2nthrt (problem 3.4.6)

Percentage Accurate: 53.1% → 86.2%

Time: 26.6s

Alternatives: 14

Speedup: 1.8×

• Could not converge on a ground truth

  1. x = -2.9006230378128678e+225
  2. n = -4.3978845535386394e-176

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: 53.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -58000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{-n}\\ \mathbf{elif}\;n \leq 1600000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -58000000.0)
   (/
    (fma
     -1.0
     (+ (log1p x) (/ (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))) n))
     (log x))
    (- n))
   (if (<= n 1600000.0)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))
     (if (<= n 6.5e+80)
       (/ (/ (exp (/ (log x) n)) n) x)
       (/ (- (log1p x) (log x)) n)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -58000000.0) {
		tmp = fma(-1.0, (log1p(x) + ((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) / n)), log(x)) / -n;
	} else if (n <= 1600000.0) {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	} else if (n <= 6.5e+80) {
		tmp = (exp((log(x) / n)) / n) / x;
	} else {
		tmp = (log1p(x) - log(x)) / n;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -58000000.0)
		tmp = Float64(fma(-1.0, Float64(log1p(x) + Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) / n)), log(x)) / Float64(-n));
	elseif (n <= 1600000.0)
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	elseif (n <= 6.5e+80)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	else
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -58000000.0], 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], If[LessEqual[n, 1600000.0], 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, 6.5e+80], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.8e7

   1. Initial program 28.9%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\frac{1}{2} \cdot {\log \left(1 + x\right)}^{2} - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]
   4. 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

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

   5. Applied rewrites 77.8%

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

   if -5.8e7 < n < 1.6e6

   1. Initial program 83.8%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
   4. 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 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.6e6 < n < 6.4999999999999998e80

   1. Initial program 12.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}{n}}{x} \]
   7. Step-by-step derivation

      1. exp-neg N/A

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

      2. neg-log N/A

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

      3. exp-neg N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

   if 6.4999999999999998e80 < n

   1. Initial program 30.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. 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 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -58000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{-n}\\ \mathbf{elif}\;n \leq 1600000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{n} \cdot \frac{x}{n} - 1}{\frac{x}{n} - 1} - t\_0\\ \end{array} \end{array} \]

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 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 1e-14)
       (/ (- (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 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 1e-14) {
		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 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 1e-14) {
		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 <= -math.inf:
		tmp = 1.0 - t_0
	elif t_1 <= 1e-14:
		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 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 1e-14)
		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, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-14], 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))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 9.99999999999999999e-15

   1. Initial program 37.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. 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 78.6

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

   5. Applied rewrites 78.6%

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

   if 9.99999999999999999e-15 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 53.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   4. 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 51.8

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

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   6. 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 62.3

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

   7. Applied rewrites 62.3%

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

Alternative 3: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+44}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;\frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\mathsf{fma}\left(\frac{x}{n}, \frac{x}{n}, 1 - \frac{x}{n}\right)} - t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e+44)
     (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
     (if (<= (/ 1.0 n) 1e-97)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1e-7)
         (/ (/ (exp (/ (log x) n)) n) x)
         (if (<= (/ 1.0 n) 1e+143)
           (-
            (/ (+ (pow (/ x n) 3.0) 1.0) (fma (/ x n) (/ x n) (- 1.0 (/ x n))))
            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) <= -5e+44) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else if ((1.0 / n) <= 1e-97) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1e-7) {
		tmp = (exp((log(x) / n)) / n) / x;
	} else if ((1.0 / n) <= 1e+143) {
		tmp = ((pow((x / n), 3.0) + 1.0) / fma((x / n), (x / n), (1.0 - (x / n)))) - 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) <= -5e+44)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e-7)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	elseif (Float64(1.0 / n) <= 1e+143)
		tmp = Float64(Float64(Float64((Float64(x / n) ^ 3.0) + 1.0) / fma(Float64(x / n), Float64(x / n), Float64(1.0 - Float64(x / n)))) - 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], -5e+44], 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-97], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-7], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+143], 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[(x / n), $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) < -4.9999999999999996e44

   1. Initial program 100.0%

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

   if -4.9999999999999996e44 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 31.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. 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 82.3

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

   5. Applied rewrites 82.3%

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

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999995e-8

   1. Initial program 12.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}{n}}{x} \]
   7. Step-by-step derivation

      1. exp-neg N/A

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

      2. neg-log N/A

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

      3. exp-neg N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

   if 9.9999999999999995e-8 < (/.f64 #s(literal 1 binary64) n) < 1e143

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   4. 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 88.0

