2nthrt (problem 3.4.6)

Percentage Accurate: 53.6% → 85.8%

Time: 18.9s

Alternatives: 14

Speedup: 1.9×

• could not determine a ground truth

  1. x = -3.0103966972848064e-145
  2. n = 6.809589291951758e-14

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: 53.6% 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: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-98}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;{\left(n \cdot \left(x \cdot e^{\frac{-\log x}{n}}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -0.002)
     t_0
     (if (<= (/ 1.0 n) 1e-98)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 4e-11)
         (pow (* n (* x (exp (/ (- (log x)) n)))) -1.0)
         t_0)))))

C

double code(double x, double n) {
	double t_0 = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-98) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 4e-11) {
		tmp = pow((n * (x * exp((-log(x) / n)))), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-98) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 4e-11) {
		tmp = Math.pow((n * (x * Math.exp((-Math.log(x) / n)))), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.002:
		tmp = t_0
	elif (1.0 / n) <= 1e-98:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 4e-11:
		tmp = math.pow((n * (x * math.exp((-math.log(x) / n)))), -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-98)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 4e-11)
		tmp = Float64(n * Float64(x * exp(Float64(Float64(-log(x)) / n)))) ^ -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-98], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-11], N[Power[N[(n * N[(x * N[Exp[N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3 or 3.99999999999999976e-11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 83.2%

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

   2. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-log1p.f64 98.4

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied rewrites 98.4%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-99

   1. Initial program 31.8%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

      4. lower-log.f64 79.5

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

   4. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 9.99999999999999939e-99 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999976e-11

   1. Initial program 13.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. 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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}} \]
   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-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. exp-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. neg-log N/A

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

      13. lower-exp.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. neg-log N/A

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

      16. lift-log.f64 N/A

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

      17. lift-neg.f64 50.6

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

   6. Applied rewrites 50.6%

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

   7. Taylor expanded in x around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. neg-log N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-exp.f64 50.7

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

   9. Applied rewrites 50.7%

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

Alternative 2: 86.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\log x}^{2}\\ t_1 := \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - t\_0\right)}{n}\\ t_2 := -t\_1\\ \mathbf{if}\;n \leq -54000000:\\ \;\;\;\;-\frac{\frac{t\_2 \cdot t\_2 - t\_0}{t\_2 - \log x}}{n}\\ \mathbf{elif}\;n \leq 34000000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(-1, t\_1, \log x\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (log x) 2.0))
        (t_1 (+ (log1p x) (/ (* 0.5 (- (pow (log1p x) 2.0) t_0)) n)))
        (t_2 (- t_1)))
   (if (<= n -54000000.0)
     (- (/ (/ (- (* t_2 t_2) t_0) (- t_2 (log x))) n))
     (if (<= n 34000000.0)
       (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))
       (- (/ (fma -1.0 t_1 (log x)) n))))))

C

double code(double x, double n) {
	double t_0 = pow(log(x), 2.0);
	double t_1 = log1p(x) + ((0.5 * (pow(log1p(x), 2.0) - t_0)) / n);
	double t_2 = -t_1;
	double tmp;
	if (n <= -54000000.0) {
		tmp = -((((t_2 * t_2) - t_0) / (t_2 - log(x))) / n);
	} else if (n <= 34000000.0) {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	} else {
		tmp = -(fma(-1.0, t_1, log(x)) / n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = log(x) ^ 2.0
	t_1 = Float64(log1p(x) + Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - t_0)) / n))
	t_2 = Float64(-t_1)
	tmp = 0.0
	if (n <= -54000000.0)
		tmp = Float64(-Float64(Float64(Float64(Float64(t_2 * t_2) - t_0) / Float64(t_2 - log(x))) / n));
	elseif (n <= 34000000.0)
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(-Float64(fma(-1.0, t_1, log(x)) / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, If[LessEqual[n, -54000000.0], (-N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$2 - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[n, 34000000.0], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(N[(-1.0 * t$95$1 + N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision])]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.4e7

