2nthrt (problem 3.4.6)

Percentage Accurate: 54.2% → 82.7%

Time: 20.5s

Alternatives: 19

Speedup: 2.6×

• could not determine a ground truth

  1. x = 1.1643866840641761e+117
  2. n = 5.666880904581514e-254

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\log x}{n} - -1}{x}\\ t_1 := \frac{x - -1}{x}\\ \mathbf{if}\;n \leq -9 \cdot 10^{+63}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{elif}\;n \leq -1500:\\ \;\;\;\;\frac{t\_0}{n}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-166}:\\ \;\;\;\;\log \left({t\_1}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;n \leq 52000000000:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 10^{+105}:\\ \;\;\;\;t\_0 \cdot \frac{1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t\_1}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (/ (log x) n) -1.0) x)) (t_1 (/ (- x -1.0) x)))
   (if (<= n -9e+63)
     (/ (- (log (/ x (- x -1.0)))) n)
     (if (<= n -1500.0)
       (/ t_0 n)
       (if (<= n 2.1e-166)
         (log (pow t_1 (/ 1.0 n)))
         (if (<= n 52000000000.0)
           (- (- (/ x n) -1.0) (pow x (/ 1.0 n)))
           (if (<= n 1e+105) (* t_0 (/ 1.0 n)) (/ (log t_1) n))))))))

C

double code(double x, double n) {
	double t_0 = ((log(x) / n) - -1.0) / x;
	double t_1 = (x - -1.0) / x;
	double tmp;
	if (n <= -9e+63) {
		tmp = -log((x / (x - -1.0))) / n;
	} else if (n <= -1500.0) {
		tmp = t_0 / n;
	} else if (n <= 2.1e-166) {
		tmp = log(pow(t_1, (1.0 / n)));
	} else if (n <= 52000000000.0) {
		tmp = ((x / n) - -1.0) - pow(x, (1.0 / n));
	} else if (n <= 1e+105) {
		tmp = t_0 * (1.0 / n);
	} else {
		tmp = log(t_1) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((log(x) / n) - (-1.0d0)) / x
    t_1 = (x - (-1.0d0)) / x
    if (n <= (-9d+63)) then
        tmp = -log((x / (x - (-1.0d0)))) / n
    else if (n <= (-1500.0d0)) then
        tmp = t_0 / n
    else if (n <= 2.1d-166) then
        tmp = log((t_1 ** (1.0d0 / n)))
    else if (n <= 52000000000.0d0) then
        tmp = ((x / n) - (-1.0d0)) - (x ** (1.0d0 / n))
    else if (n <= 1d+105) then
        tmp = t_0 * (1.0d0 / n)
    else
        tmp = log(t_1) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = ((Math.log(x) / n) - -1.0) / x;
	double t_1 = (x - -1.0) / x;
	double tmp;
	if (n <= -9e+63) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else if (n <= -1500.0) {
		tmp = t_0 / n;
	} else if (n <= 2.1e-166) {
		tmp = Math.log(Math.pow(t_1, (1.0 / n)));
	} else if (n <= 52000000000.0) {
		tmp = ((x / n) - -1.0) - Math.pow(x, (1.0 / n));
	} else if (n <= 1e+105) {
		tmp = t_0 * (1.0 / n);
	} else {
		tmp = Math.log(t_1) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((math.log(x) / n) - -1.0) / x
	t_1 = (x - -1.0) / x
	tmp = 0
	if n <= -9e+63:
		tmp = -math.log((x / (x - -1.0))) / n
	elif n <= -1500.0:
		tmp = t_0 / n
	elif n <= 2.1e-166:
		tmp = math.log(math.pow(t_1, (1.0 / n)))
	elif n <= 52000000000.0:
		tmp = ((x / n) - -1.0) - math.pow(x, (1.0 / n))
	elif n <= 1e+105:
		tmp = t_0 * (1.0 / n)
	else:
		tmp = math.log(t_1) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(log(x) / n) - -1.0) / x)
	t_1 = Float64(Float64(x - -1.0) / x)
	tmp = 0.0
	if (n <= -9e+63)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	elseif (n <= -1500.0)
		tmp = Float64(t_0 / n);
	elseif (n <= 2.1e-166)
		tmp = log((t_1 ^ Float64(1.0 / n)));
	elseif (n <= 52000000000.0)
		tmp = Float64(Float64(Float64(x / n) - -1.0) - (x ^ Float64(1.0 / n)));
	elseif (n <= 1e+105)
		tmp = Float64(t_0 * Float64(1.0 / n));
	else
		tmp = Float64(log(t_1) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((log(x) / n) - -1.0) / x;
	t_1 = (x - -1.0) / x;
	tmp = 0.0;
	if (n <= -9e+63)
		tmp = -log((x / (x - -1.0))) / n;
	elseif (n <= -1500.0)
		tmp = t_0 / n;
	elseif (n <= 2.1e-166)
		tmp = log((t_1 ^ (1.0 / n)));
	elseif (n <= 52000000000.0)
		tmp = ((x / n) - -1.0) - (x ^ (1.0 / n));
	elseif (n <= 1e+105)
		tmp = t_0 * (1.0 / n);
	else
		tmp = log(t_1) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[n, -9e+63], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[n, -1500.0], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[n, 2.1e-166], N[Log[N[Power[t$95$1, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 52000000000.0], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1e+105], N[(t$95$0 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision], N[(N[Log[t$95$1], $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -9.00000000000000034e63

