2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 82.8%

Time: 16.1s

Alternatives: 19

Speedup: 2.5×

• could not determine a ground truth

  1. x = 1.6202709024647961e+239
  2. n = 3.6204846158308346e-246

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: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(x \cdot x\right)}^{\left(\frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x \cdot \log x}{n \cdot n} \cdot 0.5}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-84)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 0.02)
     (- (/ (log (/ x (+ 1.0 x))) n))
     (if (<= (/ 1.0 n) 2e+158)
       (- (pow (+ x 1.0) (/ 1.0 n)) (pow (* x x) (/ 0.5 n)))
       (/ (* (/ (* (log x) (log x)) (* n n)) 0.5) (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 0.02) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 2e+158) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow((x * x), (0.5 / n));
	} else {
		tmp = (((log(x) * log(x)) / (n * n)) * 0.5) / (n * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-84)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 0.02d0) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 2d+158) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - ((x * x) ** (0.5d0 / n))
    else
        tmp = (((log(x) * log(x)) / (n * n)) * 0.5d0) / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 0.02) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 2e+158) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow((x * x), (0.5 / n));
	} else {
		tmp = (((Math.log(x) * Math.log(x)) / (n * n)) * 0.5) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-84:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 0.02:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 2e+158:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow((x * x), (0.5 / n))
	else:
		tmp = (((math.log(x) * math.log(x)) / (n * n)) * 0.5) / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-84)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 2e+158)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (Float64(x * x) ^ Float64(0.5 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(log(x) * log(x)) / Float64(n * n)) * 0.5) / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-84)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 0.02)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 2e+158)
		tmp = ((x + 1.0) ^ (1.0 / n)) - ((x * x) ^ (0.5 / n));
	else
		tmp = (((log(x) * log(x)) / (n * n)) * 0.5) / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-84], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+158], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[N[(x * x), $MachinePrecision], N[(0.5 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Log[x], $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e-84

   1. Initial program 82.8%

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

   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 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -2.0000000000000001e-84 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 30.9%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 79.2

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

   7. Applied rewrites 79.2%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e158

   1. Initial program 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. unpow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 77.6

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

   3. Applied rewrites 77.6%

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

   4. Taylor expanded in n around 0

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

      1. lower-/.f64 77.6

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

   6. Applied rewrites 77.6%

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

   if 1.99999999999999991e158 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.7%

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

   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 0.5

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 54.1%

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

   8. Taylor expanded in n around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   10. Applied rewrites 76.1%

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

   11. Taylor expanded in n around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-log.f64 N/A

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

       6. lift-log.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 76.1

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

   13. Applied rewrites 76.1%

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

Alternative 2: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x \cdot \log x}{n \cdot n} \cdot 0.5}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-84)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-15)
     (- (/ (log (/ x (+ 1.0 x))) n))
     (if (<= (/ 1.0 n) 2e+158)
       (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
       (/ (* (/ (* (log x) (log x)) (* n n)) 0.5) (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-15) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 2e+158) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = (((log(x) * log(x)) / (n * n)) * 0.5) / (n * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-84)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 5d-15) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 2d+158) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    else
        tmp = (((log(x) * log(x)) / (n * n)) * 0.5d0) / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-15) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 2e+158) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (((Math.log(x) * Math.log(x)) / (n * n)) * 0.5) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-84:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-15:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 2e+158:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = (((math.log(x) * math.log(x)) / (n * n)) * 0.5) / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-84)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 2e+158)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(log(x) * log(x)) / Float64(n * n)) * 0.5) / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-84)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 5e-15)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 2e+158)
		tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	else
		tmp = (((log(x) * log(x)) / (n * n)) * 0.5) / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-84], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+158], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Log[x], $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e-84

   1. Initial program 82.8%

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

   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 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -2.0000000000000001e-84 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999999e-15

   1. Initial program 30.7%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 80.3

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

   7. Applied rewrites 80.3%

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

   if 4.99999999999999999e-15 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e158