   1. Initial program 28.6%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 77.2%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}} \]

   6. Applied rewrites 77.1%

      \[\leadsto -\frac{\frac{\left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) \cdot \left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) - {\log x}^{2}}{\left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) - \log x}}{n} \]

   if -5.4e7 < n < 3.4e7

   1. Initial program 83.1%

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

   2. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-log1p.f64 98.2

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   if 3.4e7 < n

   1. Initial program 29.7%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 77.6%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{n} \cdot \frac{x}{n} - 1}{\frac{x}{n} - 1} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 -1.0)
     (- 1.0 t_0)
     (if (<= t_1 2e-13)
       (/ (- (log1p x) (log x)) n)
       (- (/ (- (* (/ x n) (/ x n)) 1.0) (- (/ x n) 1.0)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 2e-13) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 2e-13) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -1.0:
		tmp = 1.0 - t_0
	elif t_1 <= 2e-13:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 2e-13)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x / n) * Float64(x / n)) - 1.0) / Float64(Float64(x / n) - 1.0)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e-13], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(x / n), $MachinePrecision] * N[(x / n), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[(x / n), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 100.0%

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

   if -1 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2.0000000000000001e-13

   1. Initial program 43.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

      4. lower-log.f64 80.2

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

   4. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 2.0000000000000001e-13 < (-.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 52.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 50.4

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

   4. Applied rewrites 50.4%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-/.f64 61.8

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

   6. Applied rewrites 61.8%

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

Alternative 4: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -0.002)
     t_0
     (if (<= (/ 1.0 n) 2e-11)
       (-
        (/
         (fma
          -1.0
          (+ (log1p x) (/ (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))) n))
          (log x))
         n))
       t_0))))

C

double code(double x, double n) {
	double t_0 = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-11) {
		tmp = -(fma(-1.0, (log1p(x) + ((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) / n)), log(x)) / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(-Float64(fma(-1.0, Float64(log1p(x) + Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) / n)), log(x)) / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], (-N[(N[(-1.0 * N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3 or 1.99999999999999988e-11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 83.1%

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

   2. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-log1p.f64 98.3

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999988e-11

   1. Initial program 29.2%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 77.2%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-98}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;{\left(n \cdot \left(x \cdot e^{\frac{-\log x}{n}}\right)\right)}^{-1}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+222}:\\ \;\;\;\;\frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\mathsf{fma}\left(\frac{x}{n}, \frac{x}{n}, 1 - \frac{x}{n} \cdot 1\right)} - t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.002)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
     (if (<= (/ 1.0 n) 1e-98)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 4e-11)
         (pow (* n (* x (exp (/ (- (log x)) n)))) -1.0)
         (if (<= (/ 1.0 n) 1e+222)
           (-
            (/
             (+ (pow (/ x n) 3.0) 1.0)
             (fma (/ x n) (/ x n) (- 1.0 (* (/ x n) 1.0))))
            t_0)
           (- (exp (/ (log1p x) n)) 1.0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 1e-98) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 4e-11) {
		tmp = pow((n * (x * exp((-log(x) / n)))), -1.0);
	} else if ((1.0 / n) <= 1e+222) {
		tmp = ((pow((x / n), 3.0) + 1.0) / fma((x / n), (x / n), (1.0 - ((x / n) * 1.0)))) - t_0;
	} else {
		tmp = exp((log1p(x) / n)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	elseif (Float64(1.0 / n) <= 1e-98)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 4e-11)
		tmp = Float64(n * Float64(x * exp(Float64(Float64(-log(x)) / n)))) ^ -1.0;
	elseif (Float64(1.0 / n) <= 1e+222)
		tmp = Float64(Float64(Float64((Float64(x / n) ^ 3.0) + 1.0) / fma(Float64(x / n), Float64(x / n), Float64(1.0 - Float64(Float64(x / n) * 1.0)))) - t_0);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-98], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-11], N[Power[N[(n * N[(x * N[Exp[N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+222], N[(N[(N[(N[Power[N[(x / n), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(x / n), $MachinePrecision] * N[(x / n), $MachinePrecision] + N[(1.0 - N[(N[(x / n), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3