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. neg-log N/A

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

      11. lower-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lower-/.f64 58.8

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 58.8

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

   6. Applied rewrites 58.8%

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

   if -9.00000000000000034e63 < n < -1500

   1. Initial program 54.2%

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

   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 65.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 41.2

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

   7. Applied rewrites 41.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 41.2

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. log-rec N/A

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

      13. lift-log.f64 N/A

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

      14. frac-2neg N/A

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

      15. lift-/.f64 41.2

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

   9. Applied rewrites 41.2%

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

   if -1500 < n < 2.0999999999999999e-166

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. log-pow-rev N/A

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

      10. lower-log.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 52.0

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lift--.f64 52.0

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

   6. Applied rewrites 52.0%

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

   if 2.0999999999999999e-166 < n < 5.2e10

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 32.3

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

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 32.3

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

   6. Applied rewrites 32.3%

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

   if 5.2e10 < n < 9.9999999999999994e104

   1. Initial program 54.2%

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

   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 65.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 41.2

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

   7. Applied rewrites 41.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 41.2%

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

   if 9.9999999999999994e104 < n

   1. Initial program 54.2%

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

   2. Taylor expanded in 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 58.8

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

   4. Applied rewrites 58.8%

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

   6. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(x - -1\right)\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0 \cdot t\_0, 0.5, -\log \left(\sqrt{x}\right) \cdot \log x\right)}{n} - \log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \mathsf{fma}\left(x, 0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}, \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (- x -1.0))))
   (if (<= (/ 1.0 n) -2e-59)
     (/ (exp (* -1.0 (/ (log (/ 1.0 x)) n))) (* n x))
     (if (<= (/ 1.0 n) 5e-8)
       (/
        (-
         (/ (fma (* t_0 t_0) 0.5 (- (* (log (sqrt x)) (log x)))) n)
         (log (/ x (- x -1.0))))
        n)
       (-
        (+
         1.0
         (*
          x
          (fma x (- (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ 1.0 n))) (/ 1.0 n))))
        (pow x (/ 1.0 n)))))))

C

double code(double x, double n) {
	double t_0 = log((x - -1.0));
	double tmp;
	if ((1.0 / n) <= -2e-59) {
		tmp = exp((-1.0 * (log((1.0 / x)) / n))) / (n * x);
	} else if ((1.0 / n) <= 5e-8) {
		tmp = ((fma((t_0 * t_0), 0.5, -(log(sqrt(x)) * log(x))) / n) - log((x / (x - -1.0)))) / n;
	} else {
		tmp = (1.0 + (x * fma(x, ((0.5 * (1.0 / pow(n, 2.0))) - (0.5 * (1.0 / n))), (1.0 / n)))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = log(Float64(x - -1.0))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-59)
		tmp = Float64(exp(Float64(-1.0 * Float64(log(Float64(1.0 / x)) / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-8)
		tmp = Float64(Float64(Float64(fma(Float64(t_0 * t_0), 0.5, Float64(-Float64(log(sqrt(x)) * log(x)))) / n) - log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * fma(x, Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) - Float64(0.5 * Float64(1.0 / n))), Float64(1.0 / n)))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-59], N[(N[Exp[N[(-1.0 * N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-8], N[(N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5 + (-N[(N[Log[N[Sqrt[x], $MachinePrecision]], $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / n), $MachinePrecision] - N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e-59

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 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 57.9

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

   4. Applied rewrites 57.9%

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

   if -2.0000000000000001e-59 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e-8

   1. Initial program 54.2%

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

   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 65.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. add-to-fraction N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. sub-negate N/A