   1. Initial program 73.4%

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

   if 1.99999999999999991e158 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.7%

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

   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 0.5

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 54.1%

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

   8. Taylor expanded in n around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   10. Applied rewrites 76.1%

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

   11. Taylor expanded in n around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-log.f64 N/A

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

       6. lift-log.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 76.1

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

   13. Applied rewrites 76.1%

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

Alternative 3: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x \cdot \log x}{n \cdot n} \cdot 0.5}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-84)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 1.0)
     (- (/ (log (/ x (+ 1.0 x))) n))
     (if (<= (/ 1.0 n) 2e+158)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (/ (* (/ (* (log x) (log x)) (* n n)) 0.5) (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1.0) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 2e+158) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = (((log(x) * log(x)) / (n * n)) * 0.5) / (n * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-84)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 1.0d0) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 2d+158) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = (((log(x) * log(x)) / (n * n)) * 0.5d0) / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1.0) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 2e+158) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (((Math.log(x) * Math.log(x)) / (n * n)) * 0.5) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-84:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 1.0:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 2e+158:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = (((math.log(x) * math.log(x)) / (n * n)) * 0.5) / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-84)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 2e+158)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(log(x) * log(x)) / Float64(n * n)) * 0.5) / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-84)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 1.0)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 2e+158)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = (((log(x) * log(x)) / (n * n)) * 0.5) / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-84], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+158], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Log[x], $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e-84

   1. Initial program 82.8%

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

   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 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -2.0000000000000001e-84 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 31.1%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if 1 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e158

   1. Initial program 77.7%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   if 1.99999999999999991e158 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.7%

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

   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 0.5

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 54.1%

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

   8. Taylor expanded in n around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   10. Applied rewrites 76.1%

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

   11. Taylor expanded in n around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-log.f64 N/A

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

       6. lift-log.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 76.1

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

   13. Applied rewrites 76.1%

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

Alternative 4: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-84)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 1.0)
     (- (/ (log (/ x (+ 1.0 x))) n))
     (if (<= (/ 1.0 n) 2e+158)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (/ (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) x)) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1.0) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 2e+158) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-84)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 1.0d0) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 2d+158) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = -((-(((0.3333333333333333d0 / x) - 0.5d0) / x) - 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1.0) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 2e+158) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-84:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 1.0:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 2e+158:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-84)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 2e+158)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-84)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 1.0)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 2e+158)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-84], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+158], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[(N[((-N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e-84

   1. Initial program 82.8%

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

   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 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -2.0000000000000001e-84 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 31.1%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if 1 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e158

   1. Initial program 77.7%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   if 1.99999999999999991e158 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.7%

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

   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 8.5

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

   4. Applied rewrites 8.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 54.1

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

   7. Applied rewrites 54.1%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 68.1

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

   10. Applied rewrites 68.1%

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

Alternative 5: 81.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-84)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 1.0)
     (- (/ (log (/ x (+ 1.0 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) <= -2e-84) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1.0) {
		tmp = -(log((x / (1.0 + 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) <= -2e-84)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + 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], -2e-84], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $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) < -2.0000000000000001e-84

   1. Initial program 82.8%

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

   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 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -2.0000000000000001e-84 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 31.1%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 79.0

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

   7. Applied rewrites 79.0%

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

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

   1. Initial program 55.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 75.5%

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

Alternative 6: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+158}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-84)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 1.0)
     (- (/ (log (/ x (+ 1.0 x))) n))
     (if (<= (/ 1.0 n) 2e+158)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) x)) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1.0) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 2e+158) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-84)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 1.0d0) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 2d+158) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = -((-(((0.3333333333333333d0 / x) - 0.5d0) / x) - 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1.0) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 2e+158) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-84:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 1.0:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 2e+158:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-84)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 2e+158)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-84)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 1.0)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 2e+158)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-84], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+158], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[(N[((-N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e-84