   1. Initial program 99.5%

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

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999939e-99

   1. Initial program 31.8%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

      4. lower-log.f64 79.5

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

   4. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 9.99999999999999939e-99 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999976e-11

   1. Initial program 13.2%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. 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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}} \]
   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-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. exp-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. neg-log N/A

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

      13. lower-exp.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. neg-log N/A

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

      16. lift-log.f64 N/A

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

      17. lift-neg.f64 50.6

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

   6. Applied rewrites 50.6%

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

   7. Taylor expanded in x around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. neg-log N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-exp.f64 50.7

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

   9. Applied rewrites 50.7%

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

   if 3.99999999999999976e-11 < (/.f64 #s(literal 1 binary64) n) < 1e222

   1. Initial program 63.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 61.5

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

   4. Applied rewrites 61.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. flip3-+ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\mathsf{fma}\left(\frac{x}{n}, \color{blue}{\frac{x}{n}}, 1 \cdot 1 - \frac{x}{n} \cdot 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\mathsf{fma}\left(\frac{x}{n}, \frac{\color{blue}{x}}{n}, 1 \cdot 1 - \frac{x}{n} \cdot 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\mathsf{fma}\left(\frac{x}{n}, \frac{x}{\color{blue}{n}}, 1 \cdot 1 - \frac{x}{n} \cdot 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 67.7

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

   6. Applied rewrites 67.7%

      \[\leadsto \frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\color{blue}{\mathsf{fma}\left(\frac{x}{n}, \frac{x}{n}, 1 - \frac{x}{n} \cdot 1\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

   if 1e222 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 13.8%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 1.8%

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

   8. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-log1p.f64 87.9

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1 \]

   10. Applied rewrites 87.9%

       \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - 1 \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+222}:\\ \;\;\;\;\frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\mathsf{fma}\left(\frac{x}{n}, \frac{x}{n}, 1 - \frac{x}{n} \cdot 1\right)} - t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.002)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
     (if (<= (/ 1.0 n) 4e-11)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1e+222)
         (-
          (/
           (+ (pow (/ x n) 3.0) 1.0)
           (fma (/ x n) (/ x n) (- 1.0 (* (/ x n) 1.0))))
          t_0)
         (- (exp (/ (log1p x) n)) 1.0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 4e-11) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1e+222) {
		tmp = ((pow((x / n), 3.0) + 1.0) / fma((x / n), (x / n), (1.0 - ((x / n) * 1.0)))) - t_0;
	} else {
		tmp = exp((log1p(x) / n)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	elseif (Float64(1.0 / n) <= 4e-11)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e+222)
		tmp = Float64(Float64(Float64((Float64(x / n) ^ 3.0) + 1.0) / fma(Float64(x / n), Float64(x / n), Float64(1.0 - Float64(Float64(x / n) * 1.0)))) - t_0);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-11], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+222], N[(N[(N[(N[Power[N[(x / n), $MachinePrecision], 3.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(x / n), $MachinePrecision] * N[(x / n), $MachinePrecision] + N[(1.0 - N[(N[(x / n), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3

   1. Initial program 99.5%

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

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999976e-11

   1. Initial program 29.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

      4. lower-log.f64 76.7

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 3.99999999999999976e-11 < (/.f64 #s(literal 1 binary64) n) < 1e222

   1. Initial program 63.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 61.5

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

   4. Applied rewrites 61.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. flip3-+ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\mathsf{fma}\left(\frac{x}{n}, \color{blue}{\frac{x}{n}}, 1 \cdot 1 - \frac{x}{n} \cdot 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\mathsf{fma}\left(\frac{x}{n}, \frac{\color{blue}{x}}{n}, 1 \cdot 1 - \frac{x}{n} \cdot 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\mathsf{fma}\left(\frac{x}{n}, \frac{x}{\color{blue}{n}}, 1 \cdot 1 - \frac{x}{n} \cdot 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 67.7

