2nthrt (problem 3.4.6)

Percentage Accurate: 53.7% → 83.3%

Time: 19.1s

Alternatives: 19

Speedup: 2.6×

• could not determine a ground truth

  1. x = 4.118176269865806e+183
  2. n = 1.1480645382298304e-237

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: 53.7% 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: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{{\left(e^{-1}\right)}^{\left(\frac{\log x}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+170}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;--1 \cdot \frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-24)
   (/ (/ 1.0 (pow (exp -1.0) (/ (log x) n))) (* n x))
   (if (<= (/ 1.0 n) 2e-13)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 5e+170)
       (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
       (- (* -1.0 (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = (1.0 / pow(exp(-1.0), (log(x) / n))) / (n * x);
	} else if ((1.0 / n) <= 2e-13) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+170) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-24)) then
        tmp = (1.0d0 / (exp((-1.0d0)) ** (log(x) / n))) / (n * x)
    else if ((1.0d0 / n) <= 2d-13) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+170) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    else
        tmp = -((-1.0d0) * ((n / x) / (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = (1.0 / Math.pow(Math.exp(-1.0), (Math.log(x) / n))) / (n * x);
	} else if ((1.0 / n) <= 2e-13) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+170) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-24:
		tmp = (1.0 / math.pow(math.exp(-1.0), (math.log(x) / n))) / (n * x)
	elif (1.0 / n) <= 2e-13:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+170:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = -(-1.0 * ((n / x) / (n * n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64(Float64(1.0 / (exp(-1.0) ^ Float64(log(x) / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+170)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-Float64(-1.0 * Float64(Float64(n / x) / Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-24)
		tmp = (1.0 / (exp(-1.0) ^ (log(x) / n))) / (n * x);
	elseif ((1.0 / n) <= 2e-13)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+170)
		tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	else
		tmp = -(-1.0 * ((n / x) / (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[(1.0 / N[Power[N[Exp[-1.0], $MachinePrecision], N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+170], 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[(-1.0 * N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.8%

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

   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 96.6

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

   4. Applied rewrites 96.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. pow-exp N/A

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

      10. mul-1-neg N/A

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

      11. pow-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift-/.f64 96.6

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

   6. Applied rewrites 96.6%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 30.3%

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

   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 78.1

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

   4. Applied rewrites 78.1%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999977e170

   1. Initial program 70.0%

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

   if 4.99999999999999977e170 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 25.6%

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

   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 0.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. log-pow-rev N/A

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

      7. lower-/.f64 N/A

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

      8. log-pow-rev N/A

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

      9. inv-pow N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lower-*.f64 0.1

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 0.1

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

   10. Applied rewrites 0.1%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 78.4%

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

Alternative 2: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left(e^{-1}\right)}^{\left(-\frac{\log x}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+170}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;--1 \cdot \frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-24)
   (/ (pow (exp -1.0) (- (/ (log x) n))) (* n x))
   (if (<= (/ 1.0 n) 2e-13)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 5e+170)
       (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
       (- (* -1.0 (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = pow(exp(-1.0), -(log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-13) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+170) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-24)) then
        tmp = (exp((-1.0d0)) ** -(log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 2d-13) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+170) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    else
        tmp = -((-1.0d0) * ((n / x) / (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = Math.pow(Math.exp(-1.0), -(Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-13) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+170) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-24:
		tmp = math.pow(math.exp(-1.0), -(math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-13:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+170:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = -(-1.0 * ((n / x) / (n * n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64((exp(-1.0) ^ Float64(-Float64(log(x) / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+170)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-Float64(-1.0 * Float64(Float64(n / x) / Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-24)
		tmp = (exp(-1.0) ^ -(log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 2e-13)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+170)
		tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	else
		tmp = -(-1.0 * ((n / x) / (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[Power[N[Exp[-1.0], $MachinePrecision], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision])], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+170], 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[(-1.0 * N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.8%

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

   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 96.6

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

   4. Applied rewrites 96.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. pow-exp N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-/.f64 96.6