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

   5. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\left(\frac{x}{n} + 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   6. 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 93.7

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

   7. Applied rewrites 93.7%

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

   if 1e143 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 18.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 1.8%

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

   9. Taylor expanded in n around 0

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - 1 \]
   10. 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 84.7

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

   11. Applied rewrites 84.7%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+44}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;\frac{{\left(\frac{x}{n}\right)}^{3} + 1}{\mathsf{fma}\left(\frac{x}{n}, \frac{x}{n}, 1 - \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;n \leq -30500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1600000:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log1p x) (log x)) n)))
   (if (<= n -30500000000.0)
     t_0
     (if (<= n 1600000.0)
       (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))
       (if (<= n 6.5e+80) (/ (/ (exp (/ (log x) n)) n) x) t_0)))))

C

double code(double x, double n) {
	double t_0 = (log1p(x) - log(x)) / n;
	double tmp;
	if (n <= -30500000000.0) {
		tmp = t_0;
	} else if (n <= 1600000.0) {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	} else if (n <= 6.5e+80) {
		tmp = (exp((log(x) / n)) / n) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (Math.log1p(x) - Math.log(x)) / n;
	double tmp;
	if (n <= -30500000000.0) {
		tmp = t_0;
	} else if (n <= 1600000.0) {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	} else if (n <= 6.5e+80) {
		tmp = (Math.exp((Math.log(x) / n)) / n) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (math.log1p(x) - math.log(x)) / n
	tmp = 0
	if n <= -30500000000.0:
		tmp = t_0
	elif n <= 1600000.0:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	elif n <= 6.5e+80:
		tmp = (math.exp((math.log(x) / n)) / n) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (n <= -30500000000.0)
		tmp = t_0;
	elseif (n <= 1600000.0)
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	elseif (n <= 6.5e+80)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -30500000000.0], t$95$0, If[LessEqual[n, 1600000.0], 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, 6.5e+80], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.05e10 or 6.4999999999999998e80 < n

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. 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 81.7

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

   5. Applied rewrites 81.7%

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

   if -3.05e10 < n < 1.6e6

   1. Initial program 83.8%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
   4. 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 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.6e6 < n < 6.4999999999999998e80

   1. Initial program 12.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}{n}}{x} \]
   7. Step-by-step derivation

      1. exp-neg N/A

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

      2. neg-log N/A

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

      3. exp-neg N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

Alternative 5: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

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) -5e+44)
     t_0
     (if (<= (/ 1.0 n) 1e-97)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1e-7)
         (/ (/ (exp (/ (log x) n)) n) x)
         (if (<= (/ 1.0 n) 1e+143) 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) <= -5e+44) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-97) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1e-7) {
		tmp = (exp((log(x) / n)) / n) / x;
	} else if ((1.0 / n) <= 1e+143) {
		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) <= -5e+44) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-97) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e-7) {
		tmp = (Math.exp((Math.log(x) / n)) / n) / x;
	} else if ((1.0 / n) <= 1e+143) {
		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) <= -5e+44:
		tmp = t_0
	elif (1.0 / n) <= 1e-97:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1e-7:
		tmp = (math.exp((math.log(x) / n)) / n) / x
	elif (1.0 / n) <= 1e+143:
		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) <= -5e+44)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e-7)
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	elseif (Float64(1.0 / n) <= 1e+143)
		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], -5e+44], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-97], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-7], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+143], t$95$0, 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) < -4.9999999999999996e44 or 9.9999999999999995e-8 < (/.f64 #s(literal 1 binary64) n) < 1e143

   1. Initial program 98.5%

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

   if -4.9999999999999996e44 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 31.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. 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 82.3

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

   5. Applied rewrites 82.3%

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

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999995e-8

   1. Initial program 12.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}}{n}}{x} \]
   7. Step-by-step derivation

      1. exp-neg N/A

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

      2. neg-log N/A

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

      3. exp-neg N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

   if 1e143 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 18.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 1.8%

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

   9. Taylor expanded in n around 0

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - 1 \]
   10. 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 84.7

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

   11. Applied rewrites 84.7%

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

Alternative 6: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-97}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\ \;\;\;\;\frac{\frac{n + \log x}{x}}{n \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