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

   6. Applied rewrites 67.7%

      \[\leadsto \frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\color{blue}{\mathsf{fma}\left(\frac{x}{n}, \frac{x}{n}, 1 - \frac{x}{n} \cdot 1\right)}} - {x}^{\left(\frac{1}{n}\right)} \]

   if 1e222 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 13.8%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 1.8%

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

   8. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-log1p.f64 87.9

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1 \]

   10. Applied rewrites 87.9%

       \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - 1 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -0.002)
     t_0
     (if (<= (/ 1.0 n) 4e-12)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e+155) t_0 (- (exp (/ (log1p x) n)) 1.0))))))

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 ((1.0 / n) <= -0.002) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-12) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e+155) {
		tmp = t_0;
	} else {
		tmp = exp((log1p(x) / n)) - 1.0;
	}
	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 ((1.0 / n) <= -0.002) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-12) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e+155) {
		tmp = t_0;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - 1.0;
	}
	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 (1.0 / n) <= -0.002:
		tmp = t_0
	elif (1.0 / n) <= 4e-12:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e+155:
		tmp = t_0
	else:
		tmp = math.exp((math.log1p(x) / n)) - 1.0
	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 (Float64(1.0 / n) <= -0.002)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-12)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e+155)
		tmp = t_0;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - 1.0);
	end
	return 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[N[(1.0 / n), $MachinePrecision], -0.002], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-12], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+155], t$95$0, N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3 or 3.99999999999999992e-12 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000001e155

   1. Initial program 93.0%

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

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999992e-12

   1. Initial program 29.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

      4. lower-log.f64 76.8

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 2.00000000000000001e155 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 29.8%

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

   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 25.3%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 2.0%

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

   8. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-log1p.f64 76.3

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1 \]

   10. Applied rewrites 76.3%

       \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-30}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+222}:\\ \;\;\;\;\frac{\frac{x}{n} \cdot \frac{x}{n} - 1}{\frac{x}{n} - 1} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-30)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 4e-11)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 1e+222)
       (- (/ (- (* (/ x n) (/ x n)) 1.0) (- (/ x n) 1.0)) (pow x (/ 1.0 n)))
       (- (exp (/ (log1p x) n)) 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-30) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 4e-11) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1e+222) {
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - pow(x, (1.0 / n));
	} else {
		tmp = exp((log1p(x) / n)) - 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-30) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 4e-11) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e+222) {
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-30:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 4e-11:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1e+222:
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.exp((math.log1p(x) / n)) - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-30)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 4e-11)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e+222)
		tmp = Float64(Float64(Float64(Float64(Float64(x / n) * Float64(x / n)) - 1.0) / Float64(Float64(x / n) - 1.0)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-30], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-11], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+222], N[(N[(N[(N[(N[(x / n), $MachinePrecision] * N[(x / n), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[(x / n), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.99999999999999972e-30

   1. Initial program 93.1%

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

   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 95.6

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

   4. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}} \]
   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-neg.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. neg-log N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-log N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

      10. lift-log.f64 95.6

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

   6. Applied rewrites 95.6%

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

   if -4.99999999999999972e-30 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999976e-11

   1. Initial program 29.6%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

      4. lower-log.f64 78.3

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]

   4. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 3.99999999999999976e-11 < (/.f64 #s(literal 1 binary64) n) < 1e222

   1. Initial program 63.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 61.5

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

   4. Applied rewrites 61.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-/.f64 64.1

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

   6. Applied rewrites 64.1%

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

   if 1e222 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 13.8%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 1.8%

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

   8. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-log1p.f64 87.9

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1 \]