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

   6. Applied rewrites 96.6%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 30.3%

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

   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 78.1

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

   4. Applied rewrites 78.1%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999977e170

   1. Initial program 70.0%

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

   if 4.99999999999999977e170 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 25.6%

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

   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 0.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. log-pow-rev N/A

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

      7. lower-/.f64 N/A

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

      8. log-pow-rev N/A

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

      9. inv-pow N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lower-*.f64 0.1

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 0.1

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

   10. Applied rewrites 0.1%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 78.4%

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

Alternative 3: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left(e^{-1}\right)}^{\left(-\frac{\log x}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+170}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;--1 \cdot \frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-24)
   (/ (pow (exp -1.0) (- (/ (log x) n))) (* n x))
   (if (<= (/ 1.0 n) 2e-5)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 5e+170)
       (- 1.0 (pow x (/ 1.0 n)))
       (- (* -1.0 (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = pow(exp(-1.0), -(log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-5) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+170) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-24)) then
        tmp = (exp((-1.0d0)) ** -(log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 2d-5) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+170) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = -((-1.0d0) * ((n / x) / (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = Math.pow(Math.exp(-1.0), -(Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-5) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+170) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-24:
		tmp = math.pow(math.exp(-1.0), -(math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-5:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+170:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = -(-1.0 * ((n / x) / (n * n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64((exp(-1.0) ^ Float64(-Float64(log(x) / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+170)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-Float64(-1.0 * Float64(Float64(n / x) / Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-24)
		tmp = (exp(-1.0) ^ -(log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 2e-5)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+170)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = -(-1.0 * ((n / x) / (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[Power[N[Exp[-1.0], $MachinePrecision], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision])], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+170], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(-1.0 * N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.8%

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

   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 96.6

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

   4. Applied rewrites 96.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. pow-exp N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-/.f64 96.6

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

   6. Applied rewrites 96.6%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 30.4%

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

   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 77.5

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

   4. Applied rewrites 77.5%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999977e170

   1. Initial program 72.7%

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

   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 67.3%

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

   if 4.99999999999999977e170 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 25.6%

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

   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 0.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. log-pow-rev N/A

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

      7. lower-/.f64 N/A

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

      8. log-pow-rev N/A

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

      9. inv-pow N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lower-*.f64 0.1

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 0.1

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

   10. Applied rewrites 0.1%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 78.4%

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

Alternative 4: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+170}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;--1 \cdot \frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-24)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-5)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 5e+170)
       (- 1.0 (pow x (/ 1.0 n)))
       (- (* -1.0 (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-5) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+170) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-24)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 2d-5) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+170) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = -((-1.0d0) * ((n / x) / (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-5) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+170) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-24:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-5:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+170:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = -(-1.0 * ((n / x) / (n * n)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+170)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-Float64(-1.0 * Float64(Float64(n / x) / Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-24)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 2e-5)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+170)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = -(-1.0 * ((n / x) / (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+170], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(-1.0 * N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.8%

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

   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 96.6

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

   4. Applied rewrites 96.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. frac-2neg N/A

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

      7. lift-log.f64 N/A

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

      8. lift-/.f64 96.6

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

   6. Applied rewrites 96.6%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 30.4%

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

   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 77.5

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

   4. Applied rewrites 77.5%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999977e170

   1. Initial program 72.7%

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

   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 67.3%

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

   if 4.99999999999999977e170 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 25.6%

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

   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 0.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. log-pow-rev N/A

