2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 81.1%

Time: 15.2s

Alternatives: 20

Speedup: 2.9×

• could not determine a ground truth

  1. x = -4.10205790412559e+53
  2. n = -1.5159546432483987e+254

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: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\log x}{n}}{-2}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left(e^{-2}\right)}^{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{{\left(e^{-1 \cdot t\_0}\right)}^{2}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x \cdot x}{n \cdot n}, 0.16666666666666666, \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x, \frac{\mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x}{n}\right)\right) + 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ (log x) n) -2.0)))
   (if (<= (/ 1.0 n) -1e-24)
     (/ (pow (exp -2.0) t_0) (* n x))
     (if (<= (/ 1.0 n) 5e-166)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 0.2)
         (/ (pow (exp (* -1.0 t_0)) 2.0) (* n x))
         (-
          (fma
           (/
            (+
             (fma
              (/ (* x x) (* n n))
              0.16666666666666666
              (fma
               (- (* 0.3333333333333333 x) 0.5)
               x
               (/ (* (fma -0.5 x 0.5) x) n)))
             1.0)
            n)
           x
           1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = (log(x) / n) / -2.0;
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = pow(exp(-2.0), t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = pow(exp((-1.0 * t_0)), 2.0) / (n * x);
	} else {
		tmp = fma(((fma(((x * x) / (n * n)), 0.16666666666666666, fma(((0.3333333333333333 * x) - 0.5), x, ((fma(-0.5, x, 0.5) * x) / n))) + 1.0) / n), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(log(x) / n) / -2.0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64((exp(-2.0) ^ t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.2)
		tmp = Float64((exp(Float64(-1.0 * t_0)) ^ 2.0) / Float64(n * x));
	else
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(x * x) / Float64(n * n)), 0.16666666666666666, fma(Float64(Float64(0.3333333333333333 * x) - 0.5), x, Float64(Float64(fma(-0.5, x, 0.5) * x) / n))) + 1.0) / n), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] / -2.0), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[Power[N[Exp[-2.0], $MachinePrecision], t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.2], N[(N[Power[N[Exp[N[(-1.0 * t$95$0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * x + N[(N[(N[(-0.5 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-log N/A

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

      7. mul-1-neg N/A

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

      8. exp-prod N/A

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

      9. neg-log N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-log.f64 96.4

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

   6. Applied rewrites 96.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      14. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. lower-neg.f64 96.4

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

   8. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. rem-log-exp N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. pow-exp N/A

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

      9. rem-log-exp N/A

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

      10. metadata-eval 96.4

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift-neg.f64 N/A

          \[\leadsto \frac{{\left(e^{-2}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. frac-2neg N/A

          \[\leadsto \frac{{\left(e^{-2}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{\mathsf{neg}\left(n\right)}\right)}{\mathsf{neg}\left(2\right)}\right)}}{n \cdot x} \]

   10. Applied rewrites 96.4%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 5e-166

   1. Initial program 33.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if 5e-166 < (/.f64 #s(literal 1 binary64) n) < 0.20000000000000001

   1. Initial program 20.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-log N/A

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

      7. mul-1-neg N/A

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

      8. exp-prod N/A

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

      9. neg-log N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-log.f64 50.2

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

   6. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      14. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. lower-neg.f64 50.2

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

   8. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      7. unpow-prod-down N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

   10. Applied rewrites 50.2%

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

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 61.5%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{x \cdot x}{n \cdot n}, 0.16666666666666666, \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x, \frac{\mathsf{fma}\left(-0.5, x, 0.5\right) \cdot x}{n}\right)\right) + 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\log x}{n}}{-2}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left(e^{-2}\right)}^{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{{\left(e^{-1 \cdot t\_0}\right)}^{2}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-\frac{\mathsf{fma}\left(-0.3333333333333333, x, -\frac{\mathsf{fma}\left(-0.5, x, \frac{x}{n} \cdot 0.16666666666666666\right) + 0.5}{n}\right) + 0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ (log x) n) -2.0)))
   (if (<= (/ 1.0 n) -1e-24)
     (/ (pow (exp -2.0) t_0) (* n x))
     (if (<= (/ 1.0 n) 5e-166)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 0.2)
         (/ (pow (exp (* -1.0 t_0)) 2.0) (* n x))
         (-
          (fma
           (fma
            (-
             (/
              (+
               (fma
                -0.3333333333333333
                x
                (- (/ (+ (fma -0.5 x (* (/ x n) 0.16666666666666666)) 0.5) n)))
               0.5)
              n))
            x
            (/ 1.0 n))
           x
           1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = (log(x) / n) / -2.0;
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = pow(exp(-2.0), t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = pow(exp((-1.0 * t_0)), 2.0) / (n * x);
	} else {
		tmp = fma(fma(-((fma(-0.3333333333333333, x, -((fma(-0.5, x, ((x / n) * 0.16666666666666666)) + 0.5) / n)) + 0.5) / n), x, (1.0 / n)), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(log(x) / n) / -2.0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64((exp(-2.0) ^ t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.2)
		tmp = Float64((exp(Float64(-1.0 * t_0)) ^ 2.0) / Float64(n * x));
	else
		tmp = Float64(fma(fma(Float64(-Float64(Float64(fma(-0.3333333333333333, x, Float64(-Float64(Float64(fma(-0.5, x, Float64(Float64(x / n) * 0.16666666666666666)) + 0.5) / n))) + 0.5) / n)), x, Float64(1.0 / n)), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] / -2.0), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[Power[N[Exp[-2.0], $MachinePrecision], t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.2], N[(N[Power[N[Exp[N[(-1.0 * t$95$0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[(N[(N[(-0.3333333333333333 * x + (-N[(N[(N[(-0.5 * x + N[(N[(x / n), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / n), $MachinePrecision])), $MachinePrecision] + 0.5), $MachinePrecision] / n), $MachinePrecision]) * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-log N/A

