2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 85.7%

Time: 18.2s

Alternatives: 16

Speedup: 3.2×

• could not determine a ground truth

  1. x = -6.660517278126683e+260
  2. n = 365755.3225733777

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: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.0001:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.0001)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-20)
     (-
      (/
       (fma
        -1.0
        (+ (log1p x) (/ (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))) n))
        (log x))
       n))
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.0001) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-20) {
		tmp = -(fma(-1.0, (log1p(x) + ((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) / n)), log(x)) / n);
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.0001)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(-Float64(fma(-1.0, Float64(log1p(x) + Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) / n)), log(x)) / n));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.0001], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], (-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]), N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000005e-4

   1. Initial program 99.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-log.f64 99.4

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

   8. Applied rewrites 99.4%

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

   if -1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-20

   1. Initial program 29.8%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\frac{1}{2} \cdot {\log \left(1 + x\right)}^{2} - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]
   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 76.9%

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

   if 4.9999999999999999e-20 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 91.6

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

   5. Applied rewrites 91.6%

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

Alternative 2: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(\frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}, -1, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}, \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 500.0)
   (-
    (/
     (fma
      -1.0
      (+
       (log1p x)
       (/
        (fma
         (/
          (* -0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
          n)
         -1.0
         (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))))
        n))
      (log x))
     n))
   (/ (exp (/ (log x) n)) (* n x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 500.0) {
		tmp = -(fma(-1.0, (log1p(x) + (fma(((-0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n), -1.0, (0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0)))) / n)), log(x)) / n);
	} else {
		tmp = exp((log(x) / n)) / (n * x);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 500.0)
		tmp = Float64(-Float64(fma(-1.0, Float64(log1p(x) + Float64(fma(Float64(Float64(-0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n), -1.0, Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)))) / n)), log(x)) / n));
	else
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 500.0], (-N[(N[(-1.0 * N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * -1.0 + 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]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 42.6%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(\frac{-0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}, -1, 0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right)}{n}, \log x\right)}{n}} \]

   if 500 < x

   1. Initial program 67.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-log.f64 97.4

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

   8. Applied rewrites 97.4%

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

Alternative 3: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \end{array} \end{array} \]

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 -1e-6)
     (- 1.0 t_0)
     (if (<= t_1 0.0) (/ (- (log1p x) (log x)) n) (- (+ (/ 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 <= -1e-6) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = ((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 <= -1e-6) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = ((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 <= -1e-6:
		tmp = 1.0 - t_0
	elif t_1 <= 0.0:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = ((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 <= -1e-6)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(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, -1e-6], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $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))) < -9.99999999999999955e-7

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.5%

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

   if -9.99999999999999955e-7 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   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 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 79.5

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

   5. Applied rewrites 79.5%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 55.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 52.7

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

   5. Applied rewrites 52.7%

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

Alternative 4: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \end{array} \end{array} \]

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 -1e-6)
     (- 1.0 t_0)
     (if (<= t_1 0.0) (/ (log (/ (+ 1.0 x) x)) n) (- (+ (/ 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 <= -1e-6) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = ((x / n) + 1.0) - t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    if (t_1 <= (-1d-6)) then
        tmp = 1.0d0 - t_0
    else if (t_1 <= 0.0d0) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = ((x / n) + 1.0d0) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.0) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = ((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 <= -1e-6:
		tmp = 1.0 - t_0
	elif t_1 <= 0.0:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = ((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 <= -1e-6)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	tmp = 0.0;
	if (t_1 <= -1e-6)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 0.0)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = ((x / n) + 1.0) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-6], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $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))) < -9.99999999999999955e-7

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.5%

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

   if -9.99999999999999955e-7 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   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 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 79.5

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

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 79.5

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

   7. Applied rewrites 79.5%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 55.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 52.7