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

      7. lower-/.f64 N/A

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

      8. log-pow-rev N/A

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

      9. inv-pow N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lower-*.f64 0.1

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 0.1

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

   10. Applied rewrites 0.1%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 78.4%

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

Alternative 5: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + x\right)\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-16}:\\ \;\;\;\;\frac{{\left({\left(e^{-1}\right)}^{-1}\right)}^{\left(\frac{\log x}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;-\frac{\left(\left(-\frac{0.5 \cdot \left(t\_0 \cdot t\_0 - \log x \cdot \log x\right)}{n}\right) + \left(-t\_0\right)\right) + \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (+ 1.0 x))))
   (if (<= (/ 1.0 n) -1e-16)
     (/ (pow (pow (exp -1.0) -1.0) (/ (log x) n)) (* n x))
     (if (<= (/ 1.0 n) 2e-5)
       (-
        (/
         (+
          (+ (- (/ (* 0.5 (- (* t_0 t_0) (* (log x) (log x)))) n)) (- t_0))
          (log 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 t_0 = log((1.0 + x));
	double tmp;
	if ((1.0 / n) <= -1e-16) {
		tmp = pow(pow(exp(-1.0), -1.0), (log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-5) {
		tmp = -(((-((0.5 * ((t_0 * t_0) - (log(x) * log(x)))) / n) + -t_0) + log(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)
	t_0 = log(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-16)
		tmp = Float64(((exp(-1.0) ^ -1.0) ^ Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(0.5 * Float64(Float64(t_0 * t_0) - Float64(log(x) * log(x)))) / n)) + Float64(-t_0)) + log(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_] := Block[{t$95$0 = N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-16], N[(N[Power[N[Power[N[Exp[-1.0], $MachinePrecision], -1.0], $MachinePrecision], N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], (-N[(N[(N[((-N[(N[(0.5 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]) + (-t$95$0)), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.9999999999999998e-17

   1. Initial program 96.9%

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

   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 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. pow-exp N/A

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

      10. pow-unpow N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-/.f64 97.7

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

   6. Applied rewrites 97.7%

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

   if -9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 30.1%

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

   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 77.3%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.6%

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

   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 74.0%

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

Alternative 6: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left({\left(e^{-1}\right)}^{-1}\right)}^{\left(\frac{\log x}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-24)
   (/ (pow (pow (exp -1.0) -1.0) (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 2e-5)
     (/ (log (/ (+ 1.0 x) x)) n)
     (-
      (fma (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) x (/ 1.0 n)) x 1.0)
      (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = pow(pow(exp(-1.0), -1.0), (log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-5) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(fma(((0.5 / (n * n)) - (0.5 / n)), x, (1.0 / n)), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64(((exp(-1.0) ^ -1.0) ^ Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), x, Float64(1.0 / n)), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[Power[N[Power[N[Exp[-1.0], $MachinePrecision], -1.0], $MachinePrecision], N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.8%

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

   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 96.6

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

   4. Applied rewrites 96.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. pow-exp N/A

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

      10. pow-unpow N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-/.f64 96.6

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

   6. Applied rewrites 96.6%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 30.4%

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

   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 77.5

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

   4. Applied rewrites 77.5%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.6%

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

   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 74.0%

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

Alternative 7: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{{\left(e^{-1}\right)}^{\left(\frac{\log x}{n}\right)}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, x, \frac{1}{n}\right), x, 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-24)
   (/ (/ 1.0 (pow (exp -1.0) (/ (log x) n))) (* n x))
   (if (<= (/ 1.0 n) 2e-5)
     (/ (log (/ (+ 1.0 x) x)) n)
     (-
      (fma (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) x (/ 1.0 n)) x 1.0)
      (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-24) {
		tmp = (1.0 / pow(exp(-1.0), (log(x) / n))) / (n * x);
	} else if ((1.0 / n) <= 2e-5) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(fma(((0.5 / (n * n)) - (0.5 / n)), x, (1.0 / n)), x, 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-24)
		tmp = Float64(Float64(1.0 / (exp(-1.0) ^ Float64(log(x) / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), x, Float64(1.0 / n)), x, 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-24], N[(N[(1.0 / N[Power[N[Exp[-1.0], $MachinePrecision], N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -9.99999999999999924e-25

   1. Initial program 94.8%

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

   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 96.6

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

   4. Applied rewrites 96.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. mul-1-neg N/A

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

      9. pow-exp N/A

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

      10. mul-1-neg N/A

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

      11. pow-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift-/.f64 96.6

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

   6. Applied rewrites 96.6%

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

   if -9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 30.4%

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

   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 77.5

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

   4. Applied rewrites 77.5%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.6%

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

   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 74.0%

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

Alternative 8: 78.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e-10) (/ (log (/ (+ 1.0 x) x)) n) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e-10) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	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 t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= 1e-10) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= 1e-10:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e-10)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= 1e-10)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e-10], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0 or 1.00000000000000004e-10 < (-.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 77.0%