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

      7. mul-1-neg N/A

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

      8. exp-prod N/A

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

      9. neg-log N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-log.f64 96.4

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

   6. Applied rewrites 96.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      14. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. lower-neg.f64 96.4

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

   8. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. rem-log-exp N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. pow-exp N/A

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

      9. rem-log-exp N/A

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

      10. metadata-eval 96.4

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift-neg.f64 N/A

          \[\leadsto \frac{{\left(e^{-2}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. frac-2neg N/A

          \[\leadsto \frac{{\left(e^{-2}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{\mathsf{neg}\left(n\right)}\right)}{\mathsf{neg}\left(2\right)}\right)}}{n \cdot x} \]

   10. Applied rewrites 96.4%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 5e-166

   1. Initial program 33.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if 5e-166 < (/.f64 #s(literal 1 binary64) n) < 0.20000000000000001

   1. Initial program 20.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-log N/A

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

      7. mul-1-neg N/A

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

      8. exp-prod N/A

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

      9. neg-log N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-log.f64 50.2

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

   6. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      14. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. lower-neg.f64 50.2

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

   8. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      7. unpow-prod-down N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

   10. Applied rewrites 50.2%

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

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{neg}\left(\frac{\frac{1}{2} + \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \frac{x}{n}\right)}{n} + \frac{-1}{3} \cdot x\right)}{n}\right), x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 74.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-\frac{\mathsf{fma}\left(-0.3333333333333333, x, -\frac{\mathsf{fma}\left(-0.5, x, \frac{x}{n} \cdot 0.16666666666666666\right) + 0.5}{n}\right) + 0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -1e-24)
     t_0
     (if (<= (/ 1.0 n) 5e-166)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 0.2)
         t_0
         (if (<= (/ 1.0 n) 5e+187)
           (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
           (/
            (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) x))
            n)))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-166) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+187) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / 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) :: t_0
    real(8) :: tmp
    t_0 = exp((log(x) / n)) / (n * x)
    if ((1.0d0 / n) <= (-1d-24)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-166) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 0.2d0) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d+187) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = -((-(((0.3333333333333333d0 / x) - 0.5d0) / x) - 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-166) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+187) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (n * x)
	tmp = 0
	if (1.0 / n) <= -1e-24:
		tmp = t_0
	elif (1.0 / n) <= 5e-166:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 0.2:
		tmp = t_0
	elif (1.0 / n) <= 5e+187:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.2)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+187)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = exp((log(x) / n)) / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-24)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-166)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 0.2)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e+187)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.2], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+187], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $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 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25 or 5e-166 < (/.f64 #s(literal 1 binary64) n) < 0.20000000000000001

   1. Initial program 70.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

      1. flip-+ 81.4

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

      2. pow2 81.4

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

      3. metadata-eval 81.4

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

      4. pow2 81.4

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-log.f64 81.4

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

   6. Applied rewrites 81.4%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 5e-166

   1. Initial program 33.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e187

   1. Initial program 70.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 36.0

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

   4. Applied rewrites 36.0%

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

   if 5.0000000000000001e187 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 72.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 5.9

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

   4. Applied rewrites 5.9%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}\right)}{n} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{-\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{-\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{-\frac{\left(\mathsf{neg}\left(\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}\right)\right) - 1}{x}}{n} \]

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 17.3

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

   7. Applied rewrites 17.3%

      \[\leadsto \frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\log x}{n}}\\ t_1 := \frac{t\_0}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+159}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (exp (/ (log x) n))) (t_1 (/ t_0 (* n x))))
   (if (<= (/ 1.0 n) -1e-24)
     t_1
     (if (<= (/ 1.0 n) 5e-166)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 0.2)
         t_1
         (if (<= (/ 1.0 n) 2e+159)
           (- 1.0 t_0)
           (/
            (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) x))
            n)))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-166) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+159) {
		tmp = 1.0 - t_0;
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp((log(x) / n))
    t_1 = t_0 / (n * x)
    if ((1.0d0 / n) <= (-1d-24)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-166) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 0.2d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d+159) then
        tmp = 1.0d0 - t_0
    else
        tmp = -((-(((0.3333333333333333d0 / x) - 0.5d0) / x) - 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n));
	double t_1 = t_0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-166) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+159) {
		tmp = 1.0 - t_0;
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n))
	t_1 = t_0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -1e-24:
		tmp = t_1
	elif (1.0 / n) <= 5e-166:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 0.2:
		tmp = t_1
	elif (1.0 / n) <= 2e+159:
		tmp = 1.0 - t_0
	else:
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = exp(Float64(log(x) / n))
	t_1 = Float64(t_0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.2)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+159)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = exp((log(x) / n));
	t_1 = t_0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-24)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-166)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 0.2)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e+159)
		tmp = 1.0 - t_0;
	else
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.2], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+159], N[(1.0 - t$95$0), $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 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25 or 5e-166 < (/.f64 #s(literal 1 binary64) n) < 0.20000000000000001

   1. Initial program 70.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

      1. flip-+ 81.4

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

      2. pow2 81.4

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

      3. metadata-eval 81.4

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

      4. pow2 81.4

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-log.f64 81.4

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

   6. Applied rewrites 81.4%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 5e-166

   1. Initial program 33.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e159

   1. Initial program 70.6%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 40.7

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

   4. Applied rewrites 40.7%

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

   if 1.9999999999999999e159 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 76.8%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 6.2

