2nthrt (problem 3.4.6)

Percentage Accurate: 53.3% → 86.7%

Time: 17.9s

Alternatives: 15

Speedup: 3.0×

• could not determine a ground truth

  1. x = -2.5345201255206083e+47
  2. n = 1.5623955637769813e+50

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% 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: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + x\right)\\ \mathbf{if}\;x \leq 880000:\\ \;\;\;\;-\frac{\left(\left(-\frac{\left(-\frac{-0.16666666666666666 \cdot \left({t\_0}^{3} - {\log x}^{3}\right)}{n}\right) + 0.5 \cdot \left(t\_0 \cdot t\_0 - \log x \cdot \log x\right)}{n}\right) + \left(-t\_0\right)\right) + \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (+ 1.0 x))))
   (if (<= x 880000.0)
     (-
      (/
       (+
        (+
         (-
          (/
           (+
            (-
             (/
              (* -0.16666666666666666 (- (pow t_0 3.0) (pow (log x) 3.0)))
              n))
            (* 0.5 (- (* t_0 t_0) (* (log x) (log x)))))
           n))
         (- t_0))
        (log x))
       n))
     (/ (exp (/ (log x) n)) (* n x)))))

C

double code(double x, double n) {
	double t_0 = log((1.0 + x));
	double tmp;
	if (x <= 880000.0) {
		tmp = -(((-((-((-0.16666666666666666 * (pow(t_0, 3.0) - pow(log(x), 3.0))) / n) + (0.5 * ((t_0 * t_0) - (log(x) * log(x))))) / n) + -t_0) + log(x)) / n);
	} else {
		tmp = exp((log(x) / n)) / (n * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log((1.0d0 + x))
    if (x <= 880000.0d0) then
        tmp = -(((-((-(((-0.16666666666666666d0) * ((t_0 ** 3.0d0) - (log(x) ** 3.0d0))) / n) + (0.5d0 * ((t_0 * t_0) - (log(x) * log(x))))) / n) + -t_0) + log(x)) / n)
    else
        tmp = exp((log(x) / n)) / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log((1.0 + x));
	double tmp;
	if (x <= 880000.0) {
		tmp = -(((-((-((-0.16666666666666666 * (Math.pow(t_0, 3.0) - Math.pow(Math.log(x), 3.0))) / n) + (0.5 * ((t_0 * t_0) - (Math.log(x) * Math.log(x))))) / n) + -t_0) + Math.log(x)) / n);
	} else {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((1.0 + x))
	tmp = 0
	if x <= 880000.0:
		tmp = -(((-((-((-0.16666666666666666 * (math.pow(t_0, 3.0) - math.pow(math.log(x), 3.0))) / n) + (0.5 * ((t_0 * t_0) - (math.log(x) * math.log(x))))) / n) + -t_0) + math.log(x)) / n)
	else:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = log(Float64(1.0 + x))
	tmp = 0.0
	if (x <= 880000.0)
		tmp = Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(-0.16666666666666666 * Float64((t_0 ^ 3.0) - (log(x) ^ 3.0))) / n)) + Float64(0.5 * Float64(Float64(t_0 * t_0) - Float64(log(x) * log(x))))) / n)) + Float64(-t_0)) + log(x)) / n));
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log((1.0 + x));
	tmp = 0.0;
	if (x <= 880000.0)
		tmp = -(((-((-((-0.16666666666666666 * ((t_0 ^ 3.0) - (log(x) ^ 3.0))) / n) + (0.5 * ((t_0 * t_0) - (log(x) * log(x))))) / n) + -t_0) + log(x)) / n);
	else
		tmp = exp((log(x) / n)) / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 880000.0], (-N[(N[(N[((-N[(N[((-N[(N[(-0.16666666666666666 * N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]) + N[(0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]) + (-t$95$0)), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.8e5

   1. Initial program 53.3%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{-\frac{\left(\left(-\frac{\left(-\frac{-0.16666666666666666 \cdot \left({\log \left(1 + x\right)}^{3} - {\log x}^{3}\right)}{n}\right) + 0.5 \cdot \left(\log \left(1 + x\right) \cdot \log \left(1 + x\right) - \log x \cdot \log x\right)}{n}\right) + \left(-\log \left(1 + x\right)\right)\right) + \log x}{n}} \]

   if 8.8e5 < x

   1. Initial program 53.3%

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

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-log N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-log N/A

