2nthrt (problem 3.4.6)

Percentage Accurate: 53.9% → 94.4%

Time: 20.1s

Alternatives: 20

Speedup: 3.0×

• could not determine a ground truth

  1. x = -8.171748835287438e-157
  2. n = -3.1002508162817296e-300

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

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: 94.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{-1}{n}}{-x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+176}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-18)
     (/ (* t_0 (/ -1.0 n)) (- x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+176)
         (- (+ 1.0 (/ x n)) t_0)
         (* -1.0 (/ (+ 1.0 (* -1.0 (/ (log (/ 1.0 x)) n))) (* n x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = (t_0 * (-1.0 / n)) / -x;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = -1.0 * ((1.0 + (-1.0 * (log((1.0 / x)) / n))) / (n * x));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = (t_0 * (-1.0 / n)) / -x;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = -1.0 * ((1.0 + (-1.0 * (Math.log((1.0 / x)) / n))) / (n * x));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-18:
		tmp = (t_0 * (-1.0 / n)) / -x
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+176:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = -1.0 * ((1.0 + (-1.0 * (math.log((1.0 / x)) / n))) / (n * x))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-18)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / n)) / Float64(-x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+176)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(-1.0 * Float64(Float64(1.0 + Float64(-1.0 * Float64(log(Float64(1.0 / x)) / n))) / Float64(n * x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-18], N[(N[(t$95$0 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+176], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(-1.0 * N[(N[(1.0 + N[(-1.0 * N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000004e-18

   1. Initial program 53.9%

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

   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. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.2%

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

   4. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. log-rec N/A

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

      7. lift-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 57.2%

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

   6. Applied rewrites 57.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. lift-log.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 N/A

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

      21. lower-neg.f64 58.0%

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

   8. Applied rewrites 58.0%

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

   if -5.0000000000000004e-18 < (/.f64 #s(literal 1 binary64) n) < 2e-12

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. div-add N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. lower-log1p.f64 57.1%

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

   8. Applied rewrites 57.1%

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

   if 2e-12 < (/.f64 #s(literal 1 binary64) n) < 1e176

   1. Initial program 53.9%

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

   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. lower-+.f64 N/A

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

      2. lower-/.f64 31.5%

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

   4. Applied rewrites 31.5%

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

   if 1e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 22.1%

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

   7. Applied rewrites 22.1%

      \[\leadsto -1 \cdot \color{blue}{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+176}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-18)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-12)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+176)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (* -1.0 (/ (+ 1.0 (* -1.0 (/ (log (/ 1.0 x)) n))) (* n x)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = -1.0 * ((1.0 + (-1.0 * (log((1.0 / x)) / n))) / (n * x));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = -1.0 * ((1.0 + (-1.0 * (Math.log((1.0 / x)) / n))) / (n * x));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-18:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+176:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = -1.0 * ((1.0 + (-1.0 * (math.log((1.0 / x)) / n))) / (n * x))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-18)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+176)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-1.0 * Float64(Float64(1.0 + Float64(-1.0 * Float64(log(Float64(1.0 / x)) / n))) / Float64(n * x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-18], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+176], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(1.0 + N[(-1.0 * N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000004e-18

   1. Initial program 53.9%

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

   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. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.2%

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

   4. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. log-rec N/A

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

      7. lift-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 57.2%

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

   6. Applied rewrites 57.2%

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

   if -5.0000000000000004e-18 < (/.f64 #s(literal 1 binary64) n) < 2e-12

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. div-add N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. lower-log1p.f64 57.1%

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

   8. Applied rewrites 57.1%

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

   if 2e-12 < (/.f64 #s(literal 1 binary64) n) < 1e176

   1. Initial program 53.9%

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

   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. lower-+.f64 N/A

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

      2. lower-/.f64 31.5%

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

   4. Applied rewrites 31.5%

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

   if 1e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 22.1%

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

   7. Applied rewrites 22.1%

      \[\leadsto -1 \cdot \color{blue}{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-18)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-12)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+176)
       (- 1.0 (pow x (/ 1.0 n)))
       (* -1.0 (/ (+ 1.0 (* -1.0 (/ (log (/ 1.0 x)) n))) (* n x)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = -1.0 * ((1.0 + (-1.0 * (log((1.0 / x)) / n))) / (n * x));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = -1.0 * ((1.0 + (-1.0 * (Math.log((1.0 / x)) / n))) / (n * x));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-18:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+176:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = -1.0 * ((1.0 + (-1.0 * (math.log((1.0 / x)) / n))) / (n * x))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-18)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+176)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-1.0 * Float64(Float64(1.0 + Float64(-1.0 * Float64(log(Float64(1.0 / x)) / n))) / Float64(n * x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-18], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+176], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(1.0 + N[(-1.0 * N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000004e-18

