2nthrt (problem 3.4.6)

Percentage Accurate: 53.0% → 93.7%

Time: 19.0s

Alternatives: 18

Speedup: 2.4×

• could not determine a ground truth

  1. x = 5.773414570056491e+137
  2. n = 5.295711614824284e-244

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-23}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+155}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-23)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-14)
       (/ 1.0 (/ n (log1p (/ 1.0 x))))
       (if (<= (/ 1.0 n) 1e+155)
         (- (- (/ x n) -1.0) t_0)
         (/ (- (/ (log x) n) -1.0) (* (- n) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-23) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = 1.0 / (n / log1p((1.0 / x)));
	} else if ((1.0 / n) <= 1e+155) {
		tmp = ((x / n) - -1.0) - t_0;
	} else {
		tmp = ((log(x) / n) - -1.0) / (-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) <= -1e-23) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = 1.0 / (n / Math.log1p((1.0 / x)));
	} else if ((1.0 / n) <= 1e+155) {
		tmp = ((x / n) - -1.0) - t_0;
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-23:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-14:
		tmp = 1.0 / (n / math.log1p((1.0 / x)))
	elif (1.0 / n) <= 1e+155:
		tmp = ((x / n) - -1.0) - t_0
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-23)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = Float64(1.0 / Float64(n / log1p(Float64(1.0 / x))));
	elseif (Float64(1.0 / n) <= 1e+155)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(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], -1e-23], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-14], N[(1.0 / N[(n / N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+155], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999996e-24

   1. Initial program 53.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. 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-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. log-rec N/A

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

      9. lift-log.f64 N/A

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

      10. frac-2neg N/A

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

      11. mult-flip N/A

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

      12. lift-log.f64 N/A

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

      13. exp-to-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if -9.9999999999999996e-24 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e-14

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 58.5

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 58.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.5

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

   6. Applied rewrites 58.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. metadata-eval N/A

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

      5. add-flip N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. *-inverses N/A

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

      10. lift-/.f64 N/A

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

      11. lower-log1p.f64 57.2

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

   8. Applied rewrites 57.2%

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

   if 5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000001e155

   1. Initial program 53.0%

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

   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 31.3

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

   4. Applied rewrites 31.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 31.3

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

   6. Applied rewrites 31.3%

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

   if 1.00000000000000001e155 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.0%

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

   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 64.5%

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

   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 21.3

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

   7. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 21.3%

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

Alternative 2: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-23}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+155}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-23)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-14)
       (/ 1.0 (/ n (log1p (/ 1.0 x))))
       (if (<= (/ 1.0 n) 1e+155)
         (- 1.0 t_0)
         (/ (- (/ (log x) n) -1.0) (* (- n) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-23) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = 1.0 / (n / log1p((1.0 / x)));
	} else if ((1.0 / n) <= 1e+155) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((log(x) / n) - -1.0) / (-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) <= -1e-23) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = 1.0 / (n / Math.log1p((1.0 / x)));
	} else if ((1.0 / n) <= 1e+155) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-23:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-14:
		tmp = 1.0 / (n / math.log1p((1.0 / x)))
	elif (1.0 / n) <= 1e+155:
		tmp = 1.0 - t_0
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-23)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = Float64(1.0 / Float64(n / log1p(Float64(1.0 / x))));
	elseif (Float64(1.0 / n) <= 1e+155)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(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], -1e-23], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-14], N[(1.0 / N[(n / N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+155], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999996e-24

   1. Initial program 53.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. 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-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. log-rec N/A

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

      9. lift-log.f64 N/A

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

      10. frac-2neg N/A

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

      11. mult-flip N/A

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

      12. lift-log.f64 N/A

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

      13. exp-to-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if -9.9999999999999996e-24 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e-14

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 58.5

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 58.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.5

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

   6. Applied rewrites 58.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. metadata-eval N/A

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

      5. add-flip N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. *-inverses N/A

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

      10. lift-/.f64 N/A

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

      11. lower-log1p.f64 57.2

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

   8. Applied rewrites 57.2%

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

   if 5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000001e155

   1. Initial program 53.0%

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

   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 38.7%

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

   if 1.00000000000000001e155 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.0%