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

      8. frac-2neg N/A

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

      9. lift-log.f64 N/A

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

      10. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{n} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.88:\\ \;\;\;\;-\frac{\log x + \left(-\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{\log x}^{3}}{n}, -0.5 \cdot \left(\log x \cdot \log x\right)\right)}{n}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.88)
   (-
    (/
     (+
      (log x)
      (-
       (/
        (fma
         -0.16666666666666666
         (/ (pow (log x) 3.0) n)
         (* -0.5 (* (log x) (log x))))
        n)))
     n))
   (/ (exp (/ (log x) n)) (* n x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.88) {
		tmp = -((log(x) + -(fma(-0.16666666666666666, (pow(log(x), 3.0) / n), (-0.5 * (log(x) * log(x)))) / n)) / n);
	} else {
		tmp = exp((log(x) / n)) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.88)
		tmp = Float64(-Float64(Float64(log(x) + Float64(-Float64(fma(-0.16666666666666666, Float64((log(x) ^ 3.0) / n), Float64(-0.5 * Float64(log(x) * log(x)))) / n))) / n));
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.88], (-N[(N[(N[Log[x], $MachinePrecision] + (-N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision] + N[(-0.5 * N[(N[Log[x], $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision])), $MachinePrecision] / n), $MachinePrecision]), N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 53.3%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{-\frac{\left(\left(-\frac{\left(-\frac{-0.16666666666666666 \cdot \left({\log \left(1 + x\right)}^{3} - {\log x}^{3}\right)}{n}\right) + 0.5 \cdot \left(\log \left(1 + x\right) \cdot \log \left(1 + x\right) - \log x \cdot \log x\right)}{n}\right) + \left(-\log \left(1 + x\right)\right)\right) + \log x}{n}} \]

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

      \[\leadsto -\frac{\left(\log x + 0.16666666666666666 \cdot \frac{{\log x}^{3}}{n \cdot n}\right) - -0.5 \cdot \frac{\log x \cdot \log x}{n}}{n} \]

   8. Taylor expanded in n around -inf

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

      1. lower-+.f64 N/A

         \[\leadsto -\frac{\log x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\log x}^{3}}{n} - \frac{1}{2} \cdot {\log x}^{2}}{n}}{n} \]

      2. lift-log.f64 N/A

         \[\leadsto -\frac{\log x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\log x}^{3}}{n} - \frac{1}{2} \cdot {\log x}^{2}}{n}}{n} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

   10. Applied rewrites 45.8%

       \[\leadsto -\frac{\log x + \left(-\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{\log x}^{3}}{n}, -0.5 \cdot \left(\log x \cdot \log x\right)\right)}{n}\right)}{n} \]

   if 0.880000000000000004 < x

   1. Initial program 53.3%

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

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-log N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-log N/A

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

      8. frac-2neg N/A

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

      9. lift-log.f64 N/A

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

      10. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{n} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+125}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {t\_0}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -5e-153)
     (/ (exp t_0) (* n x))
     (if (<= (/ 1.0 n) 5e-16)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+125)
         (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
         (* -0.16666666666666666 (pow t_0 3.0)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-153) {
		tmp = exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-16) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+125) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = -0.16666666666666666 * pow(t_0, 3.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / n
    if ((1.0d0 / n) <= (-5d-153)) then
        tmp = exp(t_0) / (n * x)
    else if ((1.0d0 / n) <= 5d-16) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+125) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    else
        tmp = (-0.16666666666666666d0) * (t_0 ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-153) {
		tmp = Math.exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-16) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+125) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = -0.16666666666666666 * Math.pow(t_0, 3.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -5e-153:
		tmp = math.exp(t_0) / (n * x)
	elif (1.0 / n) <= 5e-16:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+125:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = -0.16666666666666666 * math.pow(t_0, 3.0)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-153)
		tmp = Float64(exp(t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-16)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+125)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-0.16666666666666666 * (t_0 ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-153)
		tmp = exp(t_0) / (n * x);
	elseif ((1.0 / n) <= 5e-16)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+125)
		tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	else
		tmp = -0.16666666666666666 * (t_0 ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-153], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-16], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+125], 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[(-0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000033e-153