   1. Initial program 53.9%

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

   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. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.2%

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

   4. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. log-rec N/A

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

      7. lift-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 57.2%

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

   6. Applied rewrites 57.2%

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

   if -5.0000000000000004e-18 < (/.f64 #s(literal 1 binary64) n) < 2e-12

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. div-add N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. lower-log1p.f64 57.1%

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

   8. Applied rewrites 57.1%

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

   if 2e-12 < (/.f64 #s(literal 1 binary64) n) < 1e176

   1. Initial program 53.9%

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

   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 39.0%

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

   if 1e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 22.1%

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

   7. Applied rewrites 22.1%

      \[\leadsto -1 \cdot \color{blue}{\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{n \cdot x}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{-1}{n}}{-x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right) - t\_0\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-18)
     (/ (* t_0 (/ -1.0 n)) (- x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log1p (/ 1.0 x)) n)
       (-
        (+
         1.0
         (*
          x
          (fma x (- (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ 1.0 n))) (/ 1.0 n))))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = (t_0 * (-1.0 / n)) / -x;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (1.0 + (x * fma(x, ((0.5 * (1.0 / pow(n, 2.0))) - (0.5 * (1.0 / n))), (1.0 / n)))) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-18)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / n)) / Float64(-x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * fma(x, Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) - Float64(0.5 * Float64(1.0 / n))), Float64(1.0 / n)))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-18], N[(N[(t$95$0 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000004e-18

   1. Initial program 53.9%

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

   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. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.2%

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

   4. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. log-rec N/A

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

      7. lift-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 57.2%

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

   6. Applied rewrites 57.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. lift-log.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. exp-to-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 N/A

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

      21. lower-neg.f64 58.0%

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

   8. Applied rewrites 58.0%

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

   if -5.0000000000000004e-18 < (/.f64 #s(literal 1 binary64) n) < 2e-12

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. div-add N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. lower-log1p.f64 57.1%

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

   8. Applied rewrites 57.1%

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

   if 2e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

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

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 23.9%

          \[\leadsto \left(1 + x \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\left(1 + x \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - -1\right) - \frac{0.5}{x}}{x}}{n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-18)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-12)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 1e+176)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (/ (- (- (/ 0.3333333333333333 (* x x)) -1.0) (/ 0.5 x)) x) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-18:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+176:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-18)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+176)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - -1.0) - Float64(0.5 / x)) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-18], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+176], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000004e-18

   1. Initial program 53.9%

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

   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. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.2%

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

   4. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. log-rec N/A

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

      7. lift-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 57.2%

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

   6. Applied rewrites 57.2%

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

   if -5.0000000000000004e-18 < (/.f64 #s(literal 1 binary64) n) < 2e-12

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. div-add N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. lower-log1p.f64 57.1%

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

   8. Applied rewrites 57.1%

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

   if 2e-12 < (/.f64 #s(literal 1 binary64) n) < 1e176

   1. Initial program 53.9%

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

   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 39.0%

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

   if 1e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 46.3%

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

   7. Applied rewrites 46.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 46.3%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 46.3%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 46.3%

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

   9. Applied rewrites 46.3%

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

Alternative 6: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+176}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - -1\right) - \frac{0.5}{x}}{x}}{n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-18)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-12)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 1e+176)
         (- 1.0 t_0)
         (/ (/ (- (- (/ 0.3333333333333333 (* x x)) -1.0) (/ 0.5 x)) x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-18) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-18:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+176:
		tmp = 1.0 - t_0
	else:
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-18)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+176)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - -1.0) - Float64(0.5 / x)) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-18], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+176], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.0000000000000004e-18

   1. Initial program 53.9%

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

   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. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 57.2%

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

   4. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. log-rec N/A

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

      7. lift-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. frac-2neg N/A

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

      12. mult-flip N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f32 57.0%

          \[\leadsto \frac{e^{\left( \left( \log x \right)_{\text{binary64}} \cdot \left( \frac{1}{n} \right)_{\text{binary64}} \right)_{\text{binary32}}}}{n \cdot x} \]

      15. lower-unsound-log.f64 N/A

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

      16. lower-unsound-*.f32 N/A

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

      17. lower-unsound-exp.f64 N/A

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

      18. pow-to-exp N/A

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

      19. lower-pow.f64 57.2%

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

   6. Applied rewrites 57.2%

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

   if -5.0000000000000004e-18 < (/.f64 #s(literal 1 binary64) n) < 2e-12

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. div-add N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. lower-log1p.f64 57.1%