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

   4. Applied rewrites 6.2%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}\right)}{n} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{-\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{-\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{-\frac{\left(\mathsf{neg}\left(\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}\right)\right) - 1}{x}}{n} \]

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 12.2

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

   7. Applied rewrites 12.2%

      \[\leadsto \frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\log x}{n}}{-2}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left(e^{-2}\right)}^{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{{\left(e^{-1 \cdot t\_0}\right)}^{2}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, x, \frac{x}{n} \cdot 0.5\right) + 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ (log x) n) -2.0)))
   (if (<= (/ 1.0 n) -1e-24)
     (/ (pow (exp -2.0) t_0) (* n x))
     (if (<= (/ 1.0 n) 5e-166)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 0.2)
         (/ (pow (exp (* -1.0 t_0)) 2.0) (* n x))
         (-
          (fma (/ (+ (fma -0.5 x (* (/ x n) 0.5)) 1.0) n) x 1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = (log(x) / n) / -2.0;
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = pow(exp(-2.0), t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = pow(exp((-1.0 * t_0)), 2.0) / (n * x);
	} else {
		tmp = fma(((fma(-0.5, x, ((x / n) * 0.5)) + 1.0) / n), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(log(x) / n) / -2.0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64((exp(-2.0) ^ t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.2)
		tmp = Float64((exp(Float64(-1.0 * t_0)) ^ 2.0) / Float64(n * x));
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.5, x, Float64(Float64(x / n) * 0.5)) + 1.0) / n), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] / -2.0), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[Power[N[Exp[-2.0], $MachinePrecision], t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.2], N[(N[Power[N[Exp[N[(-1.0 * t$95$0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * x + N[(N[(x / n), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-log N/A

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

      7. mul-1-neg N/A

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

      8. exp-prod N/A

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

      9. neg-log N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-log.f64 96.4

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

   6. Applied rewrites 96.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      14. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. lower-neg.f64 96.4

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

   8. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. rem-log-exp N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. pow-exp N/A

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

      9. rem-log-exp N/A

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

      10. metadata-eval 96.4

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift-neg.f64 N/A

          \[\leadsto \frac{{\left(e^{-2}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. frac-2neg N/A

          \[\leadsto \frac{{\left(e^{-2}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{\mathsf{neg}\left(n\right)}\right)}{\mathsf{neg}\left(2\right)}\right)}}{n \cdot x} \]

   10. Applied rewrites 96.4%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 5e-166

   1. Initial program 33.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if 5e-166 < (/.f64 #s(literal 1 binary64) n) < 0.20000000000000001

   1. Initial program 20.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-log N/A

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

      7. mul-1-neg N/A

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

      8. exp-prod N/A

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

      9. neg-log N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-log.f64 50.2

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

   6. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      14. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. lower-neg.f64 50.2

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

   8. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      7. unpow-prod-down N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

   10. Applied rewrites 50.2%

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

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.1

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

   7. Applied rewrites 74.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, x, \frac{x}{n} \cdot 0.5\right) + 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 80.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left(e^{-2}\right)}^{\left(\frac{t\_0}{-2}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, x, \frac{x}{n} \cdot 0.5\right) + 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -1e-24)
     (/ (pow (exp -2.0) (/ t_0 -2.0)) (* n x))
     (if (<= (/ 1.0 n) 5e-166)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 0.2)
         (/ (exp t_0) (* n x))
         (-
          (fma (/ (+ (fma -0.5 x (* (/ x n) 0.5)) 1.0) n) x 1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = pow(exp(-2.0), (t_0 / -2.0)) / (n * x);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = exp(t_0) / (n * x);
	} else {
		tmp = fma(((fma(-0.5, x, ((x / n) * 0.5)) + 1.0) / n), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64((exp(-2.0) ^ Float64(t_0 / -2.0)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.2)
		tmp = Float64(exp(t_0) / Float64(n * x));
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.5, x, Float64(Float64(x / n) * 0.5)) + 1.0) / n), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[Power[N[Exp[-2.0], $MachinePrecision], N[(t$95$0 / -2.0), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.2], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * x + N[(N[(x / n), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-log N/A

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

      7. mul-1-neg N/A

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

      8. exp-prod N/A

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

      9. neg-log N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-log.f64 96.4

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

   6. Applied rewrites 96.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{n}\right)}{2}\right)}}{n \cdot x} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      14. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{-1} \cdot e^{-1}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. lower-neg.f64 96.4

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

   8. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. rem-log-exp N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. pow-exp N/A

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

      9. rem-log-exp N/A

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

      10. metadata-eval 96.4

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift-neg.f64 N/A

          \[\leadsto \frac{{\left(e^{-2}\right)}^{\left(\frac{\frac{\log x}{\mathsf{neg}\left(n\right)}}{2}\right)}}{n \cdot x} \]

      15. frac-2neg N/A

          \[\leadsto \frac{{\left(e^{-2}\right)}^{\left(\frac{\mathsf{neg}\left(\frac{\log x}{\mathsf{neg}\left(n\right)}\right)}{\mathsf{neg}\left(2\right)}\right)}}{n \cdot x} \]

   10. Applied rewrites 96.4%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 5e-166

   1. Initial program 33.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if 5e-166 < (/.f64 #s(literal 1 binary64) n) < 0.20000000000000001