   1. Initial program 53.3%

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

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-log N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-log N/A

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

      8. frac-2neg N/A

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

      9. lift-log.f64 N/A

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

      10. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

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

   if -5.00000000000000033e-153 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000004e-16

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   if 5.0000000000000004e-16 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999998e125

   1. Initial program 53.3%

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

   if 1.9999999999999998e125 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{-\frac{\left(\left(-\frac{\left(-\frac{-0.16666666666666666 \cdot \left({\log \left(1 + x\right)}^{3} - {\log x}^{3}\right)}{n}\right) + 0.5 \cdot \left(\log \left(1 + x\right) \cdot \log \left(1 + x\right) - \log x \cdot \log x\right)}{n}\right) + \left(-\log \left(1 + x\right)\right)\right) + \log x}{n}} \]

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 39.6%

      \[\leadsto -0.5 \cdot \frac{\log x \cdot \log x}{n \cdot n} - \color{blue}{\mathsf{fma}\left(0.16666666666666666, {\left(\frac{\log x}{n}\right)}^{3}, \frac{\log x}{n}\right)} \]

   8. Taylor expanded in n around 0

      \[\leadsto \frac{-1}{6} \cdot \frac{{\log x}^{3}}{\color{blue}{{n}^{3}}} \]
   9. Step-by-step derivation

      1. cube-div-rev N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 31.9

         \[\leadsto -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3} \]

   10. Applied rewrites 31.9%

       \[\leadsto -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{\color{blue}{3}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\frac{n + x}{n} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {t\_0}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -5e-153)
     (/ (exp t_0) (* n x))
     (if (<= (/ 1.0 n) 5e-16)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+125)
         (- (/ (+ n x) n) (pow x (/ 1.0 n)))
         (* -0.16666666666666666 (pow t_0 3.0)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-153) {
		tmp = exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-16) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+125) {
		tmp = ((n + x) / n) - pow(x, (1.0 / n));
	} else {
		tmp = -0.16666666666666666 * pow(t_0, 3.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / n
    if ((1.0d0 / n) <= (-5d-153)) then
        tmp = exp(t_0) / (n * x)
    else if ((1.0d0 / n) <= 5d-16) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+125) then
        tmp = ((n + x) / n) - (x ** (1.0d0 / n))
    else
        tmp = (-0.16666666666666666d0) * (t_0 ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-153) {
		tmp = Math.exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-16) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+125) {
		tmp = ((n + x) / n) - Math.pow(x, (1.0 / n));
	} else {
		tmp = -0.16666666666666666 * Math.pow(t_0, 3.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -5e-153:
		tmp = math.exp(t_0) / (n * x)
	elif (1.0 / n) <= 5e-16:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+125:
		tmp = ((n + x) / n) - math.pow(x, (1.0 / n))
	else:
		tmp = -0.16666666666666666 * math.pow(t_0, 3.0)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-153)
		tmp = Float64(exp(t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-16)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+125)
		tmp = Float64(Float64(Float64(n + x) / n) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-0.16666666666666666 * (t_0 ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-153)
		tmp = exp(t_0) / (n * x);
	elseif ((1.0 / n) <= 5e-16)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+125)
		tmp = ((n + x) / n) - (x ^ (1.0 / n));
	else
		tmp = -0.16666666666666666 * (t_0 ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-153], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-16], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+125], N[(N[(N[(n + x), $MachinePrecision] / n), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000033e-153

   1. Initial program 53.3%

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

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-log N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-log N/A

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

      8. frac-2neg N/A

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

      9. lift-log.f64 N/A

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

      10. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

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

   if -5.00000000000000033e-153 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000004e-16

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   if 5.0000000000000004e-16 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999998e125

   1. Initial program 53.3%

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

   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 30.1

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

   4. Applied rewrites 30.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 30.1

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

   7. Applied rewrites 30.1%

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

   if 1.9999999999999998e125 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{-\frac{\left(\left(-\frac{\left(-\frac{-0.16666666666666666 \cdot \left({\log \left(1 + x\right)}^{3} - {\log x}^{3}\right)}{n}\right) + 0.5 \cdot \left(\log \left(1 + x\right) \cdot \log \left(1 + x\right) - \log x \cdot \log x\right)}{n}\right) + \left(-\log \left(1 + x\right)\right)\right) + \log x}{n}} \]