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

   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 64.5%

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

   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 21.3

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

   7. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 21.3%

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

Alternative 3: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-23}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-23)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-14)
       (/ 1.0 (/ n (log1p (/ 1.0 x))))
       (-
        (+
         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) <= -1e-23) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = 1.0 / (n / log1p((1.0 / x)));
	} 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) <= -1e-23)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = Float64(1.0 / Float64(n / log1p(Float64(1.0 / x))));
	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], -1e-23], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-14], N[(1.0 / N[(n / N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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) < -9.9999999999999996e-24

   1. Initial program 53.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. 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-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. log-rec N/A

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

      9. lift-log.f64 N/A

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

      10. frac-2neg N/A

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

      11. mult-flip N/A

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

      12. lift-log.f64 N/A

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

      13. exp-to-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if -9.9999999999999996e-24 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e-14

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 58.5

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 58.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.5

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

   6. Applied rewrites 58.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. metadata-eval N/A

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

      5. add-flip N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. *-inverses N/A

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

      10. lift-/.f64 N/A

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

      11. lower-log1p.f64 57.2

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

   8. Applied rewrites 57.2%

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

   if 5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.0%

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

   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 23.5

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

   4. Applied rewrites 23.5%

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

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;\frac{1 \cdot x}{\frac{n}{\log \left(\frac{x - -1}{x}\right)} \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+155}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1000.0)
   (/ (* 1.0 x) (* (/ n (log (/ (- x -1.0) x))) x))
   (if (<= (/ 1.0 n) 5e-14)
     (/ 1.0 (/ n (log1p (/ 1.0 x))))
     (if (<= (/ 1.0 n) 1e+155)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (- (/ (log x) n) -1.0) (* (- n) x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = (1.0 * x) / ((n / log(((x - -1.0) / x))) * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = 1.0 / (n / log1p((1.0 / x)));
	} else if ((1.0 / n) <= 1e+155) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = (1.0 * x) / ((n / Math.log(((x - -1.0) / x))) * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = 1.0 / (n / Math.log1p((1.0 / x)));
	} else if ((1.0 / n) <= 1e+155) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1000.0:
		tmp = (1.0 * x) / ((n / math.log(((x - -1.0) / x))) * x)
	elif (1.0 / n) <= 5e-14:
		tmp = 1.0 / (n / math.log1p((1.0 / x)))
	elif (1.0 / n) <= 1e+155:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1000.0)
		tmp = Float64(Float64(1.0 * x) / Float64(Float64(n / log(Float64(Float64(x - -1.0) / x))) * x));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = Float64(1.0 / Float64(n / log1p(Float64(1.0 / x))));
	elseif (Float64(1.0 / n) <= 1e+155)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(Float64(-n) * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1000.0], 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], 5e-14], N[(1.0 / N[(n / N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+155], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e3

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 58.5

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 58.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.5

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

   6. Applied rewrites 58.5%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-inverses N/A

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

      5. frac-times N/A

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

      6. *-inverses N/A

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

      7. lower-/.f64 N/A

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

      8. *-inverses N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 67.8

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

   8. Applied rewrites 67.8%

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

   if -1e3 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e-14

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 58.5

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 58.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.5

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

   6. Applied rewrites 58.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. metadata-eval N/A

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

      5. add-flip N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. *-inverses N/A

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

      10. lift-/.f64 N/A

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

      11. lower-log1p.f64 57.2

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

   8. Applied rewrites 57.2%

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

   if 5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000001e155

   1. Initial program 53.0%

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

   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 38.7%

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

   if 1.00000000000000001e155 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.0%

