
    else
        tmp = 0.3333333333333333d0 / ((x ** 3.0d0) * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.35e-5) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 3.1e+204) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / ((x * x) * n)) + (0.5 / x))) / n) / x;
	} else {
		tmp = 0.3333333333333333 / (Math.pow(x, 3.0) * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.35e-5:
		tmp = (x - math.log(x)) / n
	elif x <= 3.1e+204:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / ((x * x) * n)) + (0.5 / x))) / n) / x
	else:
		tmp = 0.3333333333333333 / (math.pow(x, 3.0) * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.35e-5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 3.1e+204)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(Float64(0.5 / Float64(Float64(x * x) * n)) + Float64(0.5 / x))) / n) / x);
	else
		tmp = Float64(0.3333333333333333 / Float64((x ^ 3.0) * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.35e-5)
		tmp = (x - log(x)) / n;
	elseif (x <= 3.1e+204)
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / ((x * x) * n)) + (0.5 / x))) / n) / x;
	else
		tmp = 0.3333333333333333 / ((x ^ 3.0) * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.35e-5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3.1e+204], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(0.5 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(0.3333333333333333 / N[(N[Power[x, 3.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.34999999999999986e-5

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.8%

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

   if 2.34999999999999986e-5 < x < 3.1000000000000002e204

   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 x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}}{x} + \frac{-0.5 + \frac{0.5}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{\frac{1}{3}}{n}}{x} + \frac{\frac{-1}{2} + \frac{\frac{1}{2}}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.2%

      \[\leadsto \frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{0.3333333333333333}{n}}{x} + \frac{-0.5 + \frac{0.5}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x} \]

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 64.2%

       \[\leadsto \frac{\frac{\left(\left(\mathsf{fma}\left(\frac{\log x}{x}, \frac{\frac{0.3333333333333333}{x} - 0.5}{n}, \frac{\log x}{n}\right) + \frac{\frac{0.5}{n}}{x}\right) + \left(\frac{0.3333333333333333}{x \cdot x} + 1\right)\right) - \left(\frac{0.5}{\left(x \cdot x\right) \cdot n} + \frac{0.5}{x}\right)}{n}}{x} \]

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 63.3%

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

   if 3.1000000000000002e204 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{3}}{n \cdot \color{blue}{{x}^{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.8%

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

Alternative 15: 58.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot n\\ \mathbf{if}\;x \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \left(\frac{0.5}{t\_0} + \frac{0.5}{x}\right)}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{t\_0} - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* (* x x) n)))
   (if (<= x 2.35e-5)
     (/ (- x (log x)) n)
     (if (<= x 1.1e+205)
       (/
        (/
         (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (+ (/ 0.5 t_0) (/ 0.5 x)))
         n)
        x)
       (/ (- (/ 0.3333333333333333 t_0) (/ (/ 0.5 n) x)) x)))))

C

double code(double x, double n) {
	double t_0 = (x * x) * n;
	double tmp;
	if (x <= 2.35e-5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.1e+205) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / t_0) + (0.5 / x))) / n) / x;
	} else {
		tmp = ((0.3333333333333333 / t_0) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * n
    if (x <= 2.35d-5) then
        tmp = (x - log(x)) / n
    else if (x <= 1.1d+205) then
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - ((0.5d0 / t_0) + (0.5d0 / x))) / n) / x
    else
        tmp = ((0.3333333333333333d0 / t_0) - ((0.5d0 / n) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (x * x) * n;
	double tmp;
	if (x <= 2.35e-5) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.1e+205) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / t_0) + (0.5 / x))) / n) / x;
	} else {
		tmp = ((0.3333333333333333 / t_0) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (x * x) * n
	tmp = 0
	if x <= 2.35e-5:
		tmp = (x - math.log(x)) / n
	elif x <= 1.1e+205:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / t_0) + (0.5 / x))) / n) / x
	else:
		tmp = ((0.3333333333333333 / t_0) - ((0.5 / n) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(x * x) * n)
	tmp = 0.0
	if (x <= 2.35e-5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.1e+205)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(Float64(0.5 / t_0) + Float64(0.5 / x))) / n) / x);
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 / t_0) - Float64(Float64(0.5 / n) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x * x) * n;
	tmp = 0.0;
	if (x <= 2.35e-5)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.1e+205)
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / t_0) + (0.5 / x))) / n) / x;
	else
		tmp = ((0.3333333333333333 / t_0) - ((0.5 / n) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[x, 2.35e-5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.1e+205], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(0.5 / t$95$0), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(0.3333333333333333 / t$95$0), $MachinePrecision] - N[(N[(0.5 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.34999999999999986e-5

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.8%

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

   if 2.34999999999999986e-5 < x < 1.0999999999999999e205

   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 x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}}{x} + \frac{-0.5 + \frac{0.5}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{\frac{1}{3}}{n}}{x} + \frac{\frac{-1}{2} + \frac{\frac{1}{2}}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.2%

      \[\leadsto \frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{0.3333333333333333}{n}}{x} + \frac{-0.5 + \frac{0.5}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x} \]

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 64.2%

       \[\leadsto \frac{\frac{\left(\left(\mathsf{fma}\left(\frac{\log x}{x}, \frac{\frac{0.3333333333333333}{x} - 0.5}{n}, \frac{\log x}{n}\right) + \frac{\frac{0.5}{n}}{x}\right) + \left(\frac{0.3333333333333333}{x \cdot x} + 1\right)\right) - \left(\frac{0.5}{\left(x \cdot x\right) \cdot n} + \frac{0.5}{x}\right)}{n}}{x} \]