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 39.6%

      \[\leadsto -0.5 \cdot \frac{\log x \cdot \log x}{n \cdot n} - \color{blue}{\mathsf{fma}\left(0.16666666666666666, {\left(\frac{\log x}{n}\right)}^{3}, \frac{\log x}{n}\right)} \]

   8. Taylor expanded in n around 0

      \[\leadsto \frac{-1}{6} \cdot \frac{{\log x}^{3}}{\color{blue}{{n}^{3}}} \]
   9. Step-by-step derivation

      1. cube-div-rev N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 31.9

         \[\leadsto -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3} \]

   10. Applied rewrites 31.9%

       \[\leadsto -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{\color{blue}{3}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-153)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-16)
     (/ (log (/ (+ 1.0 x) x)) 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 tmp;
	if ((1.0 / n) <= -5e-153) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-16) {
		tmp = log(((1.0 + x) / x)) / 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)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-153)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-16)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / 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_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-153], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-16], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $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 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000033e-153

   1. Initial program 53.3%

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

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-log N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-log N/A

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

      8. frac-2neg N/A

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

      9. lift-log.f64 N/A

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

      10. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

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

   if -5.00000000000000033e-153 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000004e-16

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   if 5.0000000000000004e-16 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   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 23.3%

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

Alternative 6: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+125}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {t\_0}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= (/ 1.0 n) -5e-153)
     (/ (exp t_0) (* n x))
     (if (<= (/ 1.0 n) 5e-16)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+125)
         (- 1.0 (pow x (/ 1.0 n)))
         (* -0.16666666666666666 (pow t_0 3.0)))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-153) {
		tmp = exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-16) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+125) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = -0.16666666666666666 * pow(t_0, 3.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / n
    if ((1.0d0 / n) <= (-5d-153)) then
        tmp = exp(t_0) / (n * x)
    else if ((1.0d0 / n) <= 5d-16) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+125) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (-0.16666666666666666d0) * (t_0 ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if ((1.0 / n) <= -5e-153) {
		tmp = Math.exp(t_0) / (n * x);
	} else if ((1.0 / n) <= 5e-16) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+125) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = -0.16666666666666666 * Math.pow(t_0, 3.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if (1.0 / n) <= -5e-153:
		tmp = math.exp(t_0) / (n * x)
	elif (1.0 / n) <= 5e-16:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+125:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = -0.16666666666666666 * math.pow(t_0, 3.0)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-153)
		tmp = Float64(exp(t_0) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-16)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+125)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-0.16666666666666666 * (t_0 ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-153)
		tmp = exp(t_0) / (n * x);
	elseif ((1.0 / n) <= 5e-16)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+125)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = -0.16666666666666666 * (t_0 ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-153], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-16], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+125], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000033e-153

   1. Initial program 53.3%

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

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-log N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-log N/A

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

      8. frac-2neg N/A

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

      9. lift-log.f64 N/A

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

      10. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

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

   if -5.00000000000000033e-153 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000004e-16

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   if 5.0000000000000004e-16 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999998e125

   1. Initial program 53.3%

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

   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 38.0%

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

   if 1.9999999999999998e125 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{-\frac{\left(\left(-\frac{\left(-\frac{-0.16666666666666666 \cdot \left({\log \left(1 + x\right)}^{3} - {\log x}^{3}\right)}{n}\right) + 0.5 \cdot \left(\log \left(1 + x\right) \cdot \log \left(1 + x\right) - \log x \cdot \log x\right)}{n}\right) + \left(-\log \left(1 + x\right)\right)\right) + \log x}{n}} \]

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 39.6%

      \[\leadsto -0.5 \cdot \frac{\log x \cdot \log x}{n \cdot n} - \color{blue}{\mathsf{fma}\left(0.16666666666666666, {\left(\frac{\log x}{n}\right)}^{3}, \frac{\log x}{n}\right)} \]

   8. Taylor expanded in n around 0

      \[\leadsto \frac{-1}{6} \cdot \frac{{\log x}^{3}}{\color{blue}{{n}^{3}}} \]
   9. Step-by-step derivation