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) -5e+44)
     t_0
     (if (<= (/ 1.0 n) 1e-97)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1e-7)
         (/ (/ (+ n (log x)) x) (* n n))
         (if (<= (/ 1.0 n) 1e+143) 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) <= -5e+44) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-97) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1e-7) {
		tmp = ((n + log(x)) / x) / (n * n);
	} else if ((1.0 / n) <= 1e+143) {
		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) <= -5e+44) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-97) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e-7) {
		tmp = ((n + Math.log(x)) / x) / (n * n);
	} else if ((1.0 / n) <= 1e+143) {
		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) <= -5e+44:
		tmp = t_0
	elif (1.0 / n) <= 1e-97:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1e-7:
		tmp = ((n + math.log(x)) / x) / (n * n)
	elif (1.0 / n) <= 1e+143:
		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) <= -5e+44)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-97)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e-7)
		tmp = Float64(Float64(Float64(n + log(x)) / x) / Float64(n * n));
	elseif (Float64(1.0 / n) <= 1e+143)
		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], -5e+44], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-97], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-7], N[(N[(N[(n + N[Log[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+143], t$95$0, 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) < -4.9999999999999996e44 or 9.9999999999999995e-8 < (/.f64 #s(literal 1 binary64) n) < 1e143

   1. Initial program 98.5%

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

   if -4.9999999999999996e44 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-97

   1. Initial program 31.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. 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 82.3

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

   5. Applied rewrites 82.3%

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

   if 1.00000000000000004e-97 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999995e-8

   1. Initial program 12.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. 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 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 62.6

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in n around 0

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

       1. lower-/.f64 N/A

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

       2. div-add-rev N/A

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

       3. lower-/.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-log.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 63.2

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

   11. Applied rewrites 63.2%

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

   if 1e143 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 18.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 1.8%

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

   9. Taylor expanded in n around 0

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}}} - 1 \]
   10. 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 84.7

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

   11. Applied rewrites 84.7%

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

Alternative 7: 59.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 1.55 \cdot 10^{-294}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{n} - t\_0\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+203}:\\ \;\;\;\;\frac{1 + t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= x 1.55e-294)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 1.05e-5)
       (- (/ x n) t_0)
       (if (<= x 9.2e+203) (/ (+ 1.0 t_0) (* n x)) (- 1.0 1.0))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if (x <= 1.55e-294) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.05e-5) {
		tmp = (x / n) - t_0;
	} else if (x <= 9.2e+203) {
		tmp = (1.0 + t_0) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / n
    if (x <= 1.55d-294) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.05d-5) then
        tmp = (x / n) - t_0
    else if (x <= 9.2d+203) then
        tmp = (1.0d0 + t_0) / (n * x)
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if (x <= 1.55e-294) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.05e-5) {
		tmp = (x / n) - t_0;
	} else if (x <= 9.2e+203) {
		tmp = (1.0 + t_0) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if x <= 1.55e-294:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.05e-5:
		tmp = (x / n) - t_0
	elif x <= 9.2e+203:
		tmp = (1.0 + t_0) / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 1.55e-294)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.05e-5)
		tmp = Float64(Float64(x / n) - t_0);
	elseif (x <= 9.2e+203)
		tmp = Float64(Float64(1.0 + t_0) / Float64(n * x));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / n;
	tmp = 0.0;
	if (x <= 1.55e-294)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.05e-5)
		tmp = (x / n) - t_0;
	elseif (x <= 9.2e+203)
		tmp = (1.0 + t_0) / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 1.55e-294], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e-5], N[(N[(x / n), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 9.2e+203], N[(N[(1.0 + t$95$0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.55000000000000002e-294

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.0%

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

   if 1.55000000000000002e-294 < x < 1.04999999999999994e-5

   1. Initial program 37.5%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\frac{1}{2} \cdot {\log \left(1 + x\right)}^{2} - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]
   4. 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

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. neg-log N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. neg-log N/A

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

      16. frac-2neg N/A

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

      17. lower-/.f64 N/A

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in n around inf

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 57.1

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

   11. Applied rewrites 57.1%

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

   if 1.04999999999999994e-5 < x < 9.1999999999999996e203

   1. Initial program 47.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in n around inf

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

      1. frac-2neg N/A

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

      2. neg-log N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-+.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. neg-log N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. lift-log.f64 69.0

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

   8. Applied rewrites 69.0%

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

   if 9.1999999999999996e203 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 87.0%