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

      12. sub-negate-rev N/A

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

   6. Applied rewrites 64.6%

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

   8. Applied rewrites 65.1%

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

   if 4.9999999999999998e-8 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

      1. 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 23.1

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

   4. Applied rewrites 23.1%

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

Alternative 3: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(x - -1\right)\\ \mathbf{if}\;x \leq 1.7 \cdot 10^{-112}:\\ \;\;\;\;\left(-1 \cdot \left(x \cdot \log x\right)\right) \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.25:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0 \cdot t\_0, 0.5, -\log \left(\sqrt{x}\right) \cdot \log x\right)}{n} - \log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (- x -1.0))))
   (if (<= x 1.7e-112)
     (* (* -1.0 (* x (log x))) (/ 1.0 (* n x)))
     (if (<= x 0.25)
       (/
        (-
         (/ (fma (* t_0 t_0) 0.5 (- (* (log (sqrt x)) (log x)))) n)
         (log (/ x (- x -1.0))))
        n)
       (/ (exp (* -1.0 (/ (log (/ 1.0 x)) n))) (* n x))))))

C

double code(double x, double n) {
	double t_0 = log((x - -1.0));
	double tmp;
	if (x <= 1.7e-112) {
		tmp = (-1.0 * (x * log(x))) * (1.0 / (n * x));
	} else if (x <= 0.25) {
		tmp = ((fma((t_0 * t_0), 0.5, -(log(sqrt(x)) * log(x))) / n) - log((x / (x - -1.0)))) / n;
	} else {
		tmp = exp((-1.0 * (log((1.0 / x)) / n))) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = log(Float64(x - -1.0))
	tmp = 0.0
	if (x <= 1.7e-112)
		tmp = Float64(Float64(-1.0 * Float64(x * log(x))) * Float64(1.0 / Float64(n * x)));
	elseif (x <= 0.25)
		tmp = Float64(Float64(Float64(fma(Float64(t_0 * t_0), 0.5, Float64(-Float64(log(sqrt(x)) * log(x)))) / n) - log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(log(Float64(1.0 / x)) / n))) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.7e-112], N[(N[(-1.0 * N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.25], N[(N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5 + (-N[(N[Log[N[Sqrt[x], $MachinePrecision]], $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / n), $MachinePrecision] - N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6999999999999999e-112

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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-special-/.f64 N/A

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

      4. lower-special-/.f64 58.7

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. neg-log N/A

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

      14. neg-log N/A

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

      15. lower-log.f64 N/A

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

      16. lower-/.f32 N/A

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

      17. lower-special-/.f32 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. div-flip-rev N/A

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

      22. frac-2neg N/A

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-inverses N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 53.7

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

   11. Applied rewrites 53.7%

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

   if 1.6999999999999999e-112 < x < 0.25

   1. Initial program 54.2%

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

   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 65.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. add-to-fraction N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. sub-negate N/A

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

      12. sub-negate-rev N/A

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

   6. Applied rewrites 64.6%

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

   8. Applied rewrites 65.1%

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

   if 0.25 < x

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 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 57.9

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

   4. Applied rewrites 57.9%

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

Alternative 4: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(x - -1\right)\\ \mathbf{if}\;x \leq 1.7 \cdot 10^{-109}:\\ \;\;\;\;\left(-1 \cdot \left(x \cdot \log x\right)\right) \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.25:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\log \left(\sqrt{x}\right) + n, -\log x, t\_0 \cdot \mathsf{fma}\left(t\_0, 0.5, n\right)\right)}{n}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (- x -1.0))))
   (if (<= x 1.7e-109)
     (* (* -1.0 (* x (log x))) (/ 1.0 (* n x)))
     (if (<= x 0.25)
       (/
        (/ (fma (+ (log (sqrt x)) n) (- (log x)) (* t_0 (fma t_0 0.5 n))) n)
        n)
       (/ (exp (* -1.0 (/ (log (/ 1.0 x)) n))) (* n x))))))

C

double code(double x, double n) {
	double t_0 = log((x - -1.0));
	double tmp;
	if (x <= 1.7e-109) {
		tmp = (-1.0 * (x * log(x))) * (1.0 / (n * x));
	} else if (x <= 0.25) {
		tmp = (fma((log(sqrt(x)) + n), -log(x), (t_0 * fma(t_0, 0.5, n))) / n) / n;
	} else {
		tmp = exp((-1.0 * (log((1.0 / x)) / n))) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = log(Float64(x - -1.0))
	tmp = 0.0
	if (x <= 1.7e-109)
		tmp = Float64(Float64(-1.0 * Float64(x * log(x))) * Float64(1.0 / Float64(n * x)));
	elseif (x <= 0.25)
		tmp = Float64(Float64(fma(Float64(log(sqrt(x)) + n), Float64(-log(x)), Float64(t_0 * fma(t_0, 0.5, n))) / n) / n);
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(log(Float64(1.0 / x)) / n))) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.7e-109], N[(N[(-1.0 * N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.25], N[(N[(N[(N[(N[Log[N[Sqrt[x], $MachinePrecision]], $MachinePrecision] + n), $MachinePrecision] * (-N[Log[x], $MachinePrecision]) + N[(t$95$0 * N[(t$95$0 * 0.5 + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.70000000000000006e-109