      1. cube-div-rev N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 31.9

         \[\leadsto -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{3} \]

   10. Applied rewrites 31.9%

       \[\leadsto -0.16666666666666666 \cdot {\left(\frac{\log x}{n}\right)}^{\color{blue}{3}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+125}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-153)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-16)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 2e+125)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (/ 0.3333333333333333 (* (* x x) x)) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-153) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-16) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+125) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-5d-153)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 5d-16) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+125) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (0.3333333333333333d0 / ((x * x) * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-153) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-16) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+125) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-153:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-16:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+125:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (0.3333333333333333 / ((x * x) * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-153)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-16)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+125)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5e-153)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 5e-16)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+125)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-153], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-16], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+125], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5.00000000000000033e-153

   1. Initial program 53.3%

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

   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. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-log N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-log N/A

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

      8. frac-2neg N/A

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

      9. lift-log.f64 N/A

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

      10. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

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

   if -5.00000000000000033e-153 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000004e-16

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   if 5.0000000000000004e-16 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999998e125

   1. Initial program 53.3%

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

   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 38.0%

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

   if 1.9999999999999998e125 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 46.8

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

   7. Applied rewrites 46.8%

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

   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. pow2 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. pow2 N/A

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

      6. lift-*.f64 43.5

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

   10. Applied rewrites 43.5%

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

Alternative 8: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-8}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.001:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -5e-8)
     (- 1.0 t_0)
     (if (<= t_1 0.001)
       (/ (log (/ (+ 1.0 x) x)) n)
       (/ (/ 0.3333333333333333 (* (* x x) x)) n)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -5e-8) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.001) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    if (t_1 <= (-5d-8)) then
        tmp = 1.0d0 - t_0
    else if (t_1 <= 0.001d0) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = (0.3333333333333333d0 / ((x * x) * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -5e-8) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.001) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -5e-8:
		tmp = 1.0 - t_0
	elif t_1 <= 0.001:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = (0.3333333333333333 / ((x * x) * x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -5e-8)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.001)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -5e-8)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 0.001)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-8], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.001], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $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))) < -4.9999999999999998e-8

   1. Initial program 53.3%

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

   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 38.0%

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

   if -4.9999999999999998e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 1e-3

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   if 1e-3 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 46.8

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

   7. Applied rewrites 46.8%

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

   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. pow2 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. pow2 N/A

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

      6. lift-*.f64 43.5

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

   10. Applied rewrites 43.5%

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

Alternative 9: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (/ (/ 0.3333333333333333 (* (* x x) x)) n)))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 0.001) (/ (log (/ (+ 1.0 x) x)) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = (0.3333333333333333 / ((x * x) * x)) / n;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.001) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = (0.3333333333333333 / ((x * x) * x)) / n;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 0.001) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = (0.3333333333333333 / ((x * x) * x)) / n
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 0.001:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.001)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = (0.3333333333333333 / ((x * x) * x)) / n;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 0.001)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.001], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0 or 1e-3 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 46.8

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

   7. Applied rewrites 46.8%

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

   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. pow2 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. pow2 N/A