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

   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 74.7%

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

   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))) < 1.00000000000000004e-10

   1. Initial program 44.0%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. 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 79.7

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

   4. Applied rewrites 79.7%

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

Alternative 9: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;--1 \cdot \frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) x)) n)
     (if (<= t_0 0.02)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (* -1.0 (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	} else if (t_0 <= 0.02) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	} else if (t_0 <= 0.02) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n
	elif t_0 <= 0.02:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = -(-1.0 * ((n / x) / (n * n)))
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / x)) / n);
	elseif (t_0 <= 0.02)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(-Float64(-1.0 * Float64(Float64(n / x) / Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	elseif (t_0 <= 0.02)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = -(-1.0 * ((n / x) / (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[((-N[(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, 0.02], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], (-N[(-1.0 * N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

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

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 6.1

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 0.1

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

   7. Applied rewrites 0.1%

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

   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 84.4

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

   10. Applied rewrites 84.4%

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

   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))) < 0.0200000000000000004

   1. Initial program 44.1%

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

   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 79.6

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

   4. Applied rewrites 79.6%

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

   if 0.0200000000000000004 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 52.4%

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

   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 0.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. log-pow-rev N/A

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

      7. lower-/.f64 N/A

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

      8. log-pow-rev N/A

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

      9. inv-pow N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lower-*.f64 0.4

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

   7. Applied rewrites 0.4%

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

   8. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 0.4

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

   10. Applied rewrites 0.4%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 39.8%

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

Alternative 10: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{\log x}{n} + 1}{n \cdot x}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;--1 \cdot \frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (+ (/ (log x) n) 1.0) (* n x))
     (if (<= t_0 0.02)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (* -1.0 (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((log(x) / n) + 1.0) / (n * x);
	} else if (t_0 <= 0.02) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((Math.log(x) / n) + 1.0) / (n * x);
	} else if (t_0 <= 0.02) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((math.log(x) / n) + 1.0) / (n * x)
	elif t_0 <= 0.02:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = -(-1.0 * ((n / x) / (n * n)))
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(log(x) / n) + 1.0) / Float64(n * x));
	elseif (t_0 <= 0.02)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(-Float64(-1.0 * Float64(Float64(n / x) / Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((log(x) / n) + 1.0) / (n * x);
	elseif (t_0 <= 0.02)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = -(-1.0 * ((n / x) / (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] + 1.0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], (-N[(-1.0 * N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -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 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in n around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 76.2

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

   7. Applied rewrites 76.2%

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

   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))) < 0.0200000000000000004

   1. Initial program 44.1%

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

   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 79.6

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

   4. Applied rewrites 79.6%

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

   if 0.0200000000000000004 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 52.4%

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

   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 0.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. log-pow-rev N/A

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

      7. lower-/.f64 N/A

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

      8. log-pow-rev N/A

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

      9. inv-pow N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lower-*.f64 0.4

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

   7. Applied rewrites 0.4%

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

   8. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 0.4

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

   10. Applied rewrites 0.4%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 39.8%

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

Alternative 11: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-\frac{-\log x}{\left(n \cdot n\right) \cdot x}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;--1 \cdot \frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (- (/ (- (log x)) (* (* n n) x)))
     (if (<= t_0 0.02)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (* -1.0 (/ (/ n x) (* n n))))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -(-log(x) / ((n * n) * x));
	} else if (t_0 <= 0.02) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -(-Math.log(x) / ((n * n) * x));
	} else if (t_0 <= 0.02) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = -(-1.0 * ((n / x) / (n * n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -(-math.log(x) / ((n * n) * x))
	elif t_0 <= 0.02:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = -(-1.0 * ((n / x) / (n * n)))
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-Float64(Float64(-log(x)) / Float64(Float64(n * n) * x)));
	elseif (t_0 <= 0.02)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(-Float64(-1.0 * Float64(Float64(n / x) / Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -(-log(x) / ((n * n) * x));
	elseif (t_0 <= 0.02)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = -(-1.0 * ((n / x) / (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-N[((-N[Log[x], $MachinePrecision]) / N[(N[(n * n), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$0, 0.02], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], (-N[(-1.0 * N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -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 53.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. log-pow-rev N/A