   1. Initial program 82.8%

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

   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 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -2.0000000000000001e-84 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 31.1%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if 1 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e158

   1. Initial program 77.7%

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

   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 72.7%

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

   if 1.99999999999999991e158 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.7%

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

   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 8.5

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

   4. Applied rewrites 8.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 54.1

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

   7. Applied rewrites 54.1%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 68.1

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

   10. Applied rewrites 68.1%

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

Alternative 7: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \end{array} \end{array} \]

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 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 5e-12)
       (- (/ (log (/ x (+ 1.0 x))) n))
       (/ (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) 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 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 5e-12) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 5e-12) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / 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 <= -math.inf:
		tmp = 1.0 - t_0
	elif t_1 <= 5e-12:
		tmp = -(math.log((x / (1.0 + x))) / n)
	else:
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / 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 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 5e-12)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / 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 <= -Inf)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 5e-12)
		tmp = -(log((x / (1.0 + x))) / n);
	else
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / 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, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e-12], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), N[((-N[(N[((-N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $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))) < -inf.0

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 4.9999999999999997e-12

   1. Initial program 43.4%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 79.4

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

   7. Applied rewrites 79.4%

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

   if 4.9999999999999997e-12 < (-.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 56.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 8.8

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

   4. Applied rewrites 8.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 26.3

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

   7. Applied rewrites 26.3%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 36.0

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

   10. Applied rewrites 36.0%

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

Alternative 8: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1
         (/ (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) x)) n)))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 5e-12) (- (/ (log (/ x (+ 1.0 x))) n)) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 5e-12) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 5e-12) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 5e-12:
		tmp = -(math.log((x / (1.0 + x))) / n)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / x)) / n)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 5e-12)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 5e-12)
		tmp = -(log((x / (1.0 + x))) / n);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0 or 4.9999999999999997e-12 < (-.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 78.1%

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

   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 7.2

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

   4. Applied rewrites 7.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 39.3

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 60.4

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

   10. Applied rewrites 60.4%

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

   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))) < 4.9999999999999997e-12

   1. Initial program 43.4%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 79.4

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

   7. Applied rewrites 79.4%

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

Alternative 9: 71.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (/ (+ (log x) n) x) (* n n))
     (if (<= t_0 5e-12) (- (/ (log (/ x (+ 1.0 x))) n)) (/ 1.0 (* n x))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((log(x) + n) / x) / (n * n);
	elseif (t_0 <= 5e-12)
		tmp = -(log((x / (1.0 + x))) / n);
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 5.6

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

   7. Applied rewrites 5.6%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 76.6%

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

   11. Taylor expanded in n around 0

       \[\leadsto \frac{\frac{n}{x} + \frac{\log x}{x}}{{n}^{\color{blue}{2}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{n}{x} + \frac{\log x}{x}}{{n}^{2}} \]

       2. div-add-rev N/A

          \[\leadsto \frac{\frac{n + \log x}{x}}{{n}^{2}} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{\frac{n + \log x}{x}}{{n}^{2}} \]

       4. +-commutative N/A

          \[\leadsto \frac{\frac{\log x + n}{x}}{{n}^{2}} \]

       5. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\log x + n}{x}}{{n}^{2}} \]

       6. lift-log.f64 N/A

          \[\leadsto \frac{\frac{\log x + n}{x}}{{n}^{2}} \]

       7. pow2 N/A

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

       8. lift-*.f64 76.6

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

   13. Applied rewrites 76.6%

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

   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))) < 4.9999999999999997e-12

   1. Initial program 43.4%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 79.4

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

   7. Applied rewrites 79.4%

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

   if 4.9999999999999997e-12 < (-.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 56.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 26.3%

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

Alternative 10: 71.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (log x) (* (* n n) x))
     (if (<= t_0 5e-12) (- (/ (log (/ x (+ 1.0 x))) n)) (/ 1.0 (* n x))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = log(x) / ((n * n) * x);
	elseif (t_0 <= 5e-12)
		tmp = -(log((x / (1.0 + x))) / n);
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 5.6