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

      6. lift-*.f64 43.5

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

   10. Applied rewrites 43.5%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 1e-3

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

Alternative 10: 56.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.6:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \mathbf{elif}\;n \leq 8.2 \cdot 10^{+251}:\\ \;\;\;\;\frac{x + \left(-\log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 + \frac{\log x}{n}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -5.6)
   (/ (/ 1.0 x) n)
   (if (<= n 8.5e-86)
     (/ (/ 0.3333333333333333 (* (* x x) x)) n)
     (if (<= n 8.2e+251)
       (/ (+ x (- (log x))) n)
       (- 1.0 (+ 1.0 (/ (log x) n)))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -5.6) {
		tmp = (1.0 / x) / n;
	} else if (n <= 8.5e-86) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (n <= 8.2e+251) {
		tmp = (x + -log(x)) / n;
	} else {
		tmp = 1.0 - (1.0 + (log(x) / n));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.6d0)) then
        tmp = (1.0d0 / x) / n
    else if (n <= 8.5d-86) then
        tmp = (0.3333333333333333d0 / ((x * x) * x)) / n
    else if (n <= 8.2d+251) then
        tmp = (x + -log(x)) / n
    else
        tmp = 1.0d0 - (1.0d0 + (log(x) / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -5.6) {
		tmp = (1.0 / x) / n;
	} else if (n <= 8.5e-86) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (n <= 8.2e+251) {
		tmp = (x + -Math.log(x)) / n;
	} else {
		tmp = 1.0 - (1.0 + (Math.log(x) / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -5.6:
		tmp = (1.0 / x) / n
	elif n <= 8.5e-86:
		tmp = (0.3333333333333333 / ((x * x) * x)) / n
	elif n <= 8.2e+251:
		tmp = (x + -math.log(x)) / n
	else:
		tmp = 1.0 - (1.0 + (math.log(x) / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -5.6)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (n <= 8.5e-86)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n);
	elseif (n <= 8.2e+251)
		tmp = Float64(Float64(x + Float64(-log(x))) / n);
	else
		tmp = Float64(1.0 - Float64(1.0 + Float64(log(x) / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -5.6)
		tmp = (1.0 / x) / n;
	elseif (n <= 8.5e-86)
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	elseif (n <= 8.2e+251)
		tmp = (x + -log(x)) / n;
	else
		tmp = 1.0 - (1.0 + (log(x) / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -5.6], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 8.5e-86], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 8.2e+251], N[(N[(x + (-N[Log[x], $MachinePrecision])), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.5999999999999996

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   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. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-log.f64 30.8

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

   7. Applied rewrites 30.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 41.1

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

   10. Applied rewrites 41.1%

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

   if -5.5999999999999996 < n < 8.499999999999999e-86

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 46.8

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

   7. Applied rewrites 46.8%

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

   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. pow2 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. pow2 N/A

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

      6. lift-*.f64 43.5

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

   10. Applied rewrites 43.5%

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

   if 8.499999999999999e-86 < n < 8.2000000000000002e251

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   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. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-log.f64 30.8

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

   7. Applied rewrites 30.8%

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

   if 8.2000000000000002e251 < n

   1. Initial program 53.3%

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

   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 30.1

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

   4. Applied rewrites 30.1%

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

   5. Taylor expanded in n around inf

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

      1. lower-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-/.f64 11.1

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

   7. Applied rewrites 11.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 19.1%

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

Alternative 11: 56.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -5.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{+251}:\\ \;\;\;\;\frac{x + \left(-\log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -5.6)
     t_0
     (if (<= n 8.5e-86)
       (/ (/ 0.3333333333333333 (* (* x x) x)) n)
       (if (<= n 5.2e+251) (/ (+ x (- (log x))) n) t_0)))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -5.6) {
		tmp = t_0;
	} else if (n <= 8.5e-86) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (n <= 5.2e+251) {
		tmp = (x + -log(x)) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / n
    if (n <= (-5.6d0)) then
        tmp = t_0
    else if (n <= 8.5d-86) then
        tmp = (0.3333333333333333d0 / ((x * x) * x)) / n
    else if (n <= 5.2d+251) then
        tmp = (x + -log(x)) / n
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -5.6) {
		tmp = t_0;
	} else if (n <= 8.5e-86) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (n <= 5.2e+251) {
		tmp = (x + -Math.log(x)) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -5.6:
		tmp = t_0
	elif n <= 8.5e-86:
		tmp = (0.3333333333333333 / ((x * x) * x)) / n
	elif n <= 5.2e+251:
		tmp = (x + -math.log(x)) / n
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -5.6)
		tmp = t_0;
	elseif (n <= 8.5e-86)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n);
	elseif (n <= 5.2e+251)
		tmp = Float64(Float64(x + Float64(-log(x))) / n);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -5.6)
		tmp = t_0;
	elseif (n <= 8.5e-86)
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	elseif (n <= 5.2e+251)
		tmp = (x + -log(x)) / n;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -5.6], t$95$0, If[LessEqual[n, 8.5e-86], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 5.2e+251], N[(N[(x + (-N[Log[x], $MachinePrecision])), $MachinePrecision] / n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.5999999999999996 or 5.2000000000000004e251 < n

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   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. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-log.f64 30.8