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

Alternative 8: 59.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-294}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+203}:\\ \;\;\;\;\frac{1 - 0.5 \cdot {x}^{-1}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.55e-294)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.96)
     (- (/ x n) (/ (log x) n))
     (if (<= x 9.2e+203)
       (/ (- 1.0 (* 0.5 (pow x -1.0))) (* n x))
       (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.55e-294) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.96) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 9.2e+203) {
		tmp = (1.0 - (0.5 * pow(x, -1.0))) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.55d-294) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.96d0) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 9.2d+203) then
        tmp = (1.0d0 - (0.5d0 * (x ** (-1.0d0)))) / (n * x)
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.55e-294) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.96) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 9.2e+203) {
		tmp = (1.0 - (0.5 * Math.pow(x, -1.0))) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.55e-294:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.96:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 9.2e+203:
		tmp = (1.0 - (0.5 * math.pow(x, -1.0))) / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.55e-294)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.96)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 9.2e+203)
		tmp = Float64(Float64(1.0 - Float64(0.5 * (x ^ -1.0))) / Float64(n * x));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.55e-294)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.96)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 9.2e+203)
		tmp = (1.0 - (0.5 * (x ^ -1.0))) / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.55e-294], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.96], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.2e+203], N[(N[(1.0 - N[(0.5 * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.55000000000000002e-294

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.0%

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

   if 1.55000000000000002e-294 < x < 0.95999999999999996

   1. Initial program 39.2%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\frac{1}{2} \cdot {\log \left(1 + x\right)}^{2} - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]
   4. 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

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. neg-log N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. neg-log N/A

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

      16. frac-2neg N/A

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

      17. lower-/.f64 N/A

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

   8. Applied rewrites 68.3%

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

   9. Taylor expanded in n around inf

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 55.6

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

   11. Applied rewrites 55.6%

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

   if 0.95999999999999996 < x < 9.1999999999999996e203

   1. Initial program 43.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-*.f64 72.6

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

   8. Applied rewrites 72.6%

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

   if 9.1999999999999996e203 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 87.0%

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

Alternative 9: 59.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-294}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+205}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.55e-294)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 1.05e-5)
     (- (/ x n) (/ (log x) n))
     (if (<= x 2.6e+205) (/ (pow x -1.0) n) (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.55e-294) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.05e-5) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 2.6e+205) {
		tmp = pow(x, -1.0) / 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 <= 1.55d-294) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.05d-5) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 2.6d+205) then
        tmp = (x ** (-1.0d0)) / 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 <= 1.55e-294) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.05e-5) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 2.6e+205) {
		tmp = Math.pow(x, -1.0) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.55e-294:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.05e-5:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 2.6e+205:
		tmp = math.pow(x, -1.0) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.55e-294)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.05e-5)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 2.6e+205)
		tmp = Float64((x ^ -1.0) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.55e-294)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.05e-5)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 2.6e+205)
		tmp = (x ^ -1.0) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.55e-294], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e-5], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+205], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.55000000000000002e-294

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.0%

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

   if 1.55000000000000002e-294 < x < 1.04999999999999994e-5

   1. Initial program 37.5%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\frac{1}{2} \cdot {\log \left(1 + x\right)}^{2} - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]
   4. 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

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. neg-log N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. neg-log N/A

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

      16. frac-2neg N/A

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

      17. lower-/.f64 N/A

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in n around inf

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 57.1

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

   11. Applied rewrites 57.1%

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

   if 1.04999999999999994e-5 < x < 2.5999999999999999e205

   1. Initial program 47.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 69.3

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

   8. Applied rewrites 69.3%

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

   9. Taylor expanded in n around inf

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

       1. inv-pow N/A

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

       2. lift-pow.f64 67.8

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

   11. Applied rewrites 67.8%

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

   if 2.5999999999999999e205 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 87.0%

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

Alternative 10: 59.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-294}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+205}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.55e-294)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 1.05e-5)
     (/ (- x (log x)) n)
     (if (<= x 2.6e+205) (/ (pow x -1.0) n) (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.55e-294) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.05e-5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.6e+205) {
		tmp = pow(x, -1.0) / 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 <= 1.55d-294) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.05d-5) then
        tmp = (x - log(x)) / n
    else if (x <= 2.6d+205) then
        tmp = (x ** (-1.0d0)) / 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 <= 1.55e-294) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.05e-5) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.6e+205) {
		tmp = Math.pow(x, -1.0) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.55e-294:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.05e-5:
		tmp = (x - math.log(x)) / n
	elif x <= 2.6e+205:
		tmp = math.pow(x, -1.0) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.55e-294)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.05e-5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.6e+205)
		tmp = Float64((x ^ -1.0) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.55e-294)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.05e-5)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.6e+205)
		tmp = (x ^ -1.0) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.55e-294], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e-5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.6e+205], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.55000000000000002e-294