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

   7. Applied rewrites 5.6%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 76.6%

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

   11. Taylor expanded in n around 0

       \[\leadsto \frac{\log x}{{n}^{2} \cdot \color{blue}{x}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 76.6

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

   13. Applied rewrites 76.6%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 4.9999999999999997e-12

   1. Initial program 43.4%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 79.4

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

   7. Applied rewrites 79.4%

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

   if 4.9999999999999997e-12 < (-.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 56.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 26.3%

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

Alternative 11: 71.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (log x) (* (* n n) x))
     (if (<= t_0 5e-12) (/ (log (+ 1.0 (/ 1.0 x))) n) (/ 1.0 (* n x))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = log(x) / ((n * n) * x);
	elseif (t_0 <= 5e-12)
		tmp = log((1.0 + (1.0 / x))) / n;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in n around inf

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

      1. diff-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 5.6

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

   7. Applied rewrites 5.6%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 76.6%

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

   11. Taylor expanded in n around 0

       \[\leadsto \frac{\log x}{{n}^{2} \cdot \color{blue}{x}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 76.6

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

   13. Applied rewrites 76.6%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 4.9999999999999997e-12

   1. Initial program 43.4%

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

   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 79.4

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 79.4

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

   7. Applied rewrites 79.4%

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

   if 4.9999999999999997e-12 < (-.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 56.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 26.3%

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

Alternative 12: 60.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.94:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\frac{x - 0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log 1}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.94)
   (/ (- x (log x)) n)
   (if (<= x 9.5e+145) (/ (/ (/ (- x 0.5) n) x) x) (/ (log 1.0) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.94) {
		tmp = (x - log(x)) / n;
	} else if (x <= 9.5e+145) {
		tmp = (((x - 0.5) / n) / x) / x;
	} else {
		tmp = log(1.0) / 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 <= 0.94d0) then
        tmp = (x - log(x)) / n
    else if (x <= 9.5d+145) then
        tmp = (((x - 0.5d0) / n) / x) / x
    else
        tmp = log(1.0d0) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.94) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 9.5e+145) {
		tmp = (((x - 0.5) / n) / x) / x;
	} else {
		tmp = Math.log(1.0) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.94:
		tmp = (x - math.log(x)) / n
	elif x <= 9.5e+145:
		tmp = (((x - 0.5) / n) / x) / x
	else:
		tmp = math.log(1.0) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.94)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 9.5e+145)
		tmp = Float64(Float64(Float64(Float64(x - 0.5) / n) / x) / x);
	else
		tmp = Float64(log(1.0) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.94)
		tmp = (x - log(x)) / n;
	elseif (x <= 9.5e+145)
		tmp = (((x - 0.5) / n) / x) / x;
	else
		tmp = log(1.0) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.94], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 9.5e+145], N[(N[(N[(N[(x - 0.5), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(N[Log[1.0], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.93999999999999995

   1. Initial program 43.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 22.5

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. log-pow-rev N/A

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

      4. unpow1 N/A

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

      5. lower--.f64 N/A

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

      6. lift-log.f64 51.7

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

   10. Applied rewrites 51.7%

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

   if 0.93999999999999995 < x < 9.49999999999999948e145

   1. Initial program 51.1%

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

   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 50.9

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

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{n} - \frac{1}{2} \cdot \frac{1}{n}}{{x}^{\color{blue}{2}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 64.1

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

   10. Applied rewrites 64.1%

       \[\leadsto \frac{\frac{x}{n} - \frac{0.5}{n}}{x \cdot \color{blue}{x}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-/r* N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. sub-div N/A

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

       10. lower-/.f64 N/A

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

       11. lower--.f64 64.1

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

   12. Applied rewrites 64.1%

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

   if 9.49999999999999948e145 < x

   1. Initial program 82.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. 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 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 82.3%