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

      7. lower-/.f64 N/A

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

      8. log-pow-rev N/A

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

      9. inv-pow N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lower-*.f64 76.2

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

   7. Applied rewrites 76.2%

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

   8. Taylor expanded in n around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 76.2

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

   10. Applied rewrites 76.2%

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

   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))) < 0.0200000000000000004

   1. Initial program 44.1%

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

   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 79.6

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

   4. Applied rewrites 79.6%

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

   if 0.0200000000000000004 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 52.4%

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

   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 0.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. log-pow-rev N/A

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

      7. lower-/.f64 N/A

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

      8. log-pow-rev N/A

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

      9. inv-pow N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lower-*.f64 0.4

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

   7. Applied rewrites 0.4%

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

   8. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 0.4

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

   10. Applied rewrites 0.4%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 39.8%

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

Alternative 12: 71.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := --1 \cdot \frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))
        (t_1 (- (* -1.0 (/ (/ n x) (* n n))))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 0.02) (/ (log (/ (+ 1.0 x) x)) n) t_1))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double t_1 = -(-1.0 * ((n / x) / (n * n)));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.02) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double t_1 = -(-1.0 * ((n / x) / (n * n)));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 0.02) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	t_1 = -(-1.0 * ((n / x) / (n * n)))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 0.02:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	t_1 = Float64(-Float64(-1.0 * Float64(Float64(n / x) / Float64(n * n))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.02)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	t_1 = -(-1.0 * ((n / x) / (n * n)));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 0.02)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(-1.0 * N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.02], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -inf.0 or 0.0200000000000000004 < (-.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 77.0%

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

   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 27.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. inv-pow N/A

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

      6. log-pow-rev N/A

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

      7. lower-/.f64 N/A

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

      8. log-pow-rev N/A

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

      9. inv-pow N/A

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

      10. log-rec N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lower-*.f64 39.6

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

   7. Applied rewrites 39.6%

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

   8. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 39.6

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

   10. Applied rewrites 39.6%

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

   11. Taylor expanded in n around inf

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

   13. Applied rewrites 51.5%

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

   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))) < 0.0200000000000000004

   1. Initial program 44.1%

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

   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 79.6

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

   4. Applied rewrites 79.6%

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

Alternative 13: 60.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+46}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (- (log x)) n)
   (if (<= x 8.8e+46)
     (- 1.0 1.0)
     (if (<= x 1.75e+87) (/ (/ 1.0 n) x) (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = -log(x) / n;
	} else if (x <= 8.8e+46) {
		tmp = 1.0 - 1.0;
	} else if (x <= 1.75e+87) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = -log(x) / n
    else if (x <= 8.8d+46) then
        tmp = 1.0d0 - 1.0d0
    else if (x <= 1.75d+87) then
        tmp = (1.0d0 / n) / x
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = -Math.log(x) / n;
	} else if (x <= 8.8e+46) {
		tmp = 1.0 - 1.0;
	} else if (x <= 1.75e+87) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = -math.log(x) / n
	elif x <= 8.8e+46:
		tmp = 1.0 - 1.0
	elif x <= 1.75e+87:
		tmp = (1.0 / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 8.8e+46)
		tmp = Float64(1.0 - 1.0);
	elseif (x <= 1.75e+87)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -log(x) / n;
	elseif (x <= 8.8e+46)
		tmp = 1.0 - 1.0;
	elseif (x <= 1.75e+87)
		tmp = (1.0 / n) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 8.8e+46], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[x, 1.75e+87], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   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 51.1

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

   4. Applied rewrites 51.1%

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

   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. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-log.f64 50.3