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

   8. Applied rewrites 57.1%

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

   if 2e-12 < (/.f64 #s(literal 1 binary64) n) < 1e176

   1. Initial program 53.9%

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

   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 39.0%

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

   if 1e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 46.3%

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

   7. Applied rewrites 46.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 46.3%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 46.3%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 46.3%

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

   9. Applied rewrites 46.3%

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

Alternative 7: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+162}:\\ \;\;\;\;\frac{1 \cdot x}{\frac{n}{\log \left(\frac{x - -1}{x}\right)} \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -0.04:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+176}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - -1\right) - \frac{0.5}{x}}{x}}{n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -1e+162)
     (/ (* 1.0 x) (* (/ n (log (/ (- x -1.0) x))) x))
     (if (<= (/ 1.0 n) -0.04)
       t_0
       (if (<= (/ 1.0 n) 2e-12)
         (/ (log1p (/ 1.0 x)) n)
         (if (<= (/ 1.0 n) 1e+176)
           t_0
           (/
            (/ (- (- (/ 0.3333333333333333 (* x x)) -1.0) (/ 0.5 x)) x)
            n)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+162) {
		tmp = (1.0 * x) / ((n / log(((x - -1.0) / x))) * x);
	} else if ((1.0 / n) <= -0.04) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+162) {
		tmp = (1.0 * x) / ((n / Math.log(((x - -1.0) / x))) * x);
	} else if ((1.0 / n) <= -0.04) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e+162:
		tmp = (1.0 * x) / ((n / math.log(((x - -1.0) / x))) * x)
	elif (1.0 / n) <= -0.04:
		tmp = t_0
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+176:
		tmp = t_0
	else:
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+162)
		tmp = Float64(Float64(1.0 * x) / Float64(Float64(n / log(Float64(Float64(x - -1.0) / x))) * x));
	elseif (Float64(1.0 / n) <= -0.04)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+176)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - -1.0) - Float64(0.5 / x)) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+162], N[(N[(1.0 * x), $MachinePrecision] / N[(N[(n / N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.04], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+176], t$95$0, N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999994e161

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. neg-log N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip-rev N/A

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

      7. lift-/.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. lift-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

      15. *-inverses N/A

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

      16. frac-times N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 67.0%

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

   8. Applied rewrites 67.0%

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

   if -9.9999999999999994e161 < (/.f64 #s(literal 1 binary64) n) < -0.040000000000000001 or 2e-12 < (/.f64 #s(literal 1 binary64) n) < 1e176

   1. Initial program 53.9%

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

   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 39.0%

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

   if -0.040000000000000001 < (/.f64 #s(literal 1 binary64) n) < 2e-12

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. div-add N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. lower-log1p.f64 57.1%

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

   8. Applied rewrites 57.1%

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

   if 1e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 46.3%

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

   7. Applied rewrites 46.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 46.3%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 46.3%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 46.3%

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

   9. Applied rewrites 46.3%

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

Alternative 8: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+162}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -0.04:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+176}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - -1\right) - \frac{0.5}{x}}{x}}{n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -1e+162)
     (/ (* (log (/ (- x -1.0) x)) x) (* n x))
     (if (<= (/ 1.0 n) -0.04)
       t_0
       (if (<= (/ 1.0 n) 2e-12)
         (/ (log1p (/ 1.0 x)) n)
         (if (<= (/ 1.0 n) 1e+176)
           t_0
           (/
            (/ (- (- (/ 0.3333333333333333 (* x x)) -1.0) (/ 0.5 x)) x)
            n)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+162) {
		tmp = (log(((x - -1.0) / x)) * x) / (n * x);
	} else if ((1.0 / n) <= -0.04) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+162) {
		tmp = (Math.log(((x - -1.0) / x)) * x) / (n * x);
	} else if ((1.0 / n) <= -0.04) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-12) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 1e+176) {
		tmp = t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e+162:
		tmp = (math.log(((x - -1.0) / x)) * x) / (n * x)
	elif (1.0 / n) <= -0.04:
		tmp = t_0
	elif (1.0 / n) <= 2e-12:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 1e+176:
		tmp = t_0
	else:
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+162)
		tmp = Float64(Float64(log(Float64(Float64(x - -1.0) / x)) * x) / Float64(n * x));
	elseif (Float64(1.0 / n) <= -0.04)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-12)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+176)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - -1.0) - Float64(0.5 / x)) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+162], N[(N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.04], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-12], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+176], t$95$0, N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999994e161

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. neg-log N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip-rev N/A

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

      7. lift-/.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. lift-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip-rev N/A

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

      17. *-inverses N/A

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

      18. frac-times N/A

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

      19. lift-*.f64 N/A

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

      20. lower-/.f64 N/A

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

   8. Applied rewrites 67.0%

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

   if -9.9999999999999994e161 < (/.f64 #s(literal 1 binary64) n) < -0.040000000000000001 or 2e-12 < (/.f64 #s(literal 1 binary64) n) < 1e176