   1. Initial program 20.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

      1. flip-+ 50.2

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

      2. pow2 50.2

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

      3. metadata-eval 50.2

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

      4. pow2 50.2

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-log.f64 50.2

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

   6. Applied rewrites 50.2%

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

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.1

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

   7. Applied rewrites 74.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, x, \frac{x}{n} \cdot 0.5\right) + 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left(e^{-1}\right)}^{\left(-t\_0\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, x, \frac{x}{n} \cdot 0.5\right) + 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -1e-24)
     (/ (pow (exp -1.0) (- t_0)) (* n x))
     (if (<= (/ 1.0 n) 5e-166)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 0.2)
         (/ (exp t_0) (* n x))
         (-
          (fma (/ (+ (fma -0.5 x (* (/ x n) 0.5)) 1.0) n) x 1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = pow(exp(-1.0), -t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = exp(t_0) / (n * x);
	} else {
		tmp = fma(((fma(-0.5, x, ((x / n) * 0.5)) + 1.0) / n), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64((exp(-1.0) ^ Float64(-t_0)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.2)
		tmp = Float64(exp(t_0) / Float64(n * x));
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.5, x, Float64(Float64(x / n) * 0.5)) + 1.0) / n), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[Power[N[Exp[-1.0], $MachinePrecision], (-t$95$0)], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.2], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * x + N[(N[(x / n), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-log N/A

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

      7. mul-1-neg N/A

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

      8. exp-prod N/A

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

      9. neg-log N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-log.f64 96.4

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

   6. Applied rewrites 96.4%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 5e-166

   1. Initial program 33.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if 5e-166 < (/.f64 #s(literal 1 binary64) n) < 0.20000000000000001

   1. Initial program 20.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

      1. flip-+ 50.2

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

      2. pow2 50.2

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

      3. metadata-eval 50.2

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

      4. pow2 50.2

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-log.f64 50.2

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

   6. Applied rewrites 50.2%

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

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.1

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

   7. Applied rewrites 74.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, x, \frac{x}{n} \cdot 0.5\right) + 1}{n}, x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left(e^{-1}\right)}^{\left(-t\_0\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -1e-24)
     (/ (pow (exp -1.0) (- t_0)) (* n x))
     (if (<= (/ 1.0 n) 5e-166)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 0.2)
         (/ (exp t_0) (* n x))
         (-
          (fma (fma (/ 0.5 (* n n)) x (/ 1.0 n)) x 1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = pow(exp(-1.0), -t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-166) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = exp(t_0) / (n * x);
	} else {
		tmp = fma(fma((0.5 / (n * n)), x, (1.0 / n)), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64((exp(-1.0) ^ Float64(-t_0)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.2)
		tmp = Float64(exp(t_0) / Float64(n * x));
	else
		tmp = Float64(fma(fma(Float64(0.5 / Float64(n * n)), x, Float64(1.0 / n)), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[Power[N[Exp[-1.0], $MachinePrecision], (-t$95$0)], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.2], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-log N/A

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

      7. mul-1-neg N/A

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

      8. exp-prod N/A

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

      9. neg-log N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-log.f64 96.4

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

   6. Applied rewrites 96.4%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 5e-166

   1. Initial program 33.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if 5e-166 < (/.f64 #s(literal 1 binary64) n) < 0.20000000000000001

   1. Initial program 20.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

      1. flip-+ 50.2

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

      2. pow2 50.2

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

      3. metadata-eval 50.2

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

      4. pow2 50.2

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-log.f64 50.2

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

   6. Applied rewrites 50.2%

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

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in n around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 72.1

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

   7. Applied rewrites 72.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -1e-24)
     t_0
     (if (<= (/ 1.0 n) 5e-166)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 0.2)
         t_0
         (-
          (fma (fma (/ 0.5 (* n n)) x (/ 1.0 n)) x 1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-166) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 0.2) {
		tmp = t_0;
	} else {
		tmp = fma(fma((0.5 / (n * n)), x, (1.0 / n)), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-166)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 0.2)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(Float64(0.5 / Float64(n * n)), x, Float64(1.0 / n)), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-166], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.2], t$95$0, N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25 or 5e-166 < (/.f64 #s(literal 1 binary64) n) < 0.20000000000000001

   1. Initial program 70.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

      1. flip-+ 81.4

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

      2. pow2 81.4

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

      3. metadata-eval 81.4

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

      4. pow2 81.4

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

      5. lift-neg.f64 N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{-\log x}{n}\right)}}{n \cdot x} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{-\log x}{n}\right)}}{n \cdot x} \]

      7. lift-log.f64 N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{-\log x}{n}\right)}}{n \cdot x} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}}{n \cdot x} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\log x\right)}{\mathsf{neg}\left(n\right)}}}{n \cdot x} \]

      10. frac-2neg N/A

          \[\leadsto \frac{e^{\frac{\log x}{n}}}{n \cdot x} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{e^{\frac{\log x}{n}}}{n \cdot x} \]

      12. lift-log.f64 81.4

          \[\leadsto \frac{e^{\frac{\log x}{n}}}{n \cdot x} \]

   6. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot x}} \]

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 5e-166

   1. Initial program 33.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 83.3

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]

   if 0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 70.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right) + \color{blue}{1}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right) \cdot x + 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}, \color{blue}{x}, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   5. Taylor expanded in n around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{{n}^{2}}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{n \cdot n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{2}}{n \cdot n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. lift-*.f64 20.3

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]