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

   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 64.5%

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

   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 21.3

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

   7. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 21.3%

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

Alternative 5: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+155}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1000.0)
   (/ (* (log (/ (- x -1.0) x)) x) (* n x))
   (if (<= (/ 1.0 n) 5e-14)
     (/ 1.0 (/ n (log1p (/ 1.0 x))))
     (if (<= (/ 1.0 n) 1e+155)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (- (/ (log x) n) -1.0) (* (- n) x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = (log(((x - -1.0) / x)) * x) / (n * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = 1.0 / (n / log1p((1.0 / x)));
	} else if ((1.0 / n) <= 1e+155) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = (Math.log(((x - -1.0) / x)) * x) / (n * x);
	} else if ((1.0 / n) <= 5e-14) {
		tmp = 1.0 / (n / Math.log1p((1.0 / x)));
	} else if ((1.0 / n) <= 1e+155) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1000.0:
		tmp = (math.log(((x - -1.0) / x)) * x) / (n * x)
	elif (1.0 / n) <= 5e-14:
		tmp = 1.0 / (n / math.log1p((1.0 / x)))
	elif (1.0 / n) <= 1e+155:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1000.0)
		tmp = Float64(Float64(log(Float64(Float64(x - -1.0) / x)) * x) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = Float64(1.0 / Float64(n / log1p(Float64(1.0 / x))));
	elseif (Float64(1.0 / n) <= 1e+155)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(Float64(-n) * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1000.0], 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-14], N[(1.0 / N[(n / N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+155], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e3

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 58.5

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 58.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.5

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

   6. Applied rewrites 58.5%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-inverses N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip-rev N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

   if -1e3 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e-14

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 58.5

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 58.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.5

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

   6. Applied rewrites 58.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. metadata-eval N/A

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

      5. add-flip N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. *-inverses N/A

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

      10. lift-/.f64 N/A

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

      11. lower-log1p.f64 57.2

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

   8. Applied rewrites 57.2%

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

   if 5.0000000000000002e-14 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000001e155

   1. Initial program 53.0%

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

   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 38.7%

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

   if 1.00000000000000001e155 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.0%

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

   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 64.5%

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

   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 21.3

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

   7. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 21.3%

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

Alternative 6: 81.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right) \cdot x}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+94}:\\ \;\;\;\;\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1000.0)
   (/ (* (log (/ (- x -1.0) x)) x) (* n x))
   (if (<= (/ 1.0 n) 1e+94)
     (/ 1.0 (/ n (log1p (/ 1.0 x))))
     (/ (- (/ (log x) n) -1.0) (* (- n) x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = (log(((x - -1.0) / x)) * x) / (n * x);
	} else if ((1.0 / n) <= 1e+94) {
		tmp = 1.0 / (n / log1p((1.0 / x)));
	} else {
		tmp = ((log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1000.0) {
		tmp = (Math.log(((x - -1.0) / x)) * x) / (n * x);
	} else if ((1.0 / n) <= 1e+94) {
		tmp = 1.0 / (n / Math.log1p((1.0 / x)));
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1000.0:
		tmp = (math.log(((x - -1.0) / x)) * x) / (n * x)
	elif (1.0 / n) <= 1e+94:
		tmp = 1.0 / (n / math.log1p((1.0 / x)))
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1000.0)
		tmp = Float64(Float64(log(Float64(Float64(x - -1.0) / x)) * x) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e+94)
		tmp = Float64(1.0 / Float64(n / log1p(Float64(1.0 / x))));
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(Float64(-n) * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1000.0], 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], 1e+94], N[(1.0 / N[(n / N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e3

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 58.5

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 58.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.5

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

   6. Applied rewrites 58.5%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-inverses N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip-rev N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

   if -1e3 < (/.f64 #s(literal 1 binary64) n) < 1e94

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 58.5

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 58.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.5

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

   6. Applied rewrites 58.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. metadata-eval N/A

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

      5. add-flip N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. *-inverses N/A