   1. Initial program 53.9%

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

   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 39.0%

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

   if -0.040000000000000001 < (/.f64 #s(literal 1 binary64) n) < 2e-12

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip N/A

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

      9. div-add N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. lower-log1p.f64 57.1%

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

   8. Applied rewrites 57.1%

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

   if 1e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 46.3%

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

   7. Applied rewrites 46.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 46.3%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 46.3%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 46.3%

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

   9. Applied rewrites 46.3%

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

Alternative 9: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - -1\right) - \frac{0.5}{x}}{x}}{n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -20000000.0)
   (/ (* (log (/ (- x -1.0) x)) x) (* n x))
   (if (<= (/ 1.0 n) 5e+110)
     (/ (log1p (/ 1.0 x)) n)
     (/ (/ (- (- (/ 0.3333333333333333 (* x x)) -1.0) (/ 0.5 x)) x) n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = (log(((x - -1.0) / x)) * x) / (n * x);
	} else if ((1.0 / n) <= 5e+110) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = (Math.log(((x - -1.0) / x)) * x) / (n * x);
	} else if ((1.0 / n) <= 5e+110) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -20000000.0:
		tmp = (math.log(((x - -1.0) / x)) * x) / (n * x)
	elif (1.0 / n) <= 5e+110:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = ((((0.3333333333333333 / (x * x)) - -1.0) - (0.5 / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -20000000.0)
		tmp = Float64(Float64(log(Float64(Float64(x - -1.0) / x)) * x) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e+110)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - -1.0) - Float64(0.5 / x)) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000.0], N[(N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+110], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e7

   1. Initial program 53.9%

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

   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. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 58.7%

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

   4. Applied rewrites 58.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 58.7%

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 58.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. neg-log N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip-rev N/A

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

      7. lift-/.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{x - -1}{x}\right)}}} \]

      11. mult-flip N/A

          \[\leadsto 1 \cdot \color{blue}{\frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}}} \]

      12. lift-/.f64 N/A

          \[\leadsto 1 \cdot \frac{1}{\color{blue}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}} \cdot \color{blue}{1} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}} \cdot 1 \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}} \cdot 1 \]

      16. div-flip-rev N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \cdot 1 \]

      17. *-inverses N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \cdot \frac{x}{\color{blue}{x}} \]

      18. frac-times N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{\color{blue}{n \cdot x}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{n \cdot \color{blue}{x}} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{\color{blue}{n \cdot x}} \]

   8. Applied rewrites 67.0%

      \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{\color{blue}{n \cdot x}} \]

   if -2e7 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e110

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      11. +-commutative N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      13. diff-log N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      14. lower-log.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      15. lower-/.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      17. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      18. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      19. lower--.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]
   7. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      3. neg-log N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      5. div-flip-rev N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      8. add-flip N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      9. div-add N/A

         \[\leadsto \frac{\log \left(\frac{x}{x} + \frac{1}{x}\right)}{n} \]

      10. *-inverses N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      12. lower-log1p.f64 57.1%

          \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   8. Applied rewrites 57.1%

      \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   if 4.9999999999999998e110 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      7. lower-/.f64 46.3%

         \[\leadsto \frac{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{x}}{n} \]

   7. Applied rewrites 46.3%

      \[\leadsto \frac{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{x}}{n} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{\left(\frac{\frac{1}{3}}{{x}^{2}} + 1\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      3. add-flip N/A

         \[\leadsto \frac{\frac{\left(\frac{\frac{1}{3}}{{x}^{2}} - \left(\mathsf{neg}\left(1\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{\left(\frac{\frac{1}{3}}{{x}^{2}} - -1\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      5. lower--.f64 46.3%

         \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{{x}^{2}} - -1\right) - 0.5 \cdot \frac{1}{x}}{x}}{n} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{\frac{1}{3}}{{x}^{2}} - -1\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{\left(\frac{\frac{1}{3}}{x \cdot x} - -1\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      8. lower-*.f64 46.3%

         \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - -1\right) - 0.5 \cdot \frac{1}{x}}{x}}{n} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{\frac{1}{3}}{x \cdot x} - -1\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\left(\frac{\frac{1}{3}}{x \cdot x} - -1\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}}{n} \]

      11. mult-flip-rev N/A

          \[\leadsto \frac{\frac{\left(\frac{\frac{1}{3}}{x \cdot x} - -1\right) - \frac{\frac{1}{2}}{x}}{x}}{n} \]

