2nthrt (problem 3.4.6)

Percentage Accurate: 52.5% → 80.8%

Time: 20.9s

Alternatives: 16

Speedup: 2.4×

• Could not converge on a ground truth

  1. x = -3.5722676021784613e+101
  2. n = -1.857042142859099e-241

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: 52.5% 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: 80.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-50}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.005:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+156}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -2e-167)
     t_0
     (if (<= (/ 1.0 n) 5e-50)
       (- (/ (log (/ x (+ 1.0 x))) n))
       (if (<= (/ 1.0 n) 0.005)
         t_0
         (if (<= (/ 1.0 n) 1e+156)
           (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
           (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-50) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 0.005) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+156) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((log(x) / n)) / (n * x)
    if ((1.0d0 / n) <= (-2d-167)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-50) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 0.005d0) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d+156) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-50) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 0.005) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+156) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (n * x)
	tmp = 0
	if (1.0 / n) <= -2e-167:
		tmp = t_0
	elif (1.0 / n) <= 5e-50:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 0.005:
		tmp = t_0
	elif (1.0 / n) <= 1e+156:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-167)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-50)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 0.005)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e+156)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = exp((log(x) / n)) / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-167)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-50)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 0.005)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e+156)
		tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-167], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-50], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.005], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+156], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-167 or 4.99999999999999968e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0050000000000000001

   1. Initial program 52.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -2e-167 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999968e-50

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if 0.0050000000000000001 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e155

   1. Initial program 52.5%

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

   if 9.9999999999999998e155 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 36.8

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

   10. Applied rewrites 36.8%

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

   11. Taylor expanded in n around inf

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

       1. lower-/.f64 41.6

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

   13. Applied rewrites 41.6%

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

Alternative 2: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-50}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.005:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+187}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -2e-167)
     t_0
     (if (<= (/ 1.0 n) 5e-50)
       (- (/ (log (/ x (+ 1.0 x))) n))
       (if (<= (/ 1.0 n) 0.005)
         t_0
         (if (<= (/ 1.0 n) 1e+187)
           (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
           (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-50) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 0.005) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+187) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((log(x) / n)) / (n * x)
    if ((1.0d0 / n) <= (-2d-167)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-50) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 0.005d0) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d+187) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-50) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 0.005) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+187) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (n * x)
	tmp = 0
	if (1.0 / n) <= -2e-167:
		tmp = t_0
	elif (1.0 / n) <= 5e-50:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 0.005:
		tmp = t_0
	elif (1.0 / n) <= 1e+187:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-167)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-50)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 0.005)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e+187)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = exp((log(x) / n)) / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-167)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-50)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 0.005)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e+187)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-167], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-50], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.005], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+187], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-167 or 4.99999999999999968e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0050000000000000001

   1. Initial program 52.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -2e-167 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999968e-50

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if 0.0050000000000000001 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999907e186

   1. Initial program 52.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 31.0

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

   4. Applied rewrites 31.0%

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

   if 9.99999999999999907e186 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 36.8

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

   10. Applied rewrites 36.8%

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

   11. Taylor expanded in n around inf

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

       1. lower-/.f64 41.6

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

   13. Applied rewrites 41.6%

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

Alternative 3: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ t_1 := e^{--1 \cdot t\_0}\\ t_2 := \frac{t\_1}{x \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-167}:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-50}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, t\_2, n \cdot \mathsf{fma}\left(-0.5, t\_2, \frac{t\_1}{x}\right)\right)}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)) (t_1 (exp (- (* -1.0 t_0)))) (t_2 (/ t_1 (* x x))))
   (if (<= (/ 1.0 n) -2e-167)
     (/ (exp t_0) (* n x))
     (if (<= (/ 1.0 n) 5e-50)
       (- (/ (log (/ x (+ 1.0 x))) n))
       (if (<= (/ 1.0 n) 0.005)
         (/ (fma 0.5 t_2 (* n (fma -0.5 t_2 (/ t_1 x)))) (* n n))
         (-
          (fma (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) x (/ 1.0 n)) x 1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double t_1 = exp(-(-1.0 * t_0));
	double t_2 = t_1 / (x * x);
	double tmp;
	if ((1.0 / n) <= -2e-167) {
		tmp = exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-50) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 0.005) {
		tmp = fma(0.5, t_2, (n * fma(-0.5, t_2, (t_1 / x)))) / (n * n);
	} else {
		tmp = fma(fma(((0.5 / (n * n)) - (0.5 / n)), x, (1.0 / n)), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	t_1 = exp(Float64(-Float64(-1.0 * t_0)))
	t_2 = Float64(t_1 / Float64(x * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-167)
		tmp = Float64(exp(t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-50)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 0.005)
		tmp = Float64(fma(0.5, t_2, Float64(n * fma(-0.5, t_2, Float64(t_1 / x)))) / Float64(n * n));
	else
		tmp = Float64(fma(fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), x, Float64(1.0 / n)), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-N[(-1.0 * t$95$0), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-167], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-50], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.005], N[(N[(0.5 * t$95$2 + N[(n * N[(-0.5 * t$95$2 + N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-167