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

   7. Applied rewrites 30.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 41.1

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

   10. Applied rewrites 41.1%

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

   if -5.5999999999999996 < n < 8.499999999999999e-86

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 46.8

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

   7. Applied rewrites 46.8%

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

   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. pow2 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. pow2 N/A

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

      6. lift-*.f64 43.5

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

   10. Applied rewrites 43.5%

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

   if 8.499999999999999e-86 < n < 5.2000000000000004e251

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   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. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-log.f64 30.8

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

   7. Applied rewrites 30.8%

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

Alternative 12: 56.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -5.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot x}}{n}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{+251}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -5.6)
     t_0
     (if (<= n 8.5e-86)
       (/ (/ 0.3333333333333333 (* (* x x) x)) n)
       (if (<= n 5.2e+251) (/ (- (log x)) n) t_0)))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -5.6) {
		tmp = t_0;
	} else if (n <= 8.5e-86) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (n <= 5.2e+251) {
		tmp = -log(x) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / n
    if (n <= (-5.6d0)) then
        tmp = t_0
    else if (n <= 8.5d-86) then
        tmp = (0.3333333333333333d0 / ((x * x) * x)) / n
    else if (n <= 5.2d+251) then
        tmp = -log(x) / n
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -5.6) {
		tmp = t_0;
	} else if (n <= 8.5e-86) {
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	} else if (n <= 5.2e+251) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -5.6:
		tmp = t_0
	elif n <= 8.5e-86:
		tmp = (0.3333333333333333 / ((x * x) * x)) / n
	elif n <= 5.2e+251:
		tmp = -math.log(x) / n
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -5.6)
		tmp = t_0;
	elseif (n <= 8.5e-86)
		tmp = Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * x)) / n);
	elseif (n <= 5.2e+251)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -5.6)
		tmp = t_0;
	elseif (n <= 8.5e-86)
		tmp = (0.3333333333333333 / ((x * x) * x)) / n;
	elseif (n <= 5.2e+251)
		tmp = -log(x) / n;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -5.6], t$95$0, If[LessEqual[n, 8.5e-86], N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 5.2e+251], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.5999999999999996 or 5.2000000000000004e251 < n

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   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. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-log.f64 30.8

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

   7. Applied rewrites 30.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 41.1

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

   10. Applied rewrites 41.1%

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

   if -5.5999999999999996 < n < 8.499999999999999e-86

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 46.8

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

   7. Applied rewrites 46.8%

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

   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. pow2 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. pow2 N/A

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

      6. lift-*.f64 43.5

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

   10. Applied rewrites 43.5%

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

   if 8.499999999999999e-86 < n < 5.2000000000000004e251

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   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. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-log.f64 30.8

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

   7. Applied rewrites 30.8%

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

Alternative 13: 56.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0014:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0014) (/ (- (log x)) n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0014) {
		tmp = -log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.0014d0) then
        tmp = -log(x) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.0014) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0014:
		tmp = -math.log(x) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0014)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0014)
		tmp = -log(x) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0014], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00139999999999999999

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   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. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-log.f64 30.8

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

   7. Applied rewrites 30.8%

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

   if 0.00139999999999999999 < x

   1. Initial program 53.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.2

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

   4. Applied rewrites 59.2%

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

   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. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-log.f64 30.8

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

   7. Applied rewrites 30.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 41.1

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

   10. Applied rewrites 41.1%

       \[\leadsto \frac{\frac{1}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 41.1% 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 53.3%

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

2. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. diff-log N/A

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

   3. lower-log.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 59.2

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

4. Applied rewrites 59.2%

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

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. log-rec N/A

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

   4. lower-neg.f64 N/A

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

   5. lift-log.f64 30.8

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

7. Applied rewrites 30.8%

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

8. Taylor expanded in x around inf

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

   1. lower-/.f64 41.1

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

10. Applied rewrites 41.1%

    \[\leadsto \frac{\frac{1}{x}}{n} \]
11. Add Preprocessing

Alternative 15: 40.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

double code(double x, double n) {
	return 1.0 / (n * x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * x);
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

2. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. diff-log N/A

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

   3. lower-log.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 59.2

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

4. Applied rewrites 59.2%

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

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. lower-*.f64 40.6

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

7. Applied rewrites 40.6%

   \[\leadsto \frac{1}{\color{blue}{n \cdot x}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))