      12. lower-/.f64 46.3%

          \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - -1\right) - \frac{0.5}{x}}{x}}{n} \]

   9. Applied rewrites 46.3%

      \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - -1\right) - \frac{0.5}{x}}{x}}{n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 80.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -20000000.0)
   (/ (* (log (/ (- x -1.0) x)) x) (* n x))
   (if (<= (/ 1.0 n) 5e+126)
     (/ (log1p (/ 1.0 x)) n)
     (* (/ x 1.0) (/ (/ 1.0 x) (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = (log(((x - -1.0) / x)) * x) / (n * x);
	} else if ((1.0 / n) <= 5e+126) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (x / 1.0) * ((1.0 / x) / (n * x));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = (Math.log(((x - -1.0) / x)) * x) / (n * x);
	} else if ((1.0 / n) <= 5e+126) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = (x / 1.0) * ((1.0 / x) / (n * x));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -20000000.0:
		tmp = (math.log(((x - -1.0) / x)) * x) / (n * x)
	elif (1.0 / n) <= 5e+126:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = (x / 1.0) * ((1.0 / x) / (n * x))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -20000000.0)
		tmp = Float64(Float64(log(Float64(Float64(x - -1.0) / x)) * x) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e+126)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(x / 1.0) * Float64(Float64(1.0 / x) / Float64(n * x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000.0], N[(N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+126], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(x / 1.0), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e7

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      11. +-commutative N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      13. diff-log N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      14. lower-log.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      15. lower-/.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      17. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      18. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      19. lower--.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{\color{blue}{n}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      4. neg-log N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      6. div-flip-rev N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      9. div-flip-rev N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\color{blue}{\log \left(\frac{x - -1}{x}\right)}}} \]

      11. mult-flip N/A

          \[\leadsto 1 \cdot \color{blue}{\frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}}} \]

      12. lift-/.f64 N/A

          \[\leadsto 1 \cdot \frac{1}{\color{blue}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}} \cdot \color{blue}{1} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}} \cdot 1 \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{1}{\frac{n}{\log \left(\frac{x - -1}{x}\right)}} \cdot 1 \]

      16. div-flip-rev N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \cdot 1 \]

      17. *-inverses N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \cdot \frac{x}{\color{blue}{x}} \]

      18. frac-times N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{\color{blue}{n \cdot x}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{n \cdot \color{blue}{x}} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{\color{blue}{n \cdot x}} \]

   8. Applied rewrites 67.0%

      \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{\color{blue}{n \cdot x}} \]

   if -2e7 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e126

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      11. +-commutative N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      13. diff-log N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      14. lower-log.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      15. lower-/.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      17. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      18. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      19. lower--.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]
   7. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      3. neg-log N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      5. div-flip-rev N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      8. add-flip N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      9. div-add N/A

         \[\leadsto \frac{\log \left(\frac{x}{x} + \frac{1}{x}\right)}{n} \]

      10. *-inverses N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      12. lower-log1p.f64 57.1%

          \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   8. Applied rewrites 57.1%

      \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   if 4.9999999999999998e126 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 39.5%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 39.5%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      5. lower-/.f64 40.1%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 40.1%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

       4. *-inverses N/A

          \[\leadsto \frac{\frac{x}{x}}{n \cdot x} \]

       5. mult-flip N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{n \cdot x} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{x \cdot n} \]

       7. *-lft-identity N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{\left(1 \cdot x\right) \cdot n} \]

       8. associate-*l* N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(x \cdot \color{blue}{n}\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(n \cdot x\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(n \cdot x\right)} \]

       11. times-frac N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]

       13. lower-/.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n} \cdot x} \]

       14. lower-/.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot \color{blue}{x}} \]

       15. lower-/.f64 41.8%

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot x} \]

   11. Applied rewrites 41.8%

       \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot x}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -20000000.0)
   (/ (log (/ (- x -1.0) x)) n)
   (if (<= (/ 1.0 n) 5e+126)
     (/ (log1p (/ 1.0 x)) n)
     (* (/ x 1.0) (/ (/ 1.0 x) (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+126) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = (x / 1.0) * ((1.0 / x) / (n * x));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+126) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = (x / 1.0) * ((1.0 / x) / (n * x));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -20000000.0:
		tmp = math.log(((x - -1.0) / x)) / n
	elif (1.0 / n) <= 5e+126:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = (x / 1.0) * ((1.0 / x) / (n * x))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -20000000.0)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+126)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(x / 1.0) * Float64(Float64(1.0 / x) / Float64(n * x)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000.0], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+126], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(x / 1.0), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e7

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      5. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      9. lower-/.f64 58.7%

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      11. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      13. lower--.f64 58.7%