   10. Applied rewrites 87.9%

       \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - 1 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 58.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-66}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 0.00018:\\ \;\;\;\;\frac{\frac{x}{n} \cdot \frac{x}{n} - 1}{\frac{x}{n} - 1} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+74}:\\ \;\;\;\;-\frac{-1 \cdot \frac{1 + -1 \cdot \frac{-\log x}{n}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 9.5e-66)
   (- (/ (log x) n))
   (if (<= x 0.00018)
     (- (/ (- (* (/ x n) (/ x n)) 1.0) (- (/ x n) 1.0)) (pow x (/ 1.0 n)))
     (if (<= x 3.55e+74)
       (- (/ (* -1.0 (/ (+ 1.0 (* -1.0 (/ (- (log x)) n))) x)) n))
       (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 9.5e-66) {
		tmp = -(log(x) / n);
	} else if (x <= 0.00018) {
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - pow(x, (1.0 / n));
	} else if (x <= 3.55e+74) {
		tmp = -((-1.0 * ((1.0 + (-1.0 * (-log(x) / n))) / x)) / n);
	} 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 <= 9.5d-66) then
        tmp = -(log(x) / n)
    else if (x <= 0.00018d0) then
        tmp = ((((x / n) * (x / n)) - 1.0d0) / ((x / n) - 1.0d0)) - (x ** (1.0d0 / n))
    else if (x <= 3.55d+74) then
        tmp = -(((-1.0d0) * ((1.0d0 + ((-1.0d0) * (-log(x) / n))) / x)) / n)
    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 <= 9.5e-66) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 0.00018) {
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - Math.pow(x, (1.0 / n));
	} else if (x <= 3.55e+74) {
		tmp = -((-1.0 * ((1.0 + (-1.0 * (-Math.log(x) / n))) / x)) / n);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 9.5e-66:
		tmp = -(math.log(x) / n)
	elif x <= 0.00018:
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - math.pow(x, (1.0 / n))
	elif x <= 3.55e+74:
		tmp = -((-1.0 * ((1.0 + (-1.0 * (-math.log(x) / n))) / x)) / n)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 9.5e-66)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 0.00018)
		tmp = Float64(Float64(Float64(Float64(Float64(x / n) * Float64(x / n)) - 1.0) / Float64(Float64(x / n) - 1.0)) - (x ^ Float64(1.0 / n)));
	elseif (x <= 3.55e+74)
		tmp = Float64(-Float64(Float64(-1.0 * Float64(Float64(1.0 + Float64(-1.0 * Float64(Float64(-log(x)) / n))) / x)) / n));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 9.5e-66)
		tmp = -(log(x) / n);
	elseif (x <= 0.00018)
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - (x ^ (1.0 / n));
	elseif (x <= 3.55e+74)
		tmp = -((-1.0 * ((1.0 + (-1.0 * (-log(x) / n))) / x)) / n);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 9.5e-66], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 0.00018], N[(N[(N[(N[(N[(x / n), $MachinePrecision] * N[(x / n), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[(x / n), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.55e+74], (-N[(N[(-1.0 * N[(N[(1.0 + N[(-1.0 * N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 9.5000000000000004e-66

   1. Initial program 45.3%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 61.3%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}} \]

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-log.f64 61.3

         \[\leadsto -\frac{\log x - -0.5 \cdot \frac{{\log x}^{2}}{n}}{n} \]

   7. Applied rewrites 61.3%

      \[\leadsto -\frac{\log x - -0.5 \cdot \frac{{\log x}^{2}}{n}}{n} \]

   8. Taylor expanded in n around inf

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

      1. lift-log.f64 50.3

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

   10. Applied rewrites 50.3%

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

   if 9.5000000000000004e-66 < x < 1.80000000000000011e-4

   1. Initial program 35.7%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 32.0

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

   4. Applied rewrites 32.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-/.f64 41.5

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

   6. Applied rewrites 41.5%

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

   if 1.80000000000000011e-4 < x < 3.55000000000000001e74

   1. Initial program 43.1%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 45.9%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}} \]