   1. Initial program 52.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -2e-167 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999968e-50

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if 4.99999999999999968e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0050000000000000001

   1. Initial program 52.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 36.5%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 32.9%

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

   if 0.0050000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 23.7%

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

Alternative 4: 80.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{n \cdot n} - \frac{0.5}{n}\\ t_1 := e^{-\frac{-\log x}{n}}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-167}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-50}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \frac{t\_0}{x}, \frac{t\_1}{n}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_0, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (/ 0.5 (* n n)) (/ 0.5 n))) (t_1 (exp (- (/ (- (log x)) n)))))
   (if (<= (/ 1.0 n) -2e-167)
     (/ (exp (/ (log x) n)) (* n x))
     (if (<= (/ 1.0 n) 5e-50)
       (- (/ (log (/ x (+ 1.0 x))) n))
       (if (<= (/ 1.0 n) 0.005)
         (/ (fma t_1 (/ t_0 x) (/ t_1 n)) x)
         (- (fma (fma t_0 x (/ 1.0 n)) x 1.0) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = (0.5 / (n * n)) - (0.5 / n);
	double t_1 = exp(-(-log(x) / n));
	double tmp;
	if ((1.0 / n) <= -2e-167) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-50) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 0.005) {
		tmp = fma(t_1, (t_0 / x), (t_1 / n)) / x;
	} else {
		tmp = fma(fma(t_0, x, (1.0 / n)), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n))
	t_1 = exp(Float64(-Float64(Float64(-log(x)) / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-167)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-50)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 0.005)
		tmp = Float64(fma(t_1, Float64(t_0 / x), Float64(t_1 / n)) / x);
	else
		tmp = Float64(fma(fma(t_0, x, Float64(1.0 / n)), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-167], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-50], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.005], N[(N[(t$95$1 * N[(t$95$0 / x), $MachinePrecision] + N[(t$95$1 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(t$95$0 * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-167

   1. Initial program 52.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -2e-167 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999968e-50

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if 4.99999999999999968e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0050000000000000001

   1. Initial program 52.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 36.5%

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

   if 0.0050000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 23.7%

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

Alternative 5: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-50}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.005:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+156}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -2e-167)
     t_0
     (if (<= (/ 1.0 n) 5e-50)
       (- (/ (log (/ x (+ 1.0 x))) n))
       (if (<= (/ 1.0 n) 0.005)
         t_0
         (if (<= (/ 1.0 n) 1e+156)
           (- 1.0 (pow x (/ 1.0 n)))
           (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-50) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 0.005) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+156) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((log(x) / n)) / (n * x)
    if ((1.0d0 / n) <= (-2d-167)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-50) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 0.005d0) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d+156) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-50) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 0.005) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e+156) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (n * x)
	tmp = 0
	if (1.0 / n) <= -2e-167:
		tmp = t_0
	elif (1.0 / n) <= 5e-50:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 0.005:
		tmp = t_0
	elif (1.0 / n) <= 1e+156:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-167)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-50)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 0.005)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e+156)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = exp((log(x) / n)) / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -2e-167)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-50)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 0.005)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e+156)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-167], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-50], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.005], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+156], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-167 or 4.99999999999999968e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0050000000000000001

   1. Initial program 52.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -2e-167 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999968e-50

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if 0.0050000000000000001 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e155

   1. Initial program 52.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 38.6%

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

   if 9.9999999999999998e155 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 36.8