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{\color{blue}{n}} \]

   if -2e7 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e126

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      11. +-commutative N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      13. diff-log N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      14. lower-log.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      15. lower-/.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      17. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      18. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      19. lower--.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]
   7. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\log \left(\frac{x}{x - -1}\right)\right)}{n} \]

      3. neg-log N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1}{\frac{x}{x - -1}}\right)}{n} \]

      5. div-flip-rev N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      8. add-flip N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      9. div-add N/A

         \[\leadsto \frac{\log \left(\frac{x}{x} + \frac{1}{x}\right)}{n} \]

      10. *-inverses N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\log \left(1 + \frac{1}{x}\right)}{n} \]

      12. lower-log1p.f64 57.1%

          \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   8. Applied rewrites 57.1%

      \[\leadsto \frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n} \]

   if 4.9999999999999998e126 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 39.5%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 39.5%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      5. lower-/.f64 40.1%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 40.1%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

       4. *-inverses N/A

          \[\leadsto \frac{\frac{x}{x}}{n \cdot x} \]

       5. mult-flip N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{n \cdot x} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{x \cdot n} \]

       7. *-lft-identity N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{\left(1 \cdot x\right) \cdot n} \]

       8. associate-*l* N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(x \cdot \color{blue}{n}\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(n \cdot x\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(n \cdot x\right)} \]

       11. times-frac N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]

       13. lower-/.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n} \cdot x} \]

       14. lower-/.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot \color{blue}{x}} \]

       15. lower-/.f64 41.8%

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot x} \]

   11. Applied rewrites 41.8%

       \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (* (/ x 1.0) (/ (/ 1.0 x) (* n x)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 0.01) (/ (- (log (/ x (- x -1.0)))) 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 = (x / 1.0) * ((1.0 / x) / (n * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = -log((x / (x - -1.0))) / 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 = (x / 1.0) * ((1.0 / x) / (n * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = -Math.log((x / (x - -1.0))) / 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 = (x / 1.0) * ((1.0 / x) / (n * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 0.01:
		tmp = -math.log((x / (x - -1.0))) / 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(x / 1.0) * Float64(Float64(1.0 / x) / Float64(n * x)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / 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 = (x / 1.0) * ((1.0 / x) / (n * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = -log((x / (x - -1.0))) / 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[(x / 1.0), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.01], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $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 0.01 < (-.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 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 39.5%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 39.5%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      5. lower-/.f64 40.1%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 40.1%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

       4. *-inverses N/A

          \[\leadsto \frac{\frac{x}{x}}{n \cdot x} \]

       5. mult-flip N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{n \cdot x} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{x \cdot n} \]

       7. *-lft-identity N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{\left(1 \cdot x\right) \cdot n} \]

       8. associate-*l* N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(x \cdot \color{blue}{n}\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(n \cdot x\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(n \cdot x\right)} \]

       11. times-frac N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]

       13. lower-/.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n} \cdot x} \]

       14. lower-/.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot \color{blue}{x}} \]

       15. lower-/.f64 41.8%

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot x} \]

   11. Applied rewrites 41.8%

       \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]

   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))) < 0.01

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      11. +-commutative N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      13. diff-log N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      14. lower-log.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      15. lower-/.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      17. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      18. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      19. lower--.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (* (/ x 1.0) (/ (/ 1.0 x) (* n x)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 0.01) (/ (log (/ (- x -1.0) 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 = (x / 1.0) * ((1.0 / x) / (n * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = log(((x - -1.0) / 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 = (x / 1.0) * ((1.0 / x) / (n * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = Math.log(((x - -1.0) / 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 = (x / 1.0) * ((1.0 / x) / (n * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 0.01:
		tmp = math.log(((x - -1.0) / 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(x / 1.0) * Float64(Float64(1.0 / x) / Float64(n * x)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = Float64(log(Float64(Float64(x - -1.0) / 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 = (x / 1.0) * ((1.0 / x) / (n * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = log(((x - -1.0) / 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[(x / 1.0), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.01], N[(N[Log[N[(N[(x - -1.0), $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 0.01 < (-.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 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 39.5%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 39.5%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      5. lower-/.f64 40.1%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 40.1%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

       4. *-inverses N/A

          \[\leadsto \frac{\frac{x}{x}}{n \cdot x} \]

       5. mult-flip N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{n \cdot x} \]

       6. *-commutative N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{x \cdot n} \]

       7. *-lft-identity N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{\left(1 \cdot x\right) \cdot n} \]

       8. associate-*l* N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(x \cdot \color{blue}{n}\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(n \cdot x\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{x \cdot \frac{1}{x}}{1 \cdot \left(n \cdot x\right)} \]

       11. times-frac N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]

       13. lower-/.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n} \cdot x} \]