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.0%

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

   if 1.55000000000000002e-294 < x < 1.04999999999999994e-5

   1. Initial program 37.5%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\frac{1}{2} \cdot {\log \left(1 + x\right)}^{2} - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]
   4. 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

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. neg-log N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. neg-log N/A

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

      16. frac-2neg N/A

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

      17. lower-/.f64 N/A

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in n around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lift-log.f64 57.0

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

   11. Applied rewrites 57.0%

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

   if 1.04999999999999994e-5 < x < 2.5999999999999999e205

   1. Initial program 47.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 69.3

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

   8. Applied rewrites 69.3%

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

   9. Taylor expanded in n around inf

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

       1. inv-pow N/A

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

       2. lift-pow.f64 67.8

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

   11. Applied rewrites 67.8%

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

   if 2.5999999999999999e205 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 87.0%

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

Alternative 11: 59.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+205}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.95e-299)
   (/ 1.0 (* n x))
   (if (<= x 1.05e-5)
     (/ (- x (log x)) n)
     (if (<= x 2.6e+205) (/ (pow x -1.0) n) (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.95e-299) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.05e-5) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.6e+205) {
		tmp = pow(x, -1.0) / 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 <= 1.95d-299) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 1.05d-5) then
        tmp = (x - log(x)) / n
    else if (x <= 2.6d+205) then
        tmp = (x ** (-1.0d0)) / 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 <= 1.95e-299) {
		tmp = 1.0 / (n * x);
	} else if (x <= 1.05e-5) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.6e+205) {
		tmp = Math.pow(x, -1.0) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.95e-299:
		tmp = 1.0 / (n * x)
	elif x <= 1.05e-5:
		tmp = (x - math.log(x)) / n
	elif x <= 2.6e+205:
		tmp = math.pow(x, -1.0) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.95e-299)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 1.05e-5)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.6e+205)
		tmp = Float64((x ^ -1.0) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.95e-299)
		tmp = 1.0 / (n * x);
	elseif (x <= 1.05e-5)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.6e+205)
		tmp = (x ^ -1.0) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.95e-299], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e-5], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.6e+205], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.9499999999999999e-299

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. 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 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 86.2%

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

   if 1.9499999999999999e-299 < x < 1.04999999999999994e-5

   1. Initial program 37.9%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\frac{1}{2} \cdot {\log \left(1 + x\right)}^{2} - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]
   4. 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

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. neg-log N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. neg-log N/A

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

      16. frac-2neg N/A

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

      17. lower-/.f64 N/A

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in n around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lift-log.f64 56.9

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

   11. Applied rewrites 56.9%

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

   if 1.04999999999999994e-5 < x < 2.5999999999999999e205

   1. Initial program 47.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 69.3

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

   8. Applied rewrites 69.3%

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

   9. Taylor expanded in n around inf

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

       1. inv-pow N/A

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

       2. lift-pow.f64 67.8

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

   11. Applied rewrites 67.8%

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

   if 2.5999999999999999e205 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 87.0%

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

Alternative 12: 43.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+205}:\\ \;\;\;\;\frac{{x}^{-1}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.6e+205) (/ (pow x -1.0) n) (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.6e+205) {
		tmp = pow(x, -1.0) / 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 <= 2.6d+205) then
        tmp = (x ** (-1.0d0)) / 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 <= 2.6e+205) {
		tmp = Math.pow(x, -1.0) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.6e+205:
		tmp = math.pow(x, -1.0) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.6e+205)
		tmp = Float64((x ^ -1.0) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.6e+205)
		tmp = (x ^ -1.0) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.6e+205], N[(N[Power[x, -1.0], $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5999999999999999e205

   1. Initial program 42.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. 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 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 37.9

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

   8. Applied rewrites 37.9%

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

   9. Taylor expanded in n around inf

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

       1. inv-pow N/A

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

       2. lift-pow.f64 38.4

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

   11. Applied rewrites 38.4%

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

   if 2.5999999999999999e205 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 87.0%

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

Alternative 13: 43.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{+203}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 9.2e+203) (/ 1.0 (* n x)) (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 9.2e+203) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 9.2d+203) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 9.2e+203) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 9.2e+203:
		tmp = 1.0 / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 9.2e+203)
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 9.2e+203)
		tmp = 1.0 / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 9.2e+203], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.1999999999999996e203

   1. Initial program 42.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. 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 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 38.4%

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

   if 9.1999999999999996e203 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 87.0%

      \[\leadsto 1 - \color{blue}{1} \]
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 49.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 39.0%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 25.8%

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

Reproduce

herbie shell --seed 2025073 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))