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

Alternative 13: 60.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.94:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+145}:\\ \;\;\;\;\frac{1 - \frac{0.5}{x}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log 1}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.94)
   (/ (- x (log x)) n)
   (if (<= x 6e+145) (/ (- 1.0 (/ 0.5 x)) (* n x)) (/ (log 1.0) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.94) {
		tmp = (x - log(x)) / n;
	} else if (x <= 6e+145) {
		tmp = (1.0 - (0.5 / x)) / (n * x);
	} else {
		tmp = log(1.0) / 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 <= 0.94d0) then
        tmp = (x - log(x)) / n
    else if (x <= 6d+145) then
        tmp = (1.0d0 - (0.5d0 / x)) / (n * x)
    else
        tmp = log(1.0d0) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.94) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 6e+145) {
		tmp = (1.0 - (0.5 / x)) / (n * x);
	} else {
		tmp = Math.log(1.0) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.94:
		tmp = (x - math.log(x)) / n
	elif x <= 6e+145:
		tmp = (1.0 - (0.5 / x)) / (n * x)
	else:
		tmp = math.log(1.0) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.94)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 6e+145)
		tmp = Float64(Float64(1.0 - Float64(0.5 / x)) / Float64(n * x));
	else
		tmp = Float64(log(1.0) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.94)
		tmp = (x - log(x)) / n;
	elseif (x <= 6e+145)
		tmp = (1.0 - (0.5 / x)) / (n * x);
	else
		tmp = log(1.0) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.94], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 6e+145], N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(N[Log[1.0], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.93999999999999995

   1. Initial program 43.5%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 22.5

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. log-pow-rev N/A

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

      4. unpow1 N/A

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

      5. lower--.f64 N/A

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

      6. lift-log.f64 51.7

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

   10. Applied rewrites 51.7%

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

   if 0.93999999999999995 < x < 6.0000000000000005e145

   1. Initial program 51.1%

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

   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 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 64.1

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 62.8

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

   10. Applied rewrites 62.8%

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

   if 6.0000000000000005e145 < x

   1. Initial program 82.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. 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 82.2

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 82.2%

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

Alternative 14: 60.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log 1}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.2e-6)
   (/ (- x (log x)) n)
   (if (<= x 9.5e+145) (/ (/ 1.0 n) x) (/ (log 1.0) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.2e-6) {
		tmp = (x - log(x)) / n;
	} else if (x <= 9.5e+145) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = log(1.0) / 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 <= 4.2d-6) then
        tmp = (x - log(x)) / n
    else if (x <= 9.5d+145) then
        tmp = (1.0d0 / n) / x
    else
        tmp = log(1.0d0) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.2e-6) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 9.5e+145) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = Math.log(1.0) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.2e-6:
		tmp = (x - math.log(x)) / n
	elif x <= 9.5e+145:
		tmp = (1.0 / n) / x
	else:
		tmp = math.log(1.0) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.2e-6)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 9.5e+145)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(log(1.0) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.2e-6)
		tmp = (x - log(x)) / n;
	elseif (x <= 9.5e+145)
		tmp = (1.0 / n) / x;
	else
		tmp = log(1.0) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.2e-6], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 9.5e+145], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Log[1.0], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.1999999999999996e-6

   1. Initial program 43.3%

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

   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 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 22.7

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

   7. Applied rewrites 22.7%

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

   8. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. log-pow-rev N/A

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

      4. unpow1 N/A

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

      5. lower--.f64 N/A

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

      6. lift-log.f64 52.1

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

   10. Applied rewrites 52.1%

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

   if 4.1999999999999996e-6 < x < 9.49999999999999948e145

   1. Initial program 51.1%

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

   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 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.4

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

   9. Applied rewrites 60.4%

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

   if 9.49999999999999948e145 < x

   1. Initial program 82.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. 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 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 82.3%