   7. Applied rewrites 20.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := 1 - e^{\frac{\log x}{n}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (- 1.0 (exp (/ (log x) n)))))
   (if (<= t_0 -4e+98)
     t_1
     (if (<= t_0 2e-13) (/ (log (/ (+ 1.0 x) x)) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = 1.0 - exp((log(x) / n));
	double tmp;
	if (t_0 <= -4e+98) {
		tmp = t_1;
	} else if (t_0 <= 2e-13) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    t_1 = 1.0d0 - exp((log(x) / n))
    if (t_0 <= (-4d+98)) then
        tmp = t_1
    else if (t_0 <= 2d-13) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = 1.0 - Math.exp((Math.log(x) / n));
	double tmp;
	if (t_0 <= -4e+98) {
		tmp = t_1;
	} else if (t_0 <= 2e-13) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = 1.0 - math.exp((math.log(x) / n))
	tmp = 0
	if t_0 <= -4e+98:
		tmp = t_1
	elif t_0 <= 2e-13:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(1.0 - exp(Float64(log(x) / n)))
	tmp = 0.0
	if (t_0 <= -4e+98)
		tmp = t_1;
	elseif (t_0 <= 2e-13)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = 1.0 - exp((log(x) / n));
	tmp = 0.0;
	if (t_0 <= -4e+98)
		tmp = t_1;
	elseif (t_0 <= 2e-13)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+98], t$95$1, If[LessEqual[t$95$0, 2e-13], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -3.99999999999999999e98 or 2.0000000000000001e-13 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 77.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \color{blue}{e^{\frac{\log x}{n}}} \]

      2. lower-exp.f64 N/A

         \[\leadsto 1 - e^{\frac{\log x}{n}} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 - e^{\frac{\log x}{n}} \]

      4. lower-log.f64 75.0

         \[\leadsto 1 - e^{\frac{\log x}{n}} \]

   4. Applied rewrites 75.0%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]

   if -3.99999999999999999e98 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2.0000000000000001e-13

   1. Initial program 43.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 80.1

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+98}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 -4e+98)
     t_2
     (if (<= t_1 2e-13) (/ (log (/ (+ 1.0 x) x)) n) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -4e+98) {
		tmp = t_2;
	} else if (t_1 <= 2e-13) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    t_2 = 1.0d0 - t_0
    if (t_1 <= (-4d+98)) then
        tmp = t_2
    else if (t_1 <= 2d-13) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -4e+98) {
		tmp = t_2;
	} else if (t_1 <= 2e-13) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -4e+98:
		tmp = t_2
	elif t_1 <= 2e-13:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= -4e+98)
		tmp = t_2;
	elseif (t_1 <= 2e-13)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -4e+98)
		tmp = t_2;
	elseif (t_1 <= 2e-13)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+98], t$95$2, If[LessEqual[t$95$1, 2e-13], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -3.99999999999999999e98 or 2.0000000000000001e-13 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 77.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 75.0%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if -3.99999999999999999e98 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2.0000000000000001e-13

   1. Initial program 43.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 80.1

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 74.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1
         (/ (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) x)) n)))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 0.005) (/ (log (/ (+ 1.0 x) x)) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.005) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 0.005) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 0.005:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / x)) / n)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.005)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 0.005)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[(N[((-N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.005], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0 or 0.0050000000000000001 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 77.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 6.7

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 6.7%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}\right)}{n} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{-\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{-\frac{-1 \cdot \frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{-\frac{\left(\mathsf{neg}\left(\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}\right)\right) - 1}{x}}{n} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{-\frac{\left(-\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}\right) - 1}{x}}{n} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{-\frac{\left(-\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}\right) - 1}{x}}{n} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{-\frac{\left(-\frac{\frac{1}{3} \cdot \frac{1}{x} - \frac{1}{2}}{x}\right) - 1}{x}}{n} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{-\frac{\left(-\frac{\frac{\frac{1}{3} \cdot 1}{x} - \frac{1}{2}}{x}\right) - 1}{x}}{n} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-\frac{\left(-\frac{\frac{\frac{1}{3}}{x} - \frac{1}{2}}{x}\right) - 1}{x}}{n} \]

      11. lower-/.f64 61.5

          \[\leadsto \frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n} \]

   7. Applied rewrites 61.5%

      \[\leadsto \frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n} \]

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0050000000000000001

   1. Initial program 43.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 80.0

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\log x}{n} + 1}{n \cdot x}\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 -4e+98)
     (/ (+ (/ (log x) n) 1.0) (* n x))
     (if (<= t_0 0.005) (/ (log (/ (+ 1.0 x) x)) n) (/ 1.0 (* n x))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -4e+98) {
		tmp = ((log(x) / n) + 1.0) / (n * x);
	} else if (t_0 <= 0.005) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = 1.0 / (n * 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 + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    if (t_0 <= (-4d+98)) then
        tmp = ((log(x) / n) + 1.0d0) / (n * x)
    else if (t_0 <= 0.005d0) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -4e+98) {
		tmp = ((Math.log(x) / n) + 1.0) / (n * x);
	} else if (t_0 <= 0.005) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -4e+98:
		tmp = ((math.log(x) / n) + 1.0) / (n * x)
	elif t_0 <= 0.005:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= -4e+98)
		tmp = Float64(Float64(Float64(log(x) / n) + 1.0) / Float64(n * x));
	elseif (t_0 <= 0.005)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -4e+98)
		tmp = ((log(x) / n) + 1.0) / (n * x);
	elseif (t_0 <= 0.005)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+98], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.005], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -3.99999999999999999e98

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{n \cdot x}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{n} \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}{n \cdot x} \]