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

   10. Applied rewrites 36.8%

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

   11. Taylor expanded in n around inf

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

       1. lower-/.f64 41.6

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

   13. Applied rewrites 41.6%

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

Alternative 6: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-50}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.005:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -2e-167)
     t_0
     (if (<= (/ 1.0 n) 5e-50)
       (- (/ (log (/ x (+ 1.0 x))) n))
       (if (<= (/ 1.0 n) 0.005)
         t_0
         (-
          (fma (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) x (/ 1.0 n)) x 1.0)
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -2e-167) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-50) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 0.005) {
		tmp = t_0;
	} else {
		tmp = fma(fma(((0.5 / (n * n)) - (0.5 / n)), x, (1.0 / n)), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-167)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-50)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 0.005)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), x, Float64(1.0 / n)), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-167], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-50], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.005], t$95$0, N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-167 or 4.99999999999999968e-50 < (/.f64 #s(literal 1 binary64) n) < 0.0050000000000000001

   1. Initial program 52.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -2e-167 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999968e-50

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if 0.0050000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 23.7%

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

Alternative 7: 77.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 -2e+65)
     t_2
     (if (<= t_1 0.0) (- (/ (log (/ x (+ 1.0 x))) n)) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -2e+65) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    t_2 = 1.0d0 - t_0
    if (t_1 <= (-2d+65)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -2e+65) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -2e+65:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = -(math.log((x / (1.0 + x))) / n)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= -2e+65)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -2e+65)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = -(log((x / (1.0 + x))) / n);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+65], t$95$2, If[LessEqual[t$95$1, 0.0], (-N[(N[Log[N[(x / N[(1.0 + x), $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))) < -2e65 or 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 52.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 38.6%

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

   if -2e65 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

Alternative 8: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right) \cdot n}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 -2e+65)
     (/ (/ 0.3333333333333333 (* (* x x) x)) n)
     (if (<= t_0 0.0)
       (- (/ (log (/ x (+ 1.0 x))) n))
       (/ (* (log (/ (+ 1.0 x) x)) n) (* n n))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -2e+65) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (t_0 <= 0.0) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else {
		tmp = (log(((1.0 + x) / x)) * n) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    if (t_0 <= (-2d+65)) then
        tmp = (0.3333333333333333d0 / ((x * x) * x)) / n
    else if (t_0 <= 0.0d0) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else
        tmp = (log(((1.0d0 + x) / x)) * n) / (n * n)
    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+65) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (t_0 <= 0.0) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else {
		tmp = (Math.log(((1.0 + x) / x)) * n) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -2e+65:
		tmp = (0.3333333333333333 / ((x * x) * x)) / n
	elif t_0 <= 0.0:
		tmp = -(math.log((x / (1.0 + x))) / n)
	else:
		tmp = (math.log(((1.0 + x) / x)) * n) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= -2e+65)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n);
	elseif (t_0 <= 0.0)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	else
		tmp = Float64(Float64(log(Float64(Float64(1.0 + x) / x)) * n) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -2e+65)
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	elseif (t_0 <= 0.0)
		tmp = -(log((x / (1.0 + x))) / n);
	else
		tmp = (log(((1.0 + x) / x)) * n) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+65], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), N[(N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -2e65

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 47.3

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{3}}{{x}^{3}}}{n} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 42.6

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

   10. Applied rewrites 42.6%

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

   if -2e65 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 36.8

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

   10. Applied rewrites 36.8%

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

   11. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lift-+.f64 49.1

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

   13. Applied rewrites 49.1%

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

Alternative 9: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 -2e+65)
     (/ (/ 0.3333333333333333 (* (* x x) x)) n)
     (if (<= t_0 0.0) (- (/ (log (/ x (+ 1.0 x))) n)) (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -2e+65) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (t_0 <= 0.0) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    if (t_0 <= (-2d+65)) then
        tmp = (0.3333333333333333d0 / ((x * x) * x)) / n
    else if (t_0 <= 0.0d0) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else
        tmp = (n / x) / (n * n)
    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+65) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (t_0 <= 0.0) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -2e+65:
		tmp = (0.3333333333333333 / ((x * x) * x)) / n
	elif t_0 <= 0.0:
		tmp = -(math.log((x / (1.0 + x))) / n)
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= -2e+65)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n);
	elseif (t_0 <= 0.0)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -2e+65)
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	elseif (t_0 <= 0.0)
		tmp = -(log((x / (1.0 + x))) / n);
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+65], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -2e65