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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-special-/.f64 N/A

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

      4. lower-special-/.f64 58.7

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. neg-log N/A

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

      14. neg-log N/A

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

      15. lower-log.f64 N/A

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

      16. lower-/.f32 N/A

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

      17. lower-special-/.f32 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. div-flip-rev N/A

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

      22. frac-2neg N/A

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-inverses N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 53.7

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

   11. Applied rewrites 53.7%

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

   if 1.70000000000000006e-109 < x < 0.25

   1. Initial program 54.2%

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

   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 65.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. add-to-fraction N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. sub-negate N/A

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

      12. sub-negate-rev N/A

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

   6. Applied rewrites 64.6%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\log \left(x - -1\right), n, \left(\log \left(x - -1\right) \cdot \log \left(x - -1\right)\right) \cdot 0.5\right) - \log x \cdot \left(n + \log x \cdot 0.5\right)}{n}}{n} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-log.f64 N/A

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

      8. log-rec N/A

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

      9. lift-/.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lower-fma.f64 38.8

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

   8. Applied rewrites 38.8%

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

   if 0.25 < x

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 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 57.9

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

   4. Applied rewrites 57.9%

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

Alternative 5: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{-97}:\\ \;\;\;\;\left(-1 \cdot \left(x \cdot \log x\right)\right) \cdot \frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 17.5:\\ \;\;\;\;\log \left({\left(\frac{x - -1}{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.9e-97)
   (* (* -1.0 (* x (log x))) (/ 1.0 (* n x)))
   (if (<= x 17.5)
     (log (pow (/ (- x -1.0) x) (/ 1.0 n)))
     (/ (exp (* -1.0 (/ (log (/ 1.0 x)) n))) (* n x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.9e-97) {
		tmp = (-1.0 * (x * log(x))) * (1.0 / (n * x));
	} else if (x <= 17.5) {
		tmp = log(pow(((x - -1.0) / x), (1.0 / n)));
	} else {
		tmp = exp((-1.0 * (log((1.0 / x)) / n))) / (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 <= 4.9d-97) then
        tmp = ((-1.0d0) * (x * log(x))) * (1.0d0 / (n * x))
    else if (x <= 17.5d0) then
        tmp = log((((x - (-1.0d0)) / x) ** (1.0d0 / n)))
    else
        tmp = exp(((-1.0d0) * (log((1.0d0 / x)) / n))) / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.9e-97) {
		tmp = (-1.0 * (x * Math.log(x))) * (1.0 / (n * x));
	} else if (x <= 17.5) {
		tmp = Math.log(Math.pow(((x - -1.0) / x), (1.0 / n)));
	} else {
		tmp = Math.exp((-1.0 * (Math.log((1.0 / x)) / n))) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.9e-97:
		tmp = (-1.0 * (x * math.log(x))) * (1.0 / (n * x))
	elif x <= 17.5:
		tmp = math.log(math.pow(((x - -1.0) / x), (1.0 / n)))
	else:
		tmp = math.exp((-1.0 * (math.log((1.0 / x)) / n))) / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.9e-97)
		tmp = Float64(Float64(-1.0 * Float64(x * log(x))) * Float64(1.0 / Float64(n * x)));
	elseif (x <= 17.5)
		tmp = log((Float64(Float64(x - -1.0) / x) ^ Float64(1.0 / n)));
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(log(Float64(1.0 / x)) / n))) / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.9e-97)
		tmp = (-1.0 * (x * log(x))) * (1.0 / (n * x));
	elseif (x <= 17.5)
		tmp = log((((x - -1.0) / x) ^ (1.0 / n)));
	else
		tmp = exp((-1.0 * (log((1.0 / x)) / n))) / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.9e-97], N[(N[(-1.0 * N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 17.5], N[Log[N[Power[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.8999999999999997e-97

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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-special-/.f64 N/A

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

      4. lower-special-/.f64 58.7

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. neg-log N/A

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

      14. neg-log N/A

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

      15. lower-log.f64 N/A

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

      16. lower-/.f32 N/A

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

      17. lower-special-/.f32 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. div-flip-rev N/A

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

      22. frac-2neg N/A

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-inverses N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 53.7

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

   11. Applied rewrites 53.7%

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

   if 4.8999999999999997e-97 < x < 17.5

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. diff-log N/A

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

      9. log-pow-rev N/A

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

      10. lower-log.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lower-/.f64 52.0