      4. log-rec N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}}{n \cdot x} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}}{n \cdot x} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{e^{-\frac{-1 \cdot \log x}{n}}}{n \cdot x} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{e^{-\frac{-1 \cdot \log x}{n}}}{n \cdot x} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{e^{-\frac{\mathsf{neg}\left(\log x\right)}{n}}}{n \cdot x} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot x} \]

      10. lower-log.f64 N/A

          \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot x} \]

      11. lower-*.f64 100.0

          \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot \color{blue}{x}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \frac{1 + \frac{\log x}{n}}{\color{blue}{n} \cdot x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\log x}{n} + 1}{n \cdot x} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\log x}{n} + 1}{n \cdot x} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\log x}{n} + 1}{n \cdot x} \]

      4. lift-log.f64 77.7

         \[\leadsto \frac{\frac{\log x}{n} + 1}{n \cdot x} \]

   7. Applied rewrites 77.7%

      \[\leadsto \frac{\frac{\log x}{n} + 1}{\color{blue}{n} \cdot x} \]

   if -3.99999999999999999e98 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0050000000000000001

   1. Initial program 43.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 80.0

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]

   if 0.0050000000000000001 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 54.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{n \cdot x}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{n} \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}{n \cdot x} \]

      4. log-rec N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}}{n \cdot x} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}}{n \cdot x} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{e^{-\frac{-1 \cdot \log x}{n}}}{n \cdot x} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{e^{-\frac{-1 \cdot \log x}{n}}}{n \cdot x} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{e^{-\frac{\mathsf{neg}\left(\log x\right)}{n}}}{n \cdot x} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot x} \]

      10. lower-log.f64 N/A

          \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot x} \]

      11. lower-*.f64 1.6

          \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot \color{blue}{x}} \]

   4. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \frac{1}{\color{blue}{n} \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 27.8%

      \[\leadsto \frac{1}{\color{blue}{n} \cdot x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 68.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{1}{n \cdot x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (/ 1.0 (* n x))))
   (if (<= t_0 -4e+98)
     t_1
     (if (<= t_0 0.005) (/ (log (/ (+ 1.0 x) x)) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = 1.0 / (n * x);
	double tmp;
	if (t_0 <= -4e+98) {
		tmp = t_1;
	} else if (t_0 <= 0.005) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    t_1 = 1.0d0 / (n * x)
    if (t_0 <= (-4d+98)) then
        tmp = t_1
    else if (t_0 <= 0.005d0) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = 1.0 / (n * x);
	double tmp;
	if (t_0 <= -4e+98) {
		tmp = t_1;
	} else if (t_0 <= 0.005) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = 1.0 / (n * x)
	tmp = 0
	if t_0 <= -4e+98:
		tmp = t_1
	elif t_0 <= 0.005:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (t_0 <= -4e+98)
		tmp = t_1;
	elseif (t_0 <= 0.005)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = 1.0 / (n * x);
	tmp = 0.0;
	if (t_0 <= -4e+98)
		tmp = t_1;
	elseif (t_0 <= 0.005)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+98], t$95$1, If[LessEqual[t$95$0, 0.005], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -3.99999999999999999e98 or 0.0050000000000000001 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 77.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{n \cdot x}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{n} \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}{n \cdot x} \]

      4. log-rec N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}}{n \cdot x} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}}{n \cdot x} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{e^{-\frac{-1 \cdot \log x}{n}}}{n \cdot x} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{e^{-\frac{-1 \cdot \log x}{n}}}{n \cdot x} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{e^{-\frac{\mathsf{neg}\left(\log x\right)}{n}}}{n \cdot x} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot x} \]

      10. lower-log.f64 N/A

          \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot x} \]

      11. lower-*.f64 50.8

          \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot \color{blue}{x}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \frac{1}{\color{blue}{n} \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.9%

      \[\leadsto \frac{1}{\color{blue}{n} \cdot x} \]

   if -3.99999999999999999e98 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0050000000000000001

   1. Initial program 43.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 80.0

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 59.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.98)
   (/ (- x (log x)) n)
   (if (<= x 1.5e+183) (/ (/ (- 1.0 (/ 0.5 x)) x) n) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.5e+183) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	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 <= 0.98d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.5d+183) then
        tmp = ((1.0d0 - (0.5d0 / x)) / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.5e+183) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.98:
		tmp = (x - math.log(x)) / n
	elif x <= 1.5e+183:
		tmp = ((1.0 - (0.5 / x)) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.5e+183)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / x) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.98)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.5e+183)
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.5e+183], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.97999999999999998

   1. Initial program 43.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 51.3

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x + -1 \cdot \log x}{n} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \log x + x}{n} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-1 \cdot \log x + x}{n} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + x}{n} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\left(-\log x\right) + x}{n} \]

      5. lift-log.f64 51.0

         \[\leadsto \frac{\left(-\log x\right) + x}{n} \]

   7. Applied rewrites 51.0%

      \[\leadsto \frac{\left(-\log x\right) + x}{n} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\left(-\log x\right) + x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\left(-\log x\right) + x}{n} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + x}{n} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \log x + x}{n} \]

      5. +-commutative N/A

         \[\leadsto \frac{x + -1 \cdot \log x}{n} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{x - \left(\mathsf{neg}\left(-1\right)\right) \cdot \log x}{n} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x - 1 \cdot \log x}{n} \]

      8. log-pow-rev N/A

         \[\leadsto \frac{x - \log \left({x}^{1}\right)}{n} \]

      9. unpow1 N/A

         \[\leadsto \frac{x - \log x}{n} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x - \log x}{n} \]

      11. lift-log.f64 51.0

          \[\leadsto \frac{x - \log x}{n} \]