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 47.3

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{3}}{{x}^{3}}}{n} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 42.6

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

   10. Applied rewrites 42.6%

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

   if -2e65 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 36.8

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

   10. Applied rewrites 36.8%

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

   11. Taylor expanded in n around inf

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

       1. lower-/.f64 41.6

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

   13. Applied rewrites 41.6%

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

Alternative 10: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 -2e+65)
     (/ (/ 0.3333333333333333 (* (* x x) x)) n)
     (if (<= t_0 0.0) (/ (log (/ (+ 1.0 x) x)) n) (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -2e+65) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (t_0 <= 0.0) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    if (t_0 <= (-2d+65)) then
        tmp = (0.3333333333333333d0 / ((x * x) * x)) / n
    else if (t_0 <= 0.0d0) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = (n / x) / (n * n)
    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+65) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (t_0 <= 0.0) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -2e+65:
		tmp = (0.3333333333333333 / ((x * x) * x)) / n
	elif t_0 <= 0.0:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= -2e+65)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n);
	elseif (t_0 <= 0.0)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -2e+65)
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	elseif (t_0 <= 0.0)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+65], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -2e65

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 47.3

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{3}}{{x}^{3}}}{n} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 42.6

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

   10. Applied rewrites 42.6%

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

   if -2e65 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 36.8

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

   10. Applied rewrites 36.8%

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

   11. Taylor expanded in n around inf

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

       1. lower-/.f64 41.6

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

   13. Applied rewrites 41.6%

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

Alternative 11: 59.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{x + \left(-\log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2.0)
   (/ (/ 0.3333333333333333 (* (* x x) x)) n)
   (if (<= (/ 1.0 n) -5e-193)
     (/ (/ 1.0 x) n)
     (if (<= (/ 1.0 n) 2e-73) (/ (+ x (- (log x))) n) (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2.0) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if ((1.0 / n) <= -5e-193) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-73) {
		tmp = (x + -log(x)) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2.0d0)) then
        tmp = (0.3333333333333333d0 / ((x * x) * x)) / n
    else if ((1.0d0 / n) <= (-5d-193)) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d-73) then
        tmp = (x + -log(x)) / n
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2.0) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if ((1.0 / n) <= -5e-193) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-73) {
		tmp = (x + -Math.log(x)) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2.0:
		tmp = (0.3333333333333333 / ((x * x) * x)) / n
	elif (1.0 / n) <= -5e-193:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-73:
		tmp = (x + -math.log(x)) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n);
	elseif (Float64(1.0 / n) <= -5e-193)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-73)
		tmp = Float64(Float64(x + Float64(-log(x))) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2.0)
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	elseif ((1.0 / n) <= -5e-193)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e-73)
		tmp = (x + -log(x)) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.0], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-193], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-73], N[(N[(x + (-N[Log[x], $MachinePrecision])), $MachinePrecision] / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 47.3

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{3}}{{x}^{3}}}{n} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 42.6

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

   10. Applied rewrites 42.6%

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

   if -2 < (/.f64 #s(literal 1 binary64) n) < -5.0000000000000005e-193

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 47.3

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 41.3%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 41.3

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

   13. Applied rewrites 41.3%

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

   if -5.0000000000000005e-193 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999999e-73

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. lower-+.f64 N/A

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

      4. neg-log N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-log.f64 31.2

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

   7. Applied rewrites 31.2%

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

   if 1.99999999999999999e-73 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 36.8

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

   10. Applied rewrites 36.8%

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

   11. Taylor expanded in n around inf

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

       1. lower-/.f64 41.6

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

   13. Applied rewrites 41.6%

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

Alternative 12: 56.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2.0)
   (/ (/ 0.3333333333333333 (* (* x x) x)) n)
   (if (<= (/ 1.0 n) -5e-193)
     (/ (/ 1.0 x) n)
     (if (<= (/ 1.0 n) 2e-73) (/ (- (log x)) n) (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2.0) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if ((1.0 / n) <= -5e-193) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-73) {
		tmp = -log(x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2.0d0)) then
        tmp = (0.3333333333333333d0 / ((x * x) * x)) / n
    else if ((1.0d0 / n) <= (-5d-193)) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d-73) then
        tmp = -log(x) / n
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2.0) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if ((1.0 / n) <= -5e-193) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-73) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2.0:
		tmp = (0.3333333333333333 / ((x * x) * x)) / n
	elif (1.0 / n) <= -5e-193:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-73:
		tmp = -math.log(x) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n);
	elseif (Float64(1.0 / n) <= -5e-193)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-73)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2.0)
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	elseif ((1.0 / n) <= -5e-193)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e-73)
		tmp = -log(x) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.0], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-193], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-73], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 47.3