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

      16. lift-+.f64 N/A

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

      17. add-flip N/A

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

      18. metadata-eval N/A

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

      19. lift--.f64 52.0

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

   6. Applied rewrites 52.0%

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

   if 17.5 < x

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 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 57.9

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

   4. Applied rewrites 57.9%

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

Alternative 6: 78.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;1 - e^{\frac{\log x}{n}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} - -1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -0.2)
     (- 1.0 (exp (/ (log x) n)))
     (if (<= t_1 2e-9)
       (/ (- (log (/ x (- x -1.0)))) n)
       (- (- (/ x n) -1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -0.2) {
		tmp = 1.0 - exp((log(x) / n));
	} else if (t_1 <= 2e-9) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = ((x / n) - -1.0) - t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    if (t_1 <= (-0.2d0)) then
        tmp = 1.0d0 - exp((log(x) / n))
    else if (t_1 <= 2d-9) then
        tmp = -log((x / (x - (-1.0d0)))) / n
    else
        tmp = ((x / n) - (-1.0d0)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -0.2) {
		tmp = 1.0 - Math.exp((Math.log(x) / n));
	} else if (t_1 <= 2e-9) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = ((x / n) - -1.0) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -0.2:
		tmp = 1.0 - math.exp((math.log(x) / n))
	elif t_1 <= 2e-9:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = ((x / n) - -1.0) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(1.0 - exp(Float64(log(x) / n)));
	elseif (t_1 <= 2e-9)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = Float64(Float64(Float64(x / n) - -1.0) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = 1.0 - exp((log(x) / n));
	elseif (t_1 <= 2e-9)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = ((x / n) - -1.0) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(1.0 - N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-9], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(N[(N[(x / n), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $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))) < -0.20000000000000001

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 39.9

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

   4. Applied rewrites 39.9%

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

   if -0.20000000000000001 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2.00000000000000012e-9

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. neg-log N/A

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

      11. lower-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lower-/.f64 58.8

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 58.8

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

   6. Applied rewrites 58.8%

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

   if 2.00000000000000012e-9 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 32.3

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

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 32.3

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

   6. Applied rewrites 32.3%

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

Alternative 7: 78.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;1 - e^{\frac{\log x}{n}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -0.2)
     (- 1.0 (exp (/ (log x) n)))
     (if (<= t_1 2e-9) (/ (- (log (/ x (- x -1.0)))) n) (- 1.0 t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -0.2) {
		tmp = 1.0 - exp((log(x) / n));
	} else if (t_1 <= 2e-9) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    if (t_1 <= (-0.2d0)) then
        tmp = 1.0d0 - exp((log(x) / n))
    else if (t_1 <= 2d-9) then
        tmp = -log((x / (x - (-1.0d0)))) / n
    else
        tmp = 1.0d0 - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -0.2) {
		tmp = 1.0 - Math.exp((Math.log(x) / n));
	} else if (t_1 <= 2e-9) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -0.2:
		tmp = 1.0 - math.exp((math.log(x) / n))
	elif t_1 <= 2e-9:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(1.0 - exp(Float64(log(x) / n)));
	elseif (t_1 <= 2e-9)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = 1.0 - exp((log(x) / n));
	elseif (t_1 <= 2e-9)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = 1.0 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(1.0 - N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-9], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(1.0 - t$95$0), $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))) < -0.20000000000000001

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 39.9

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

   4. Applied rewrites 39.9%

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

   if -0.20000000000000001 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2.00000000000000012e-9

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. neg-log N/A

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

      11. lower-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lower-/.f64 58.8

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 58.8

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

   6. Applied rewrites 58.8%

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

   if 2.00000000000000012e-9 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 39.9%

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

Alternative 8: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 -0.2)
     t_2
     (if (<= t_1 2e-9) (/ (- (log (/ x (- x -1.0)))) n) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 2e-9) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    t_2 = 1.0d0 - t_0
    if (t_1 <= (-0.2d0)) then
        tmp = t_2
    else if (t_1 <= 2d-9) then
        tmp = -log((x / (x - (-1.0d0)))) / n
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 2e-9) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -0.2:
		tmp = t_2
	elif t_1 <= 2e-9:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 2e-9)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 2e-9)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 2e-9], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -0.20000000000000001 or 2.00000000000000012e-9 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 39.9%

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

   if -0.20000000000000001 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2.00000000000000012e-9