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

   5. Applied rewrites 52.7%

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

Alternative 5: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ t_2 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0))
        (t_2 (- 1.0 t_0)))
   (if (<= t_1 -1e-6) t_2 (if (<= t_1 0.0) (/ (log (/ (+ 1.0 x) x)) n) t_2))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = ((x + 1.0d0) ** (1.0d0 / n)) - t_0
    t_2 = 1.0d0 - t_0
    if (t_1 <= (-1d-6)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double t_2 = 1.0 - t_0;
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	t_2 = 1.0 - t_0
	tmp = 0
	if t_1 <= -1e-6:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	t_2 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= -1e-6)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = ((x + 1.0) ^ (1.0 / n)) - t_0;
	t_2 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= -1e-6)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-6], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -9.99999999999999955e-7 or 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 76.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.3%

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

   if -9.99999999999999955e-7 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   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 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 79.5

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

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 79.5

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

   7. Applied rewrites 79.5%

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

Alternative 6: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-47}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-47)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-20)
     (/ (log (/ (+ 1.0 x) x)) n)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-47) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-20) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-47) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-20) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-47:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-20:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-47)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-20)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-47], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-20], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-47

   1. Initial program 88.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-log.f64 92.8

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

   8. Applied rewrites 92.8%

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

   if -1.9999999999999999e-47 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e-20

   1. Initial program 30.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 78.7

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

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if 4.9999999999999999e-20 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 91.6

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

   5. Applied rewrites 91.6%

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

Alternative 7: 81.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-47}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-19}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+208}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-47)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 1e-19)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 5e+208)
       (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))
       (- (exp (/ x n)) 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-47) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-19) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+208) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = exp((x / n)) - 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 ((1.0d0 / n) <= (-2d-47)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 1d-19) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+208) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    else
        tmp = exp((x / n)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-47) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-19) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+208) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.exp((x / n)) - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-47:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 1e-19:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+208:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.exp((x / n)) - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-47)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-19)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+208)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(exp(Float64(x / n)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-47)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 1e-19)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+208)
		tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	else
		tmp = exp((x / n)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-47], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-19], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+208], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-47

   1. Initial program 88.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-log.f64 92.8

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

   8. Applied rewrites 92.8%

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

   if -1.9999999999999999e-47 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-20

   1. Initial program 30.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 78.7

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

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if 9.9999999999999998e-20 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000004e208

   1. Initial program 64.1%

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

   if 5.0000000000000004e208 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 22.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 18.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 1.9%

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

   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 81.9

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

   11. Applied rewrites 81.9%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 81.9%

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

Alternative 8: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-47}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-19}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+229}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-47)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 1e-19)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 5e+229)
       (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
       (- (exp (/ x n)) 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-47) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-19) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+229) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else {
		tmp = exp((x / n)) - 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 ((1.0d0 / n) <= (-2d-47)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 1d-19) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+229) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else
        tmp = exp((x / n)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-47) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-19) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+229) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.exp((x / n)) - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-47:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 1e-19:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+229:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	else:
		tmp = math.exp((x / n)) - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-47)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-19)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+229)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(exp(Float64(x / n)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-47)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 1e-19)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+229)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	else
		tmp = exp((x / n)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-47], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-19], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+229], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-47

   1. Initial program 88.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-log.f64 92.8

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

   8. Applied rewrites 92.8%

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

   if -1.9999999999999999e-47 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-20

   1. Initial program 30.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 78.7

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

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if 9.9999999999999998e-20 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000005e229

   1. Initial program 62.3%

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

   3. Taylor expanded in x around 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 58.4

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

   5. Applied rewrites 58.4%

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

   if 5.0000000000000005e229 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 18.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 15.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 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 85.0