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{3}}{{x}^{3}}}{n} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 42.6

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

   10. Applied rewrites 42.6%

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

   if -2 < (/.f64 #s(literal 1 binary64) n) < -5.0000000000000005e-193

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 47.3

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 41.3%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 41.3

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

   13. Applied rewrites 41.3%

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

   if -5.0000000000000005e-193 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999999e-73

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 31.2

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

   7. Applied rewrites 31.2%

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

   if 1.99999999999999999e-73 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 36.8

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

   10. Applied rewrites 36.8%

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

   11. Taylor expanded in n around inf

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

       1. lower-/.f64 41.6

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

   13. Applied rewrites 41.6%

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

Alternative 13: 56.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55)
   (/ (- (log x)) n)
   (if (<= x 3.7e+168) (/ (/ 1.0 x) n) (/ (/ n x) (* n n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -log(x) / n;
	} else if (x <= 3.7e+168) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (n / x) / (n * 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 <= 0.55d0) then
        tmp = -log(x) / n
    else if (x <= 3.7d+168) then
        tmp = (1.0d0 / x) / n
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -Math.log(x) / n;
	} else if (x <= 3.7e+168) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -math.log(x) / n
	elif x <= 3.7e+168:
		tmp = (1.0 / x) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 3.7e+168)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -log(x) / n;
	elseif (x <= 3.7e+168)
		tmp = (1.0 / x) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 3.7e+168], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.55000000000000004

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 31.2

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

   7. Applied rewrites 31.2%

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

   if 0.55000000000000004 < x < 3.70000000000000009e168

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 47.3

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 41.3%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 41.3

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

   13. Applied rewrites 41.3%

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

   if 3.70000000000000009e168 < x

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 36.8

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

   10. Applied rewrites 36.8%

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

   11. Taylor expanded in n around inf

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

       1. lower-/.f64 41.6

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

   13. Applied rewrites 41.6%

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

Alternative 14: 43.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -3e+144) (/ (/ 1.0 x) n) (/ (/ n x) (* n n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -3e+144) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3d+144)) then
        tmp = (1.0d0 / x) / n
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -3e+144) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -3e+144:
		tmp = (1.0 / x) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -3e+144)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -3e+144)
		tmp = (1.0 / x) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -3e+144], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.9999999999999999e144

   1. Initial program 52.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 47.3

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 41.3%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 41.3

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

   13. Applied rewrites 41.3%

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

   if -2.9999999999999999e144 < n

   1. Initial program 52.5%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. log-pow-rev N/A

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

      5. lower-*.f64 N/A

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

      6. log-pow-rev N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-log.f64 36.8

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

   10. Applied rewrites 36.8%

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

   11. Taylor expanded in n around inf

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

       1. lower-/.f64 41.6

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

   13. Applied rewrites 41.6%

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

Alternative 15: 41.3% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double n) {
	return (1.0 / x) / n;
}

Python

def code(x, n):
	return (1.0 / x) / n

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

2. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. diff-log N/A

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

   3. lower-log.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 58.3

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

4. Applied rewrites 58.3%

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

5. Taylor expanded in x around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 47.3

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

7. Applied rewrites 47.3%

   \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}{n} \]

8. Taylor expanded in x around inf

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

10. Applied rewrites 41.3%

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

11. Taylor expanded in x around inf

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

    1. lower-/.f64 41.3

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

13. Applied rewrites 41.3%

    \[\leadsto \frac{\frac{1}{x}}{n} \]
14. Add Preprocessing

Alternative 16: 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 52.5%

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

2. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. diff-log N/A

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

   3. lower-log.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 58.3

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

4. Applied rewrites 58.3%

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

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. lift-*.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 2025134 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))