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. neg-log N/A

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

      11. lower-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lower-/.f64 58.8

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 58.8

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

   6. Applied rewrites 58.8%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 128000:\\ \;\;\;\;\frac{x}{\frac{x}{\log \left(\frac{x - -1}{x}\right)} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - 0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 128000.0)
   (/ x (* (/ x (log (/ (- x -1.0) x))) n))
   (/ (/ (- x 0.5) (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 128000.0) {
		tmp = x / ((x / log(((x - -1.0) / x))) * n);
	} else {
		tmp = ((x - 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 <= 128000.0d0) then
        tmp = x / ((x / log(((x - (-1.0d0)) / x))) * n)
    else
        tmp = ((x - 0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 128000.0) {
		tmp = x / ((x / Math.log(((x - -1.0) / x))) * n);
	} else {
		tmp = ((x - 0.5) / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 128000.0:
		tmp = x / ((x / math.log(((x - -1.0) / x))) * n)
	else:
		tmp = ((x - 0.5) / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 128000.0)
		tmp = Float64(x / Float64(Float64(x / log(Float64(Float64(x - -1.0) / x))) * n));
	else
		tmp = Float64(Float64(Float64(x - 0.5) / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 128000.0)
		tmp = x / ((x / log(((x - -1.0) / x))) * n);
	else
		tmp = ((x - 0.5) / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 128000.0], N[(x / N[(N[(x / N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 0.5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 128000

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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-special-/.f64 N/A

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

      4. lower-special-/.f64 58.7

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. neg-log N/A

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

      14. neg-log N/A

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

      15. lower-log.f64 N/A

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

      16. lower-/.f32 N/A

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

      17. lower-special-/.f32 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. div-flip-rev N/A

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

      22. frac-2neg N/A

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-inverses N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   10. Applied rewrites 67.8%

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

   if 128000 < x

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   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 28.9

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

   7. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

      5. remove-double-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. sub-to-fraction N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 10: 72.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\left(-1 \cdot \left(x \cdot \log x\right)\right) \cdot \frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - 0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.68)
   (* (* -1.0 (* x (log x))) (/ 1.0 (* n x)))
   (/ (/ (- x 0.5) (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = (-1.0 * (x * log(x))) * (1.0 / (n * x));
	} else {
		tmp = ((x - 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 <= 0.68d0) then
        tmp = ((-1.0d0) * (x * log(x))) * (1.0d0 / (n * x))
    else
        tmp = ((x - 0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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-special-/.f64 N/A

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

      4. lower-special-/.f64 58.7

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. neg-log N/A

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

      14. neg-log N/A

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

      15. lower-log.f64 N/A

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

      16. lower-/.f32 N/A

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

      17. lower-special-/.f32 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. div-flip-rev N/A

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

      22. frac-2neg N/A

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-inverses N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 53.7

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

   11. Applied rewrites 53.7%

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

   if 0.680000000000000049 < x

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   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 28.9

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

   7. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

      5. remove-double-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. sub-to-fraction N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 11: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\frac{\log x}{n} - -1}{x}}{n}\\ \mathbf{elif}\;t\_0 \leq 10^{-8}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 -2e+157)
     (/ (/ (- (/ (log x) n) -1.0) x) n)
     (if (<= t_0 1e-8)
       (/ (- (log (/ x (- x -1.0)))) n)
       (/ (* x 1.0) (* x (* 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 <= -2e+157) {
		tmp = (((log(x) / n) - -1.0) / x) / n;
	} else if (t_0 <= 1e-8) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = (x * 1.0) / (x * (n * x));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    if (t_0 <= (-2d+157)) then
        tmp = (((log(x) / n) - (-1.0d0)) / x) / n
    else if (t_0 <= 1d-8) then
        tmp = -log((x / (x - (-1.0d0)))) / n
    else
        tmp = (x * 1.0d0) / (x * (n * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -2e+157) {
		tmp = (((Math.log(x) / n) - -1.0) / x) / n;
	} else if (t_0 <= 1e-8) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = (x * 1.0) / (x * (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 <= -2e+157:
		tmp = (((math.log(x) / n) - -1.0) / x) / n
	elif t_0 <= 1e-8:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = (x * 1.0) / (x * (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 <= -2e+157)
		tmp = Float64(Float64(Float64(Float64(log(x) / n) - -1.0) / x) / n);
	elseif (t_0 <= 1e-8)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = Float64(Float64(x * 1.0) / Float64(x * 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 <= -2e+157)
		tmp = (((log(x) / n) - -1.0) / x) / n;
	elseif (t_0 <= 1e-8)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = (x * 1.0) / (x * (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, -2e+157], N[(N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] - -1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[t$95$0, 1e-8], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $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))) < -1.99999999999999997e157

   1. Initial program 54.2%

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

   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 65.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 41.2

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

   7. Applied rewrites 41.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 41.2

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. log-rec N/A

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

      13. lift-log.f64 N/A

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

      14. frac-2neg N/A

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

      15. lift-/.f64 41.2

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

   9. Applied rewrites 41.2%

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

   if -1.99999999999999997e157 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 1e-8