   9. Applied rewrites 51.0%

      \[\leadsto \frac{x - \log x}{n} \]

   if 0.97999999999999998 < x < 1.49999999999999998e183

   1. Initial program 55.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 55.7

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1 - \frac{\frac{1}{2} \cdot 1}{x}}{x}}{n} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{1 - \frac{\frac{1}{2}}{x}}{x}}{n} \]

      5. lower-/.f64 62.4

         \[\leadsto \frac{\frac{1 - \frac{0.5}{x}}{x}}{n} \]

   7. Applied rewrites 62.4%

      \[\leadsto \frac{\frac{1 - \frac{0.5}{x}}{x}}{n} \]

   if 1.49999999999999998e183 < x

   1. Initial program 86.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 53.5%

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} - 1 \]
   6. Step-by-step derivation

      1. flip-+ 86.1

         \[\leadsto 1 - 1 \]

      2. pow2 86.1

         \[\leadsto 1 - 1 \]

      3. metadata-eval 86.1

         \[\leadsto 1 - 1 \]

      4. pow2 86.1

         \[\leadsto 1 - 1 \]

   7. Applied rewrites 86.1%

      \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 59.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (- x (log x)) n)
   (if (<= x 1.5e+183) (/ (/ 1.0 x) n) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.5e+183) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	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 <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.5d+183) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.5e+183) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 1.5e+183:
		tmp = (1.0 / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.5e+183)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.5e+183)
		tmp = (1.0 / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.5e+183], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 43.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 51.3

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x + -1 \cdot \log x}{n} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \log x + x}{n} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-1 \cdot \log x + x}{n} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + x}{n} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\left(-\log x\right) + x}{n} \]

      5. lift-log.f64 51.0

         \[\leadsto \frac{\left(-\log x\right) + x}{n} \]

   7. Applied rewrites 51.0%

      \[\leadsto \frac{\left(-\log x\right) + x}{n} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\left(-\log x\right) + x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\left(-\log x\right) + x}{n} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\log x\right)\right) + x}{n} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \log x + x}{n} \]

      5. +-commutative N/A

         \[\leadsto \frac{x + -1 \cdot \log x}{n} \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{x - \left(\mathsf{neg}\left(-1\right)\right) \cdot \log x}{n} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x - 1 \cdot \log x}{n} \]

      8. log-pow-rev N/A

         \[\leadsto \frac{x - \log \left({x}^{1}\right)}{n} \]

      9. unpow1 N/A

         \[\leadsto \frac{x - \log x}{n} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x - \log x}{n} \]

      11. lift-log.f64 51.0

          \[\leadsto \frac{x - \log x}{n} \]

   9. Applied rewrites 51.0%

      \[\leadsto \frac{x - \log x}{n} \]

   if 1 < x < 1.49999999999999998e183

   1. Initial program 55.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 55.7

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{x}}{n} \]
   6. Step-by-step derivation

      1. lower-/.f64 61.1

         \[\leadsto \frac{\frac{1}{x}}{n} \]

   7. Applied rewrites 61.1%

      \[\leadsto \frac{\frac{1}{x}}{n} \]

   if 1.49999999999999998e183 < x

   1. Initial program 86.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 53.5%

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} - 1 \]
   6. Step-by-step derivation

      1. flip-+ 86.1

         \[\leadsto 1 - 1 \]

      2. pow2 86.1

         \[\leadsto 1 - 1 \]

      3. metadata-eval 86.1

         \[\leadsto 1 - 1 \]

      4. pow2 86.1

         \[\leadsto 1 - 1 \]

   7. Applied rewrites 86.1%

      \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 59.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55)
   (/ (- (log x)) n)
   (if (<= x 1.5e+183) (/ (/ 1.0 x) n) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -log(x) / n;
	} else if (x <= 1.5e+183) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	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 <= 0.55d0) then
        tmp = -log(x) / n
    else if (x <= 1.5d+183) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.5e+183) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -math.log(x) / n
	elif x <= 1.5e+183:
		tmp = (1.0 / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.5e+183)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -log(x) / n;
	elseif (x <= 1.5e+183)
		tmp = (1.0 / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.5e+183], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.55000000000000004

   1. Initial program 43.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 51.3

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{-1 \cdot \log x}{n} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log x\right)}{n} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{-\log x}{n} \]

      3. lift-log.f64 50.6

         \[\leadsto \frac{-\log x}{n} \]

   7. Applied rewrites 50.6%

      \[\leadsto \frac{-\log x}{n} \]

   if 0.55000000000000004 < x < 1.49999999999999998e183

   1. Initial program 55.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 55.7

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{x}}{n} \]
   6. Step-by-step derivation

      1. lower-/.f64 61.0

         \[\leadsto \frac{\frac{1}{x}}{n} \]

   7. Applied rewrites 61.0%

      \[\leadsto \frac{\frac{1}{x}}{n} \]

   if 1.49999999999999998e183 < x

   1. Initial program 86.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 53.5%

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} - 1 \]
   6. Step-by-step derivation

      1. flip-+ 86.1

         \[\leadsto 1 - 1 \]

      2. pow2 86.1

         \[\leadsto 1 - 1 \]

      3. metadata-eval 86.1

         \[\leadsto 1 - 1 \]

      4. pow2 86.1

         \[\leadsto 1 - 1 \]