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

   7. Applied rewrites 50.3%

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

   if 1 < x < 8.8000000000000001e46 or 1.74999999999999993e87 < x

   1. Initial program 69.7%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 38.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 69.7%

      \[\leadsto \color{blue}{1} - 1 \]

   if 8.8000000000000001e46 < x < 1.74999999999999993e87

   1. Initial program 50.5%

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

   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 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 61.3%

      \[\leadsto \frac{1}{\color{blue}{n} \cdot x} \]
   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 62.9

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

   9. Applied rewrites 62.9%

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

Alternative 14: 60.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;\frac{x + \left(-\log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - 0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.95) (/ (+ x (- (log x))) n) (/ (/ (- x 0.5) (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.95) {
		tmp = (x + -log(x)) / n;
	} else {
		tmp = ((x - 0.5) / (x * x)) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.95d0) then
        tmp = (x + -log(x)) / n
    else
        tmp = ((x - 0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

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

Python

def code(x, n):
	tmp = 0
	if x <= 0.95:
		tmp = (x + -math.log(x)) / n
	else:
		tmp = ((x - 0.5) / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.95)
		tmp = Float64(Float64(x + Float64(-log(x))) / n);
	else
		tmp = Float64(Float64(Float64(x - 0.5) / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.95)
		tmp = (x + -log(x)) / n;
	else
		tmp = ((x - 0.5) / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   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 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-log.f64 50.7

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

   7. Applied rewrites 50.7%

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

   if 0.94999999999999996 < x

   1. Initial program 67.2%

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

   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 67.9

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 65.3

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

   7. Applied rewrites 65.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.1

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

   10. Applied rewrites 74.1%

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

Alternative 15: 58.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - 0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.7) (/ (- (log x)) n) (/ (/ (- x 0.5) (* x x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -log(x) / n;
	} else {
		tmp = ((x - 0.5) / (x * x)) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = -log(x) / n
    else
        tmp = ((x - 0.5d0) / (x * x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = ((x - 0.5) / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.7:
		tmp = -math.log(x) / n
	else:
		tmp = ((x - 0.5) / (x * x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(Float64(x - 0.5) / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = -log(x) / n;
	else
		tmp = ((x - 0.5) / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.7], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(N[(x - 0.5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   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 51.1

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

   4. Applied rewrites 51.1%

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

   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. log-rec N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-log.f64 50.3

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

   7. Applied rewrites 50.3%

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

   if 0.69999999999999996 < x

   1. Initial program 67.2%

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

   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 67.9

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 65.3

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

   7. Applied rewrites 65.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.1

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

   10. Applied rewrites 74.1%

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

Alternative 16: 47.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500000.0) (- 1.0 1.0) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 1.0 - 1.0;
	} 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 ((1.0d0 / n) <= (-500000.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500000.0:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -500000.0)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e5

   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

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 49.3%

      \[\leadsto \color{blue}{1} - 1 \]

   if -5e5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.1%

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

   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 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 45.7%

      \[\leadsto \frac{1}{\color{blue}{n} \cdot x} \]
   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 46.5

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

   9. Applied rewrites 46.5%

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

Alternative 17: 47.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500000.0) (- 1.0 1.0) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-500000.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (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) <= -500000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500000.0:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -500000.0)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e5

   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

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 49.3%

      \[\leadsto \color{blue}{1} - 1 \]

   if -5e5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.1%

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

   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 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 46.5

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

   7. Applied rewrites 46.5%

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

Alternative 18: 46.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500000.0) (- 1.0 1.0) (/ 1.0 (* n x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 1.0 - 1.0;
	} 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 ((1.0d0 / n) <= (-500000.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500000.0:
		tmp = 1.0 - 1.0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -500000.0)
		tmp = 1.0 - 1.0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000.0], N[(1.0 - 1.0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e5

   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

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 49.3%

      \[\leadsto \color{blue}{1} - 1 \]

   if -5e5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.1%

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

   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 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 45.7

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

   7. Applied rewrites 45.7%

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

Alternative 19: 31.1% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

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

C

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

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 - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, n_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 53.7%

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

2. Taylor expanded in n around inf

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

4. Applied rewrites 18.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 31.1%

   \[\leadsto \color{blue}{1} - 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))