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 63.3%

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

   if 1.0999999999999999e205 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{1}{3}}{n \cdot {x}^{2}} - \frac{\frac{\frac{1}{2}}{n}}{x}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.6%

       \[\leadsto \frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} - \frac{\frac{0.5}{n}}{x}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot n\\ \mathbf{if}\;x \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \left(\frac{0.5}{t\_0} + \frac{0.5}{x}\right)}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{t\_0} - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (* (* x x) n)))
   (if (<= x 1.8e-5)
     (/ (- (log x)) n)
     (if (<= x 1.1e+205)
       (/
        (/
         (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (+ (/ 0.5 t_0) (/ 0.5 x)))
         n)
        x)
       (/ (- (/ 0.3333333333333333 t_0) (/ (/ 0.5 n) x)) x)))))

C

double code(double x, double n) {
	double t_0 = (x * x) * n;
	double tmp;
	if (x <= 1.8e-5) {
		tmp = -log(x) / n;
	} else if (x <= 1.1e+205) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / t_0) + (0.5 / x))) / n) / x;
	} else {
		tmp = ((0.3333333333333333 / t_0) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * n
    if (x <= 1.8d-5) then
        tmp = -log(x) / n
    else if (x <= 1.1d+205) then
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - ((0.5d0 / t_0) + (0.5d0 / x))) / n) / x
    else
        tmp = ((0.3333333333333333d0 / t_0) - ((0.5d0 / n) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (x * x) * n;
	double tmp;
	if (x <= 1.8e-5) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.1e+205) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / t_0) + (0.5 / x))) / n) / x;
	} else {
		tmp = ((0.3333333333333333 / t_0) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (x * x) * n
	tmp = 0
	if x <= 1.8e-5:
		tmp = -math.log(x) / n
	elif x <= 1.1e+205:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / t_0) + (0.5 / x))) / n) / x
	else:
		tmp = ((0.3333333333333333 / t_0) - ((0.5 / n) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(x * x) * n)
	tmp = 0.0
	if (x <= 1.8e-5)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.1e+205)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(Float64(0.5 / t_0) + Float64(0.5 / x))) / n) / x);
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 / t_0) - Float64(Float64(0.5 / n) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x * x) * n;
	tmp = 0.0;
	if (x <= 1.8e-5)
		tmp = -log(x) / n;
	elseif (x <= 1.1e+205)
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - ((0.5 / t_0) + (0.5 / x))) / n) / x;
	else
		tmp = ((0.3333333333333333 / t_0) - ((0.5 / n) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[x, 1.8e-5], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.1e+205], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(0.5 / t$95$0), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(0.3333333333333333 / t$95$0), $MachinePrecision] - N[(N[(0.5 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.80000000000000005e-5

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.5%

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

   if 1.80000000000000005e-5 < x < 1.0999999999999999e205

   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 x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}}{x} + \frac{-0.5 + \frac{0.5}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{\frac{1}{3}}{n}}{x} + \frac{\frac{-1}{2} + \frac{\frac{1}{2}}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.2%

      \[\leadsto \frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{0.3333333333333333}{n}}{x} + \frac{-0.5 + \frac{0.5}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x} \]

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 64.2%

       \[\leadsto \frac{\frac{\left(\left(\mathsf{fma}\left(\frac{\log x}{x}, \frac{\frac{0.3333333333333333}{x} - 0.5}{n}, \frac{\log x}{n}\right) + \frac{\frac{0.5}{n}}{x}\right) + \left(\frac{0.3333333333333333}{x \cdot x} + 1\right)\right) - \left(\frac{0.5}{\left(x \cdot x\right) \cdot n} + \frac{0.5}{x}\right)}{n}}{x} \]

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 63.3%

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

   if 1.0999999999999999e205 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{1}{3}}{n \cdot {x}^{2}} - \frac{\frac{\frac{1}{2}}{n}}{x}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.6%

       \[\leadsto \frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} - \frac{\frac{0.5}{n}}{x}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 40.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-log.f64 60.2

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

5. Applied rewrites 60.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 39.1%

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

9. Final simplification 39.1%

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

Alternative 18: 39.8% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x n) :precision binary64 (pow (* n x) -1.0))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-log.f64 60.2

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

5. Applied rewrites 60.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 39.1%

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

10. Applied rewrites 38.8%

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

11. Final simplification 38.8%

    \[\leadsto {\left(n \cdot x\right)}^{-1} \]
12. Add Preprocessing

Alternative 19: 47.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.1e+205)
   (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)
   (/ (- (/ 0.3333333333333333 (* (* x x) n)) (/ (/ 0.5 n) x)) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.1e+205) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = ((0.3333333333333333 / ((x * x) * n)) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.1d+205) then
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
    else
        tmp = ((0.3333333333333333d0 / ((x * x) * n)) - ((0.5d0 / n) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.1e+205) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = ((0.3333333333333333 / ((x * x) * n)) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.1e+205:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x
	else:
		tmp = ((0.3333333333333333 / ((x * x) * n)) - ((0.5 / n) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.1e+205)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) - Float64(Float64(0.5 / n) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.1e+205)
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	else
		tmp = ((0.3333333333333333 / ((x * x) * n)) - ((0.5 / n) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.1e+205], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0999999999999999e205

   1. Initial program 45.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{\frac{\log x}{n}}, \frac{\frac{\frac{0.16666666666666666}{{n}^{3}} - \frac{-0.3333333333333333 + \frac{0.5}{n}}{n}}{x} + \frac{-0.5 + \frac{0.5}{n}}{n}}{x}, \frac{e^{\frac{\log x}{n}}}{n}\right)}{x}} \]

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 39.6%

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

   if 1.0999999999999999e205 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{1}{3}}{n \cdot {x}^{2}} - \frac{\frac{\frac{1}{2}}{n}}{x}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.6%

       \[\leadsto \frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} - \frac{\frac{0.5}{n}}{x}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2024347 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))