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

      10. lift-/.f64 N/A

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

      11. lower-log1p.f64 57.2

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

   8. Applied rewrites 57.2%

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

   if 1e94 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.0%

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

   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 64.5%

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

   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 21.3

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

   7. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 21.3%

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

Alternative 7: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+100}:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1000:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+94}:\\ \;\;\;\;\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e+100)
   (/ (* x 1.0) (* x (* n x)))
   (if (<= (/ 1.0 n) -1000.0)
     (/ (- (log (/ x (- x -1.0)))) n)
     (if (<= (/ 1.0 n) 1e+94)
       (/ 1.0 (/ n (log1p (/ 1.0 x))))
       (/ (- (/ (log x) n) -1.0) (* (- n) x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+100) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if ((1.0 / n) <= -1000.0) {
		tmp = -log((x / (x - -1.0))) / n;
	} else if ((1.0 / n) <= 1e+94) {
		tmp = 1.0 / (n / log1p((1.0 / x)));
	} else {
		tmp = ((log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+100) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if ((1.0 / n) <= -1000.0) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else if ((1.0 / n) <= 1e+94) {
		tmp = 1.0 / (n / Math.log1p((1.0 / x)));
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e+100:
		tmp = (x * 1.0) / (x * (n * x))
	elif (1.0 / n) <= -1000.0:
		tmp = -math.log((x / (x - -1.0))) / n
	elif (1.0 / n) <= 1e+94:
		tmp = 1.0 / (n / math.log1p((1.0 / x)))
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+100)
		tmp = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (Float64(1.0 / n) <= -1000.0)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	elseif (Float64(1.0 / n) <= 1e+94)
		tmp = Float64(1.0 / Float64(n / log1p(Float64(1.0 / x))));
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(Float64(-n) * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+100], N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1000.0], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+94], N[(1.0 / N[(n / N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000003e100

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 40.1

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 40.6

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

   9. Applied rewrites 40.6%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   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. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.0

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

   11. Applied rewrites 41.0%

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

   if -2.00000000000000003e100 < (/.f64 #s(literal 1 binary64) n) < -1e3

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 58.6

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

      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 58.6

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

   6. Applied rewrites 58.6%

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

   if -1e3 < (/.f64 #s(literal 1 binary64) n) < 1e94

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 58.5

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. lower-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 58.5

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 58.5

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

   6. Applied rewrites 58.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. metadata-eval N/A

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

      5. add-flip N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. *-inverses N/A

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

      10. lift-/.f64 N/A

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

      11. lower-log1p.f64 57.2

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

   8. Applied rewrites 57.2%

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

   if 1e94 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.0%

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

   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 64.5%

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

   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 21.3

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

   7. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 21.3%

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

Alternative 8: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n} - -1}{\left(-n\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (* x 1.0) (* x (* n x)))
     (if (<= t_0 0.1)
       (/ (- (log (/ x (- x -1.0)))) n)
       (/ (- (/ (log x) n) -1.0) (* (- n) x))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 0.1) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = ((log(x) / n) - -1.0) / (-n * x);
	}
	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 tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 0.1) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = ((Math.log(x) / n) - -1.0) / (-n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x * 1.0) / (x * (n * x))
	elif t_0 <= 0.1:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = ((math.log(x) / n) - -1.0) / (-n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (t_0 <= 0.1)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = Float64(Float64(Float64(log(x) / n) - -1.0) / Float64(Float64(-n) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * 1.0) / (x * (n * x));
	elseif (t_0 <= 0.1)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = ((log(x) / n) - -1.0) / (-n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / N[((-n) * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 40.1

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 40.6

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

   9. Applied rewrites 40.6%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   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. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.0

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

   11. Applied rewrites 41.0%

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

   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))) < 0.10000000000000001

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 58.6

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

      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 58.6

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

   6. Applied rewrites 58.6%

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

   if 0.10000000000000001 < (-.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 53.0%

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

   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 64.5%

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

   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 21.3

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

   7. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 21.3%

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

Alternative 9: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(n \cdot \log x\right)}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (* x 1.0) (* x (* n x)))
     (if (<= t_0 5e-13)
       (/ (- (log (/ x (- x -1.0)))) n)
       (/ (* -1.0 (* n (log x))) (* n n))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 5e-13) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = (-1.0 * (n * log(x))) / (n * n);
	}
	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 tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 5e-13) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = (-1.0 * (n * Math.log(x))) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x * 1.0) / (x * (n * x))
	elif t_0 <= 5e-13:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = (-1.0 * (n * math.log(x))) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (t_0 <= 5e-13)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = Float64(Float64(-1.0 * Float64(n * log(x))) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * 1.0) / (x * (n * x));
	elseif (t_0 <= 5e-13)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = (-1.0 * (n * log(x))) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-13], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(N[(-1.0 * N[(n * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 40.1