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

   11. Applied rewrites 85.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 85.0%

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

Alternative 9: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-124}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+163}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x} - 0.3333333333333333}{x}\right) - 0.5}{x}\right) - 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.4e-265)
   (/ (- (log x)) n)
   (if (<= x 9.5e-124)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.9)
       (/ (- x (log x)) n)
       (if (<= x 4.7e+163)
         (/
          (-
           (/
            (-
             (- (/ (- (- (/ (- (/ 0.25 x) 0.3333333333333333) x)) 0.5) x))
             1.0)
            x))
          n)
         (- 1.0 1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.4e-265) {
		tmp = -log(x) / n;
	} else if (x <= 9.5e-124) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.7e+163) {
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.4d-265) then
        tmp = -log(x) / n
    else if (x <= 9.5d-124) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4.7d+163) then
        tmp = -((-((-(((0.25d0 / x) - 0.3333333333333333d0) / x) - 0.5d0) / x) - 1.0d0) / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.4e-265) {
		tmp = -Math.log(x) / n;
	} else if (x <= 9.5e-124) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.7e+163) {
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.4e-265:
		tmp = -math.log(x) / n
	elif x <= 9.5e-124:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.9:
		tmp = (x - math.log(x)) / n
	elif x <= 4.7e+163:
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.4e-265)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 9.5e-124)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.7e+163)
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.25 / x) - 0.3333333333333333) / x)) - 0.5) / x)) - 1.0) / x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.4e-265)
		tmp = -log(x) / n;
	elseif (x <= 9.5e-124)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.9)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.7e+163)
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.4e-265], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 9.5e-124], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.7e+163], N[((-N[(N[((-N[(N[((-N[(N[(N[(0.25 / x), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]) - 0.5), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 3.4000000000000001e-265

   1. Initial program 53.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 49.7

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

   8. Applied rewrites 49.7%

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

   if 3.4000000000000001e-265 < x < 9.49999999999999989e-124

   1. Initial program 44.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 44.1%

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

   if 9.49999999999999989e-124 < x < 0.900000000000000022

   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 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.5%

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

   if 0.900000000000000022 < x < 4.70000000000000019e163

   1. Initial program 52.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 3.8

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

   8. Applied rewrites 3.8%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}\right)}{n} \]

       2. lower-neg.f64 N/A

          \[\leadsto \frac{-\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{-\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

   11. Applied rewrites 65.4%

       \[\leadsto \frac{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x} - 0.3333333333333333}{x}\right) - 0.5}{x}\right) - 1}{x}}{n} \]

   if 4.70000000000000019e163 < x

   1. Initial program 84.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 84.3%

      \[\leadsto 1 - \color{blue}{1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 61.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+163}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x} - 0.3333333333333333}{x}\right) - 0.5}{x}\right) - 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.9)
   (/ (- x (log x)) n)
   (if (<= x 4.7e+163)
     (/
      (-
       (/
        (- (- (/ (- (- (/ (- (/ 0.25 x) 0.3333333333333333) x)) 0.5) x)) 1.0)
        x))
      n)
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.7e+163) {
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4.7d+163) then
        tmp = -((-((-(((0.25d0 / x) - 0.3333333333333333d0) / x) - 0.5d0) / x) - 1.0d0) / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.7e+163) {
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.9:
		tmp = (x - math.log(x)) / n
	elif x <= 4.7e+163:
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.7e+163)
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.25 / x) - 0.3333333333333333) / x)) - 0.5) / x)) - 1.0) / x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.9)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.7e+163)
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.7e+163], N[((-N[(N[((-N[(N[((-N[(N[(N[(0.25 / x), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]) - 0.5), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.900000000000000022

   1. Initial program 42.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.9%

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

   if 0.900000000000000022 < x < 4.70000000000000019e163

   1. Initial program 52.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 3.8

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

   8. Applied rewrites 3.8%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}\right)}{n} \]

       2. lower-neg.f64 N/A

          \[\leadsto \frac{-\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{-\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

   11. Applied rewrites 65.4%

       \[\leadsto \frac{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x} - 0.3333333333333333}{x}\right) - 0.5}{x}\right) - 1}{x}}{n} \]