   7. Applied rewrites 86.1%

      \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 44.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.85:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-171}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -0.85)
   (/ 1.0 (* n x))
   (if (<= n -1.35e-171) (- 1.0 1.0) (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -0.85) {
		tmp = 1.0 / (n * x);
	} else if (n <= -1.35e-171) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / 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 (n <= (-0.85d0)) then
        tmp = 1.0d0 / (n * x)
    else if (n <= (-1.35d-171)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -0.85) {
		tmp = 1.0 / (n * x);
	} else if (n <= -1.35e-171) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -0.85:
		tmp = 1.0 / (n * x)
	elif n <= -1.35e-171:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -0.85)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= -1.35e-171)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -0.85)
		tmp = 1.0 / (n * x);
	elseif (n <= -1.35e-171)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -0.85], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.35e-171], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.849999999999999978

   1. Initial program 29.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{n \cdot x}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{n} \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}{n \cdot x} \]

      4. log-rec N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}}{n \cdot x} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}}{n \cdot x} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{e^{-\frac{-1 \cdot \log x}{n}}}{n \cdot x} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{e^{-\frac{-1 \cdot \log x}{n}}}{n \cdot x} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{e^{-\frac{\mathsf{neg}\left(\log x\right)}{n}}}{n \cdot x} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot x} \]

      10. lower-log.f64 N/A

          \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot x} \]

      11. lower-*.f64 48.8

          \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot \color{blue}{x}} \]

   4. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \frac{1}{\color{blue}{n} \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.3%

      \[\leadsto \frac{1}{\color{blue}{n} \cdot x} \]

   if -0.849999999999999978 < n < -1.35000000000000007e-171

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 2.4%

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} - 1 \]
   6. Step-by-step derivation

      1. flip-+ 48.8

         \[\leadsto 1 - 1 \]

      2. pow2 48.8

         \[\leadsto 1 - 1 \]

      3. metadata-eval 48.8

         \[\leadsto 1 - 1 \]

      4. pow2 48.8

         \[\leadsto 1 - 1 \]

   7. Applied rewrites 48.8%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1.35000000000000007e-171 < n

   1. Initial program 52.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{\color{blue}{n}} \]

      2. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      4. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      6. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. lower-+.f64 52.5

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

   4. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1 + x}{x}\right)}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{x}}{n} \]
   6. Step-by-step derivation

      1. lower-/.f64 42.5

         \[\leadsto \frac{\frac{1}{x}}{n} \]

   7. Applied rewrites 42.5%

      \[\leadsto \frac{\frac{1}{x}}{n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;n \leq -0.85:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-171}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))))
   (if (<= n -0.85) t_0 (if (<= n -1.35e-171) (- 1.0 1.0) t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -0.85) {
		tmp = t_0;
	} else if (n <= -1.35e-171) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = t_0;
	}
	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 = 1.0d0 / (n * x)
    if (n <= (-0.85d0)) then
        tmp = t_0
    else if (n <= (-1.35d-171)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -0.85) {
		tmp = t_0;
	} else if (n <= -1.35e-171) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	tmp = 0
	if n <= -0.85:
		tmp = t_0
	elif n <= -1.35e-171:
		tmp = 1.0 - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (n <= -0.85)
		tmp = t_0;
	elseif (n <= -1.35e-171)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	tmp = 0.0;
	if (n <= -0.85)
		tmp = t_0;
	elseif (n <= -1.35e-171)
		tmp = 1.0 - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -0.85], t$95$0, If[LessEqual[n, -1.35e-171], N[(1.0 - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -0.849999999999999978 or -1.35000000000000007e-171 < n

   1. Initial program 45.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{n \cdot x}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{n} \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}{n \cdot x} \]

      4. log-rec N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}}{n \cdot x} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}}{n \cdot x} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{e^{-\frac{-1 \cdot \log x}{n}}}{n \cdot x} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{e^{-\frac{-1 \cdot \log x}{n}}}{n \cdot x} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{e^{-\frac{\mathsf{neg}\left(\log x\right)}{n}}}{n \cdot x} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot x} \]

      10. lower-log.f64 N/A

          \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot x} \]

      11. lower-*.f64 49.1

          \[\leadsto \frac{e^{-\frac{-\log x}{n}}}{n \cdot \color{blue}{x}} \]

   4. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \frac{1}{\color{blue}{n} \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.8%

      \[\leadsto \frac{1}{\color{blue}{n} \cdot x} \]

   if -0.849999999999999978 < n < -1.35000000000000007e-171

   1. Initial program 100.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 2.4%

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} - 1 \]
   6. Step-by-step derivation

      1. flip-+ 48.8

         \[\leadsto 1 - 1 \]

      2. pow2 48.8

         \[\leadsto 1 - 1 \]

      3. metadata-eval 48.8

         \[\leadsto 1 - 1 \]

      4. pow2 48.8

         \[\leadsto 1 - 1 \]

   7. Applied rewrites 48.8%

      \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 30.4% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (- 1.0 1.0))

C

double code(double x, double n) {
	return 1.0 - 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 = 1.0d0 - 1.0d0
end function

Java

public static double code(double x, double n) {
	return 1.0 - 1.0;
}

Python

def code(x, n):
	return 1.0 - 1.0

Julia

function code(x, n)
	return Float64(1.0 - 1.0)
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 - 1.0;
end

Wolfram

code[x_, n_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in n around inf

   \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 17.3%

   \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{1} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} - 1 \]
6. Step-by-step derivation

   1. flip-+ 30.4

      \[\leadsto 1 - 1 \]

   2. pow2 30.4

      \[\leadsto 1 - 1 \]

   3. metadata-eval 30.4

      \[\leadsto 1 - 1 \]

   4. pow2 30.4

      \[\leadsto 1 - 1 \]

7. Applied rewrites 30.4%

   \[\leadsto \color{blue}{1} - 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))