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 40.6

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

   9. Applied rewrites 40.6%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   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. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.0

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

   11. Applied rewrites 41.0%

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

   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))) < 4.9999999999999999e-13

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 58.6

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

      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 58.6

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

   6. Applied rewrites 58.6%

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

   if 4.9999999999999999e-13 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 47.9

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 47.9

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

   6. Applied rewrites 47.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 37.4

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

   9. Applied rewrites 37.4%

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

Alternative 10: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (* x 1.0) (* x (* n x)))
     (if (<= t_0 0.1) (/ (- (log (/ x (- x -1.0)))) n) (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 0.1) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	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 tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 0.1) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x * 1.0) / (x * (n * x))
	elif t_0 <= 0.1:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (t_0 <= 0.1)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * 1.0) / (x * (n * x));
	elseif (t_0 <= 0.1)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 40.1

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 40.6

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

   9. Applied rewrites 40.6%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   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. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.0

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

   11. Applied rewrites 41.0%

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

   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))) < 0.10000000000000001

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 58.6

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

      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 58.6

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

   6. Applied rewrites 58.6%

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

   if 0.10000000000000001 < (-.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 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 47.9

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 47.9

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

   6. Applied rewrites 47.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 40.9

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.9%

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

Alternative 11: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (* x 1.0) (* x (* n x)))
     (if (<= t_0 0.1) (/ (log (/ (- x -1.0) x)) n) (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 0.1) {
		tmp = log(((x - -1.0) / x)) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	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 tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * 1.0) / (x * (n * x));
	} else if (t_0 <= 0.1) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x * 1.0) / (x * (n * x))
	elif t_0 <= 0.1:
		tmp = math.log(((x - -1.0) / x)) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)));
	elseif (t_0 <= 0.1)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * 1.0) / (x * (n * x));
	elseif (t_0 <= 0.1)
		tmp = log(((x - -1.0) / x)) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 40.1

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 40.6

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

   9. Applied rewrites 40.6%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
   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. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 41.0

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

   11. Applied rewrites 41.0%

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

   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))) < 0.10000000000000001

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 58.6

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

      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 58.6

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

   6. Applied rewrites 58.6%

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

   if 0.10000000000000001 < (-.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 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 47.9

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 47.9

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

   6. Applied rewrites 47.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 40.9

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.9%

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

Alternative 12: 60.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x} - -1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.95)
   (/ (- x (log x)) n)
   (if (<= x 2.9e+155)
     (/ (/ (- (/ -0.5 x) -1.0) x) n)
     (/ (/ (/ -0.5 x) x) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.9e+155) {
		tmp = (((-0.5 / x) - -1.0) / x) / n;
	} else {
		tmp = ((-0.5 / x) / 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 <= 0.95d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.9d+155) then
        tmp = ((((-0.5d0) / x) - (-1.0d0)) / x) / n
    else
        tmp = (((-0.5d0) / x) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.9e+155) {
		tmp = (((-0.5 / x) - -1.0) / x) / n;
	} else {
		tmp = ((-0.5 / x) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.95:
		tmp = (x - math.log(x)) / n
	elif x <= 2.9e+155:
		tmp = (((-0.5 / x) - -1.0) / x) / n
	else:
		tmp = ((-0.5 / x) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.9e+155)
		tmp = Float64(Float64(Float64(Float64(-0.5 / x) - -1.0) / x) / n);
	else
		tmp = Float64(Float64(Float64(-0.5 / x) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.95)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.9e+155)
		tmp = (((-0.5 / x) - -1.0) / x) / n;
	else
		tmp = ((-0.5 / x) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.9e+155], N[(N[(N[(N[(-0.5 / x), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(-0.5 / x), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.94999999999999996