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. neg-log N/A

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

      11. lower-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lower-/.f64 58.8

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 58.8

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

   6. Applied rewrites 58.8%

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

   if 1e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   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 40.8

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

   7. Applied rewrites 40.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-inverses N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 41.7

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

   9. Applied rewrites 41.7%

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

Alternative 12: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-8}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x - -1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (/ (* x 1.0) (* x (* n x)))))
   (if (<= t_0 -2e+157)
     t_1
     (if (<= t_0 1e-8) (/ (- (log (/ x (- x -1.0)))) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = (x * 1.0) / (x * (n * x));
	double tmp;
	if (t_0 <= -2e+157) {
		tmp = t_1;
	} else if (t_0 <= 1e-8) {
		tmp = -log((x / (x - -1.0))) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    t_1 = (x * 1.0d0) / (x * (n * x))
    if (t_0 <= (-2d+157)) then
        tmp = t_1
    else if (t_0 <= 1d-8) then
        tmp = -log((x / (x - (-1.0d0)))) / n
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = (x * 1.0) / (x * (n * x));
	double tmp;
	if (t_0 <= -2e+157) {
		tmp = t_1;
	} else if (t_0 <= 1e-8) {
		tmp = -Math.log((x / (x - -1.0))) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = (x * 1.0) / (x * (n * x))
	tmp = 0
	if t_0 <= -2e+157:
		tmp = t_1
	elif t_0 <= 1e-8:
		tmp = -math.log((x / (x - -1.0))) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)))
	tmp = 0.0
	if (t_0 <= -2e+157)
		tmp = t_1;
	elseif (t_0 <= 1e-8)
		tmp = Float64(Float64(-log(Float64(x / Float64(x - -1.0)))) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = (x * 1.0) / (x * (n * x));
	tmp = 0.0;
	if (t_0 <= -2e+157)
		tmp = t_1;
	elseif (t_0 <= 1e-8)
		tmp = -log((x / (x - -1.0))) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+157], t$95$1, If[LessEqual[t$95$0, 1e-8], N[((-N[Log[N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -1.99999999999999997e157 or 1e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   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 40.8

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

   7. Applied rewrites 40.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-inverses N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 41.7

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

   9. Applied rewrites 41.7%

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

   if -1.99999999999999997e157 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 1e-8

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. neg-log N/A

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

      11. lower-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lower-/.f64 58.8

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. metadata-eval N/A

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

      20. lift--.f64 58.8

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

   6. Applied rewrites 58.8%

      \[\leadsto \frac{-\log \left(\frac{x}{x - -1}\right)}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{x \cdot 1}{x \cdot \left(n \cdot x\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{x - -1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (/ (* x 1.0) (* x (* n x)))))
   (if (<= t_0 -2e+157)
     t_1
     (if (<= t_0 1e-8) (/ (log (/ (- x -1.0) x)) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = (x * 1.0) / (x * (n * x));
	double tmp;
	if (t_0 <= -2e+157) {
		tmp = t_1;
	} else if (t_0 <= 1e-8) {
		tmp = log(((x - -1.0) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    t_1 = (x * 1.0d0) / (x * (n * x))
    if (t_0 <= (-2d+157)) then
        tmp = t_1
    else if (t_0 <= 1d-8) then
        tmp = log(((x - (-1.0d0)) / x)) / n
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = (x * 1.0) / (x * (n * x));
	double tmp;
	if (t_0 <= -2e+157) {
		tmp = t_1;
	} else if (t_0 <= 1e-8) {
		tmp = Math.log(((x - -1.0) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = (x * 1.0) / (x * (n * x))
	tmp = 0
	if t_0 <= -2e+157:
		tmp = t_1
	elif t_0 <= 1e-8:
		tmp = math.log(((x - -1.0) / x)) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(x * 1.0) / Float64(x * Float64(n * x)))
	tmp = 0.0
	if (t_0 <= -2e+157)
		tmp = t_1;
	elseif (t_0 <= 1e-8)
		tmp = Float64(log(Float64(Float64(x - -1.0) / x)) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = (x * 1.0) / (x * (n * x));
	tmp = 0.0;
	if (t_0 <= -2e+157)
		tmp = t_1;
	elseif (t_0 <= 1e-8)
		tmp = log(((x - -1.0) / x)) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 1.0), $MachinePrecision] / N[(x * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+157], t$95$1, If[LessEqual[t$95$0, 1e-8], N[(N[Log[N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -1.99999999999999997e157 or 1e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   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 40.8

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

   7. Applied rewrites 40.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-inverses N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 41.7

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

   9. Applied rewrites 41.7%

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

   if -1.99999999999999997e157 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 1e-8