       14. lower-/.f64 N/A

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot \color{blue}{x}} \]

       15. lower-/.f64 41.8%

           \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{n \cdot x} \]

   11. Applied rewrites 41.8%

       \[\leadsto \frac{x}{1} \cdot \frac{\frac{1}{x}}{\color{blue}{n \cdot x}} \]

   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))) < 0.01

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      5. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      9. lower-/.f64 58.7%

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      11. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      13. lower--.f64 58.7%

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{\color{blue}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 72.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{x}{x \cdot \left(n \cdot x\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (/ x (* x (* n x)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 0.01) (/ (log (/ (- x -1.0) 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 = x / (x * (n * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = log(((x - -1.0) / 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 = x / (x * (n * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = Math.log(((x - -1.0) / 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 = x / (x * (n * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 0.01:
		tmp = math.log(((x - -1.0) / 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(x / Float64(x * Float64(n * x)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = Float64(log(Float64(Float64(x - -1.0) / 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 = x / (x * (n * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = log(((x - -1.0) / 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[(x / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.01], N[(N[Log[N[(N[(x - -1.0), $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 0.01 < (-.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 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 39.5%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 39.5%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      5. lower-/.f64 40.1%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 40.1%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{n \cdot x} \]

       5. mult-flip N/A

          \[\leadsto 1 \cdot \frac{1}{\color{blue}{n \cdot x}} \]

       6. *-inverses N/A

          \[\leadsto \frac{x}{x} \cdot \frac{1}{\color{blue}{n} \cdot x} \]

       7. frac-times N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       8. *-rgt-identity N/A

          \[\leadsto \frac{x}{x \cdot \left(\color{blue}{n} \cdot x\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{x}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       10. lower-*.f64 41.2%

           \[\leadsto \frac{x}{x \cdot \left(n \cdot \color{blue}{x}\right)} \]

   11. Applied rewrites 41.2%

       \[\leadsto \frac{x}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

   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))) < 0.01

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. diff-log N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      5. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{1 + x}{x}\right)}{n} \]

      7. +-commutative N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      9. lower-/.f64 58.7%

         \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\log \left(\frac{x + 1}{x}\right)}{n} \]

      11. add-flip N/A

          \[\leadsto \frac{\log \left(\frac{x - \left(\mathsf{neg}\left(1\right)\right)}{x}\right)}{n} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

      13. lower--.f64 58.7%

          \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{n} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{\log \left(\frac{x - -1}{x}\right)}{\color{blue}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.000112:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.000112) (/ (- x (log x)) n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.000112) {
		tmp = (x - log(x)) / n;
	} 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 (x <= 0.000112d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.000112) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.000112:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.000112)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.000112)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.000112], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.12e-4

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x - \log x}{n} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x - \log x}{n} \]

      2. lower-log.f64 30.3%

         \[\leadsto \frac{x - \log x}{n} \]

   7. Applied rewrites 30.3%

      \[\leadsto \frac{x - \log x}{n} \]

   if 1.12e-4 < x

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 39.5%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 39.5%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot n} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{1}{x}}{n} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{n} \]

      6. lower-/.f64 40.1%

         \[\leadsto \frac{\frac{1}{x}}{n} \]

   9. Applied rewrites 40.1%

      \[\leadsto \frac{\frac{1}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 56.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.000112:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.000112) (/ (- (log x)) n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.000112) {
		tmp = -log(x) / n;
	} 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 (x <= 0.000112d0) then
        tmp = -log(x) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.000112) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.000112:
		tmp = -math.log(x) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.000112)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.000112)
		tmp = -log(x) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.000112], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.12e-4

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      2. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\log x - \log \left(1 + x\right)\right)\right)}{n} \]

      3. sub-negate-rev N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)\right)}{n} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(\left(\log \left(1 + x\right) - \log x\right)\right)\right)}{n} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      9. lift-log.f64 N/A

         \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(1 + x\right)\right)}{n} \]

      11. +-commutative N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{-\left(\log x - \log \left(x + 1\right)\right)}{n} \]

      13. diff-log N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      14. lower-log.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      15. lower-/.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x + 1}\right)}{n} \]

      17. add-flip N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)}{n} \]

      18. metadata-eval N/A

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

      19. lower--.f64 58.7%

          \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{-\log x}{n} \]
   8. Step-by-step derivation

   9. Applied rewrites 30.4%

      \[\leadsto \frac{-\log x}{n} \]

   if 1.12e-4 < x

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 39.5%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 39.5%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot n} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{1}{x}}{n} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{n} \]

      6. lower-/.f64 40.1%

         \[\leadsto \frac{\frac{1}{x}}{n} \]