   6. Applied rewrites 45.7%

      \[\leadsto -\frac{\frac{\left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) \cdot \left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) - {\log x}^{2}}{\left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) - \log x}}{n} \]

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. neg-log N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 56.7

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

   9. Applied rewrites 56.7%

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

   if 3.55000000000000001e74 < x

   1. Initial program 75.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 75.5%

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

Alternative 10: 58.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-18}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+74}:\\ \;\;\;\;-\frac{-1 \cdot \frac{1 + -1 \cdot \frac{-\log x}{n}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.6e-18)
   (- (/ (log x) n))
   (if (<= x 3.55e+74)
     (- (/ (* -1.0 (/ (+ 1.0 (* -1.0 (/ (- (log x)) n))) x)) n))
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.6e-18) {
		tmp = -(log(x) / n);
	} else if (x <= 3.55e+74) {
		tmp = -((-1.0 * ((1.0 + (-1.0 * (-log(x) / n))) / x)) / n);
	} 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 <= 1.6d-18) then
        tmp = -(log(x) / n)
    else if (x <= 3.55d+74) then
        tmp = -(((-1.0d0) * ((1.0d0 + ((-1.0d0) * (-log(x) / n))) / x)) / n)
    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 <= 1.6e-18) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 3.55e+74) {
		tmp = -((-1.0 * ((1.0 + (-1.0 * (-Math.log(x) / n))) / x)) / n);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.6e-18:
		tmp = -(math.log(x) / n)
	elif x <= 3.55e+74:
		tmp = -((-1.0 * ((1.0 + (-1.0 * (-math.log(x) / n))) / x)) / n)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.6e-18)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 3.55e+74)
		tmp = Float64(-Float64(Float64(-1.0 * Float64(Float64(1.0 + Float64(-1.0 * Float64(Float64(-log(x)) / n))) / x)) / n));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.6e-18)
		tmp = -(log(x) / n);
	elseif (x <= 3.55e+74)
		tmp = -((-1.0 * ((1.0 + (-1.0 * (-log(x) / n))) / x)) / n);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.6e-18], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 3.55e+74], (-N[(N[(-1.0 * N[(N[(1.0 + N[(-1.0 * N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6e-18

   1. Initial program 43.0%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 61.5%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}} \]

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-log.f64 61.5

         \[\leadsto -\frac{\log x - -0.5 \cdot \frac{{\log x}^{2}}{n}}{n} \]

   7. Applied rewrites 61.5%

      \[\leadsto -\frac{\log x - -0.5 \cdot \frac{{\log x}^{2}}{n}}{n} \]

   8. Taylor expanded in n around inf

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

      1. lift-log.f64 50.6

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

   10. Applied rewrites 50.6%

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

   if 1.6e-18 < x < 3.55000000000000001e74

   1. Initial program 44.6%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 49.9%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}} \]

   6. Applied rewrites 49.5%

      \[\leadsto -\frac{\frac{\left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) \cdot \left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) - {\log x}^{2}}{\left(-\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}\right)\right) - \log x}}{n} \]

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. neg-log N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 49.7

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

   9. Applied rewrites 49.7%

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

   if 3.55000000000000001e74 < x

   1. Initial program 75.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 75.5%

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

Alternative 11: 46.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500000000.0) (- 1.0 1.0) (/ (pow x -1.0) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = pow(x, -1.0) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-500000000.0d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (x ** (-1.0d0)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = Math.pow(x, -1.0) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500000000.0:
		tmp = 1.0 - 1.0
	else:
		tmp = math.pow(x, -1.0) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500000000.0)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64((x ^ -1.0) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -500000000.0)
		tmp = 1.0 - 1.0;
	else
		tmp = (x ^ -1.0) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000000.0], N[(1.0 - 1.0), $MachinePrecision], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e8

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 50.0%

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

   if -5e8 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{-\frac{-\log x}{n}}, \frac{{n}^{-2} \cdot 0.5 - \frac{0.5}{n}}{x}, \frac{e^{-\frac{-\log x}{n}}}{n}\right)}{x}} \]