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 31.6

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

   7. Applied rewrites 31.6%

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

   if 0.94999999999999996 < x < 2.8999999999999999e155

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 27.7

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

   7. Applied rewrites 27.7%

      \[\leadsto \frac{\frac{1 - 0.5 \cdot \frac{1}{x}}{x}}{n} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. distribute-neg-frac N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 27.7

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

   9. Applied rewrites 27.7%

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

   if 2.8999999999999999e155 < x

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 27.7

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 21.7

         \[\leadsto \frac{\frac{\frac{-0.5}{x}}{x}}{n} \]

   10. Applied rewrites 21.7%

       \[\leadsto \frac{\frac{\frac{-0.5}{x}}{x}}{n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 59.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+155}:\\ \;\;\;\;\frac{x \cdot \frac{1}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.8e-8)
   (/ (- x (log x)) n)
   (if (<= x 9e+155) (/ (* x (/ 1.0 n)) (* x x)) (/ (/ (/ -0.5 x) x) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.8e-8) {
		tmp = (x - log(x)) / n;
	} else if (x <= 9e+155) {
		tmp = (x * (1.0 / n)) / (x * x);
	} else {
		tmp = ((-0.5 / x) / 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 <= 2.8d-8) then
        tmp = (x - log(x)) / n
    else if (x <= 9d+155) then
        tmp = (x * (1.0d0 / n)) / (x * x)
    else
        tmp = (((-0.5d0) / x) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.8e-8) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 9e+155) {
		tmp = (x * (1.0 / n)) / (x * x);
	} else {
		tmp = ((-0.5 / x) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.8e-8:
		tmp = (x - math.log(x)) / n
	elif x <= 9e+155:
		tmp = (x * (1.0 / n)) / (x * x)
	else:
		tmp = ((-0.5 / x) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.8e-8)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 9e+155)
		tmp = Float64(Float64(x * Float64(1.0 / n)) / Float64(x * x));
	else
		tmp = Float64(Float64(Float64(-0.5 / x) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.8e-8)
		tmp = (x - log(x)) / n;
	elseif (x <= 9e+155)
		tmp = (x * (1.0 / n)) / (x * x);
	else
		tmp = ((-0.5 / x) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.8e-8], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 9e+155], N[(N[(x * N[(1.0 / n), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 / x), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.7999999999999999e-8

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 31.6

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

   7. Applied rewrites 31.6%

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

   if 2.7999999999999999e-8 < x < 8.99999999999999947e155

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 40.1

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 40.6

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

   9. Applied rewrites 40.6%

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

       1. lift-/.f64 N/A

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

       2. *-lft-identity N/A

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

       3. associate-/l* N/A

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

       4. *-inverses N/A

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

       5. frac-times N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 39.9

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

   11. Applied rewrites 39.9%

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

   if 8.99999999999999947e155 < x

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 27.7

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 21.7

         \[\leadsto \frac{\frac{\frac{-0.5}{x}}{x}}{n} \]

   10. Applied rewrites 21.7%

       \[\leadsto \frac{\frac{\frac{-0.5}{x}}{x}}{n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.8e-8)
   (/ (- x (log x)) n)
   (if (<= x 2.9e+155) (/ (/ 1.0 n) x) (/ (/ (/ -0.5 x) x) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.8e-8) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.9e+155) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = ((-0.5 / x) / 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 <= 2.8d-8) then
        tmp = (x - log(x)) / n
    else if (x <= 2.9d+155) then
        tmp = (1.0d0 / n) / x
    else
        tmp = (((-0.5d0) / x) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.8e-8) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.9e+155) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = ((-0.5 / x) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.8e-8:
		tmp = (x - math.log(x)) / n
	elif x <= 2.9e+155:
		tmp = (1.0 / n) / x
	else:
		tmp = ((-0.5 / x) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.8e-8)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.9e+155)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(Float64(Float64(-0.5 / x) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.8e-8)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.9e+155)
		tmp = (1.0 / n) / x;
	else
		tmp = ((-0.5 / x) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.8e-8], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.9e+155], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(-0.5 / x), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.7999999999999999e-8