   9. Applied rewrites 40.1%

      \[\leadsto \frac{\frac{1}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 46.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40000000000:\\ \;\;\;\;\frac{x}{x \cdot \left(n \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -40000000000.0) (/ x (* x (* n x))) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40000000000.0) {
		tmp = x / (x * (n * x));
	} 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 ((1.0d0 / n) <= (-40000000000.0d0)) then
        tmp = x / (x * (n * x))
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40000000000.0) {
		tmp = x / (x * (n * x));
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -40000000000.0:
		tmp = x / (x * (n * x))
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40000000000.0)
		tmp = Float64(x / Float64(x * Float64(n * x)));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -40000000000.0)
		tmp = x / (x * (n * x));
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -40000000000.0], N[(x / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e10

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 39.5%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 39.5%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{n}}{x} \]

      5. lower-/.f64 40.1%

         \[\leadsto \frac{\frac{1}{n}}{x} \]

   9. Applied rewrites 40.1%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{n}}{x} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{n \cdot x} \]

       5. mult-flip N/A

          \[\leadsto 1 \cdot \frac{1}{\color{blue}{n \cdot x}} \]

       6. *-inverses N/A

          \[\leadsto \frac{x}{x} \cdot \frac{1}{\color{blue}{n} \cdot x} \]

       7. frac-times N/A

          \[\leadsto \frac{x \cdot 1}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       8. *-rgt-identity N/A

          \[\leadsto \frac{x}{x \cdot \left(\color{blue}{n} \cdot x\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{x}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

       10. lower-*.f64 41.2%

           \[\leadsto \frac{x}{x \cdot \left(n \cdot \color{blue}{x}\right)} \]

   11. Applied rewrites 41.2%

       \[\leadsto \frac{x}{x \cdot \color{blue}{\left(n \cdot x\right)}} \]

   if -4e10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   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. lower--.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      3. lower-log.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

      5. lower-log.f64 58.7%

         \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lower-*.f64 39.5%

         \[\leadsto \frac{1}{n \cdot x} \]

   7. Applied rewrites 39.5%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{n \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot n} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{1}{x}}{n} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{n} \]

      6. lower-/.f64 40.1%

         \[\leadsto \frac{\frac{1}{x}}{n} \]

   9. Applied rewrites 40.1%

      \[\leadsto \frac{\frac{1}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 40.1% accurate, 5.8× speedup

Math

\[\frac{\frac{1}{x}}{n} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 x) n))

C

double code(double x, double n) {
	return (1.0 / x) / 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 = (1.0d0 / x) / n
end function

Java

public static double code(double x, double n) {
	return (1.0 / x) / n;
}

Python

def code(x, n):
	return (1.0 / x) / n

Julia

function code(x, n)
	return Float64(Float64(1.0 / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

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. lower--.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   3. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   5. lower-log.f64 58.7%

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 58.7%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lower-*.f64 39.5%

      \[\leadsto \frac{1}{n \cdot x} \]

7. Applied rewrites 39.5%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{n \cdot x} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot n} \]

   4. associate-/l/ N/A

      \[\leadsto \frac{\frac{1}{x}}{n} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1}{x}}{n} \]

   6. lower-/.f64 40.1%

      \[\leadsto \frac{\frac{1}{x}}{n} \]

9. Applied rewrites 40.1%

   \[\leadsto \frac{\frac{1}{x}}{n} \]
10. Add Preprocessing

Alternative 19: 40.1% accurate, 5.8× speedup

Math

\[\frac{\frac{1}{n}}{x} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

double code(double x, double n) {
	return (1.0 / n) / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) / x
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

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. lower--.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   3. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   5. lower-log.f64 58.7%

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 58.7%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lower-*.f64 39.5%

      \[\leadsto \frac{1}{n \cdot x} \]

7. Applied rewrites 39.5%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{n \cdot x} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{n}}{x} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{n}}{x} \]

   5. lower-/.f64 40.1%

      \[\leadsto \frac{\frac{1}{n}}{x} \]

9. Applied rewrites 40.1%

   \[\leadsto \frac{\frac{1}{n}}{x} \]
10. Add Preprocessing

Alternative 20: 39.5% accurate, 6.1× speedup

Math

\[\frac{1}{n \cdot x} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

double code(double x, double n) {
	return 1.0 / (n * x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 1.0d0 / (n * x)
end function

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * x);
end

Wolfram

code[x_, n_] := N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

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. lower--.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   3. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

   5. lower-log.f64 58.7%

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 58.7%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{n \cdot \color{blue}{x}} \]

   2. lower-*.f64 39.5%

      \[\leadsto \frac{1}{n \cdot x} \]

7. Applied rewrites 39.5%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025196 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))