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 34.5%

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

   8. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow-flip N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 38.0

         \[\leadsto \frac{{x}^{-1} - 0.5 \cdot {x}^{-2}}{n} \]

   10. Applied rewrites 38.0%

       \[\leadsto \frac{{x}^{-1} - 0.5 \cdot {x}^{-2}}{n} \]

   11. Taylor expanded in x around inf

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

       1. inv-pow N/A

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

       2. lift-pow.f64 44.9

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

   13. Applied rewrites 44.9%

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

Alternative 12: 59.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{x - 0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.68)
   (- (/ (log x) n))
   (if (<= x 3.55e+74) (/ (/ (- x 0.5) (* x x)) n) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = -(log(x) / n);
	} else if (x <= 3.55e+74) {
		tmp = ((x - 0.5) / (x * x)) / n;
	} 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 <= 0.68d0) then
        tmp = -(log(x) / n)
    else if (x <= 3.55d+74) then
        tmp = ((x - 0.5d0) / (x * x)) / n
    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 <= 0.68) {
		tmp = -(Math.log(x) / n);
	} else if (x <= 3.55e+74) {
		tmp = ((x - 0.5) / (x * x)) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.68:
		tmp = -(math.log(x) / n)
	elif x <= 3.55e+74:
		tmp = ((x - 0.5) / (x * x)) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64(-Float64(log(x) / n));
	elseif (x <= 3.55e+74)
		tmp = Float64(Float64(Float64(x - 0.5) / Float64(x * x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = -(log(x) / n);
	elseif (x <= 3.55e+74)
		tmp = ((x - 0.5) / (x * x)) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.68], (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 3.55e+74], N[(N[(N[(x - 0.5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.680000000000000049

   1. Initial program 43.4%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 61.8%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}} \]

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-log.f64 60.9

         \[\leadsto -\frac{\log x - -0.5 \cdot \frac{{\log x}^{2}}{n}}{n} \]

   7. Applied rewrites 60.9%

      \[\leadsto -\frac{\log x - -0.5 \cdot \frac{{\log x}^{2}}{n}}{n} \]

   8. Taylor expanded in n around inf

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

      1. lift-log.f64 49.9

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

   10. Applied rewrites 49.9%

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

   if 0.680000000000000049 < x < 3.55000000000000001e74

   1. Initial program 42.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{-\frac{-\log x}{n}}, \frac{{n}^{-2} \cdot 0.5 - \frac{0.5}{n}}{x}, \frac{e^{-\frac{-\log x}{n}}}{n}\right)}{x}} \]

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 57.3%

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

   8. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow-flip N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 61.1

         \[\leadsto \frac{{x}^{-1} - 0.5 \cdot {x}^{-2}}{n} \]

   10. Applied rewrites 61.1%

       \[\leadsto \frac{{x}^{-1} - 0.5 \cdot {x}^{-2}}{n} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{x - \frac{1}{2}}{{x}^{2}}}{n} \]
   12. 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. pow2 N/A

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

       4. lower-*.f64 61.0

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

   13. Applied rewrites 61.0%

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

   if 3.55000000000000001e74 < x

   1. Initial program 75.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 75.5%

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

Alternative 13: 45.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000000:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500000000.0) (- 1.0 1.0) (/ 1.0 (* n x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000000.0) {
		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) <= (-500000000.0d0)) 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) <= -500000000.0) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500000000.0:
		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) <= -500000000.0)
		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) <= -500000000.0)
		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], -500000000.0], 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) < -5e8

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 50.0%

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

   if -5e8 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.9%

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

   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 40.8

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

   4. Applied rewrites 40.8%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 44.3%

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

Alternative 14: 30.6% accurate, 57.8× 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 53.6%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 38.8%

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

5. Taylor expanded in n around inf

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

7. Applied rewrites 30.6%

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

Reproduce

herbie shell --seed 2025093 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))