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

Alternative 15: 60.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log 1}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.2e-6)
   (/ (- (log x)) n)
   (if (<= x 9.5e+145) (/ (/ 1.0 n) x) (/ (log 1.0) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.2e-6) {
		tmp = -log(x) / n;
	} else if (x <= 9.5e+145) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = log(1.0) / 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 <= 4.2d-6) then
        tmp = -log(x) / n
    else if (x <= 9.5d+145) then
        tmp = (1.0d0 / n) / x
    else
        tmp = log(1.0d0) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.2e-6) {
		tmp = -Math.log(x) / n;
	} else if (x <= 9.5e+145) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = Math.log(1.0) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.2e-6:
		tmp = -math.log(x) / n
	elif x <= 9.5e+145:
		tmp = (1.0 / n) / x
	else:
		tmp = math.log(1.0) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.2e-6)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 9.5e+145)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(log(1.0) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.2e-6)
		tmp = -log(x) / n;
	elseif (x <= 9.5e+145)
		tmp = (1.0 / n) / x;
	else
		tmp = log(1.0) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.2e-6], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 9.5e+145], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Log[1.0], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.1999999999999996e-6

   1. Initial program 43.3%

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

   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 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 51.8

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

   7. Applied rewrites 51.8%

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

   if 4.1999999999999996e-6 < x < 9.49999999999999948e145

   1. Initial program 51.1%

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

   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 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.4

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

   9. Applied rewrites 60.4%

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

   if 9.49999999999999948e145 < x

   1. Initial program 82.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. 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 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 82.3%

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

Alternative 16: 45.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.45:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -2.4 \cdot 10^{-224}:\\ \;\;\;\;\frac{\log 1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -3.45)
   (/ 1.0 (* n x))
   (if (<= n -2.4e-224) (/ (log 1.0) n) (/ (/ 1.0 n) x))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -3.45) {
		tmp = 1.0 / (n * x);
	} else if (n <= -2.4e-224) {
		tmp = log(1.0) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.45d0)) then
        tmp = 1.0d0 / (n * x)
    else if (n <= (-2.4d-224)) then
        tmp = log(1.0d0) / n
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -3.45) {
		tmp = 1.0 / (n * x);
	} else if (n <= -2.4e-224) {
		tmp = Math.log(1.0) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -3.45:
		tmp = 1.0 / (n * x)
	elif n <= -2.4e-224:
		tmp = math.log(1.0) / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -3.45)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= -2.4e-224)
		tmp = Float64(log(1.0) / n);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -3.45)
		tmp = 1.0 / (n * x);
	elseif (n <= -2.4e-224)
		tmp = log(1.0) / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -3.45], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.4e-224], N[(N[Log[1.0], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.4500000000000002

   1. Initial program 29.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. 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 49.1

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 47.6%

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

   if -3.4500000000000002 < n < -2.40000000000000014e-224

   1. Initial program 99.9%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. 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 48.5

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 50.2%

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

   if -2.40000000000000014e-224 < n

   1. Initial program 47.4%

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

   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 43.9

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

   4. Applied rewrites 43.9%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.6

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

   9. Applied rewrites 42.6%

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

Alternative 17: 39.9% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. mul-1-neg N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 57.4

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

4. Applied rewrites 57.4%

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

5. Taylor expanded in n around inf

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

7. Applied rewrites 39.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 39.9

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

9. Applied rewrites 39.9%

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

Alternative 18: 39.9% 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 53.5%

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

2. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. diff-log N/A

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

   3. lower-log.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 58.4

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

4. Applied rewrites 58.4%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 39.9

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

7. Applied rewrites 39.9%

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

Alternative 19: 39.3% 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 53.5%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. mul-1-neg N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 57.4

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

4. Applied rewrites 57.4%

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

5. Taylor expanded in n around inf

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

7. Applied rewrites 39.3%

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

Reproduce

herbie shell --seed 2025095 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))