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 31.6

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

   7. Applied rewrites 31.6%

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

   if 2.7999999999999999e-8 < x < 2.8999999999999999e155

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 40.1

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 40.6

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

   9. Applied rewrites 40.6%

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

   if 2.8999999999999999e155 < x

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 27.7

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 21.7

         \[\leadsto \frac{\frac{\frac{-0.5}{x}}{x}}{n} \]

   10. Applied rewrites 21.7%

       \[\leadsto \frac{\frac{\frac{-0.5}{x}}{x}}{n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 45.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6500:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6500.0)
   (/ (/ n x) (* n n))
   (if (<= x 2.9e+155) (/ (/ 1.0 n) x) (/ (/ (/ -0.5 x) x) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6500.0) {
		tmp = (n / x) / (n * n);
	} else if (x <= 2.9e+155) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = ((-0.5 / x) / 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 <= 6500.0d0) then
        tmp = (n / x) / (n * n)
    else if (x <= 2.9d+155) then
        tmp = (1.0d0 / n) / x
    else
        tmp = (((-0.5d0) / x) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6500.0) {
		tmp = (n / x) / (n * n);
	} else if (x <= 2.9e+155) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = ((-0.5 / x) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6500.0:
		tmp = (n / x) / (n * n)
	elif x <= 2.9e+155:
		tmp = (1.0 / n) / x
	else:
		tmp = ((-0.5 / x) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6500.0)
		tmp = Float64(Float64(n / x) / Float64(n * n));
	elseif (x <= 2.9e+155)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(Float64(Float64(-0.5 / x) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6500.0)
		tmp = (n / x) / (n * n);
	elseif (x <= 2.9e+155)
		tmp = (1.0 / n) / x;
	else
		tmp = ((-0.5 / x) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6500.0], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e+155], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(-0.5 / x), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6500

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 47.9

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 47.9

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

   6. Applied rewrites 47.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 40.9

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.9%

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

   if 6500 < x < 2.8999999999999999e155

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 40.1

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 40.6

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

   9. Applied rewrites 40.6%

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

   if 2.8999999999999999e155 < x

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 27.7

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 21.7

         \[\leadsto \frac{\frac{\frac{-0.5}{x}}{x}}{n} \]

   10. Applied rewrites 21.7%

       \[\leadsto \frac{\frac{\frac{-0.5}{x}}{x}}{n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 42.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.5e-6) (/ (/ n x) (* n n)) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.5e-6) {
		tmp = (n / x) / (n * n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.5d-6) then
        tmp = (n / x) / (n * n)
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.5e-6) {
		tmp = (n / x) / (n * n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.5e-6:
		tmp = (n / x) / (n * n)
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.5e-6)
		tmp = Float64(Float64(n / x) / Float64(n * n));
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.5e-6)
		tmp = (n / x) / (n * n);
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.5e-6], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.49999999999999995e-6

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 47.9

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 47.9

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

   6. Applied rewrites 47.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 40.9

         \[\leadsto \frac{\frac{n}{x}}{n \cdot n} \]

   9. Applied rewrites 40.9%

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

   if 3.49999999999999995e-6 < x

   1. Initial program 53.0%

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

   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 58.5

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

   4. Applied rewrites 58.5%

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

   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 40.1

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

   7. Applied rewrites 40.1%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 40.6

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

   9. Applied rewrites 40.6%

      \[\leadsto \frac{\frac{1}{n}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 40.6% accurate, 5.8× speedup

Math

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

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 53.0%

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

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 58.5

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

4. Applied rewrites 58.5%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

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 40.1

      \[\leadsto \frac{1}{n \cdot x} \]

7. Applied rewrites 40.1%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
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 40.6

      \[\leadsto \frac{\frac{1}{n}}{x} \]

9. Applied rewrites 40.6%

   \[\leadsto \frac{\frac{1}{n}}{x} \]
10. Add Preprocessing

Alternative 18: 40.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n \cdot x} \end{array} \]

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 53.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

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 58.5

      \[\leadsto \frac{\log \left(1 + x\right) - \log x}{n} \]

4. Applied rewrites 58.5%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

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 40.1

      \[\leadsto \frac{1}{n \cdot x} \]

7. Applied rewrites 40.1%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025162 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))