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   6. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{x - -1}{x}\right)}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 61.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.25)
   (/ (- x (log x)) n)
   (if (<= x 5.2e+161) (/ (/ 1.0 n) x) (/ (/ (/ -0.5 x) x) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.25) {
		tmp = (x - log(x)) / n;
	} else if (x <= 5.2e+161) {
		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 <= 0.25d0) then
        tmp = (x - log(x)) / n
    else if (x <= 5.2d+161) 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 <= 0.25) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 5.2e+161) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = ((-0.5 / x) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.25:
		tmp = (x - math.log(x)) / n
	elif x <= 5.2e+161:
		tmp = (1.0 / n) / x
	else:
		tmp = ((-0.5 / x) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.25)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 5.2e+161)
		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 <= 0.25)
		tmp = (x - log(x)) / n;
	elseif (x <= 5.2e+161)
		tmp = (1.0 / n) / x;
	else
		tmp = ((-0.5 / x) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.25], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.2e+161], 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 < 0.25

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-log.f64 30.7

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

   7. Applied rewrites 30.7%

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

   if 0.25 < x < 5.1999999999999996e161

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   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 40.8

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

   7. Applied rewrites 40.8%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.5

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

   9. Applied rewrites 41.5%

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

   if 5.1999999999999996e161 < x

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   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 28.9

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

   7. Applied rewrites 28.9%

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

   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 22.9

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

   10. Applied rewrites 22.9%

       \[\leadsto \frac{\frac{\frac{-0.5}{x}}{x}}{n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - 0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.96) (/ (- x (log x)) n) (/ (/ (- x 0.5) (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.96) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((x - 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 <= 0.96d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((x - 0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

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

Python

def code(x, n):
	tmp = 0
	if x <= 0.96:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((x - 0.5) / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.96)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(x - 0.5) / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.96)
		tmp = (x - log(x)) / n;
	else
		tmp = ((x - 0.5) / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.96], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x - 0.5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.95999999999999996

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-log.f64 30.7

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

   7. Applied rewrites 30.7%

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

   if 0.95999999999999996 < x

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   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 28.9

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

   7. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

      5. remove-double-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. sub-to-fraction N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 16: 45.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{\frac{x}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5.2e+161) (/ (/ (/ x n) x) x) (/ (/ (/ -0.5 x) x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5.2e+161) {
		tmp = ((x / 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 <= 5.2d+161) then
        tmp = ((x / 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 <= 5.2e+161) {
		tmp = ((x / n) / x) / x;
	} else {
		tmp = ((-0.5 / x) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5.2e+161:
		tmp = ((x / n) / x) / x
	else:
		tmp = ((-0.5 / x) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5.2e+161)
		tmp = Float64(Float64(Float64(x / n) / 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 <= 5.2e+161)
		tmp = ((x / n) / x) / x;
	else
		tmp = ((-0.5 / x) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5.2e+161], N[(N[(N[(x / n), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(-0.5 / x), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.1999999999999996e161

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   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 40.8

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

   7. Applied rewrites 40.8%

      \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
   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 41.5

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

   9. Applied rewrites 41.5%

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

       1. *-lft-identity N/A

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

       2. *-inverses N/A

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

       3. associate-*l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. mult-flip-rev N/A

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

       7. lower-/.f64 41.6

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

   11. Applied rewrites 41.6%

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

   if 5.1999999999999996e161 < x

   1. Initial program 54.2%

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

   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 58.8

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

   4. Applied rewrites 58.8%

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

   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 28.9

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

   7. Applied rewrites 28.9%

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

   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 22.9

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

   10. Applied rewrites 22.9%

       \[\leadsto \frac{\frac{\frac{-0.5}{x}}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 41.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double x, double n) {
	return ((x / n) / x) / 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 = ((x / n) / x) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

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 58.8

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

4. Applied rewrites 58.8%

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

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 40.8

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

7. Applied rewrites 40.8%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
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 41.5

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

9. Applied rewrites 41.5%

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

    1. *-lft-identity N/A

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

    2. *-inverses N/A

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

    3. associate-*l/ N/A

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

    4. lower-/.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. mult-flip-rev N/A

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

    7. lower-/.f64 41.6

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

11. Applied rewrites 41.6%

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

Alternative 18: 41.5% 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 54.2%

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

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 58.8

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

4. Applied rewrites 58.8%

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

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 40.8

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

7. Applied rewrites 40.8%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
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 41.5

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

9. Applied rewrites 41.5%

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

Alternative 19: 40.8% 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 54.2%

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

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 58.8

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

4. Applied rewrites 58.8%

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

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 40.8

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

7. Applied rewrites 40.8%

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

Reproduce

herbie shell --seed 2025151 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))