   if 4.70000000000000019e163 < x

   1. Initial program 84.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 84.3%

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

Alternative 11: 61.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{+163}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x} - 0.3333333333333333}{x}\right) - 0.5}{x}\right) - 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.7)
   (/ (- (log x)) n)
   (if (<= x 4.7e+163)
     (/
      (-
       (/
        (- (- (/ (- (- (/ (- (/ 0.25 x) 0.3333333333333333) x)) 0.5) x)) 1.0)
        x))
      n)
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -log(x) / n;
	} else if (x <= 4.7e+163) {
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = -log(x) / n
    else if (x <= 4.7d+163) then
        tmp = -((-((-(((0.25d0 / x) - 0.3333333333333333d0) / x) - 0.5d0) / x) - 1.0d0) / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.7) {
		tmp = -Math.log(x) / n;
	} else if (x <= 4.7e+163) {
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.7:
		tmp = -math.log(x) / n
	elif x <= 4.7e+163:
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 4.7e+163)
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.25 / x) - 0.3333333333333333) / x)) - 0.5) / x)) - 1.0) / x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = -log(x) / n;
	elseif (x <= 4.7e+163)
		tmp = -((-((-(((0.25 / x) - 0.3333333333333333) / x) - 0.5) / x) - 1.0) / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.7], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 4.7e+163], N[((-N[(N[((-N[(N[((-N[(N[(N[(0.25 / x), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]) - 0.5), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.69999999999999996

   1. Initial program 42.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if 0.69999999999999996 < x < 4.70000000000000019e163

   1. Initial program 52.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 3.8

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

   8. Applied rewrites 3.8%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}\right)}{n} \]

       2. lower-neg.f64 N/A

          \[\leadsto \frac{-\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{-\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}}{n} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x} - 0.3333333333333333}{x}\right) - 0.5}{x}\right) - 1}{x}}{n} \]

   if 4.70000000000000019e163 < x

   1. Initial program 84.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 84.3%

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

Alternative 12: 54.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.2)
   (/ (/ 0.3333333333333333 (* n (* x x))) x)
   (/ (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) x)) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = (0.3333333333333333 / (n * (x * x))) / x;
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-0.2d0)) then
        tmp = (0.3333333333333333d0 / (n * (x * x))) / x
    else
        tmp = -((-(((0.3333333333333333d0 / x) - 0.5d0) / x) - 1.0d0) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = (0.3333333333333333 / (n * (x * x))) / x;
	} else {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.2:
		tmp = (0.3333333333333333 / (n * (x * x))) / x
	else:
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.2)
		tmp = Float64(Float64(0.3333333333333333 / Float64(n * Float64(x * x))) / x);
	else
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.2)
		tmp = (0.3333333333333333 / (n * (x * x))) / x;
	else
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.2], N[(N[(0.3333333333333333 / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[((-N[(N[((-N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.20000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 19.1

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

   8. Applied rewrites 19.1%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 69.8

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

   11. Applied rewrites 69.8%

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

   if -0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 41.1

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

   8. Applied rewrites 41.1%

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

   9. Taylor expanded in x around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower--.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower--.f64 N/A

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. lower-/.f64 47.8

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

   11. Applied rewrites 47.8%

       \[\leadsto \frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n \cdot \left(x \cdot x\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.2)
   (/ (/ 0.3333333333333333 (* n (* x x))) x)
   (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = (0.3333333333333333 / (n * (x * x))) / x;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-0.2d0)) then
        tmp = (0.3333333333333333d0 / (n * (x * x))) / x
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = (0.3333333333333333 / (n * (x * x))) / x;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.2:
		tmp = (0.3333333333333333 / (n * (x * x))) / x
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.2)
		tmp = Float64(Float64(0.3333333333333333 / Float64(n * Float64(x * x))) / x);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.2)
		tmp = (0.3333333333333333 / (n * (x * x))) / x;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.2], N[(N[(0.3333333333333333 / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.20000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   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 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 19.1

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

   8. Applied rewrites 19.1%

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

   9. Taylor expanded in x around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 69.8

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

   11. Applied rewrites 69.8%

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

   if -0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 41.3

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 45.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 46.0

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

   10. Applied rewrites 46.0%

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

Alternative 14: 47.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.2) (- 1.0 1.0) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-0.2d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.2:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.2)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.2)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.2], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.20000000000000001

   1. Initial program 99.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 49.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 52.6%

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

   if -0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 41.3

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 45.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 46.0

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

   10. Applied rewrites 46.0%

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

Alternative 15: 47.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.2) (- 1.0 1.0) (/ 1.0 (* n x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-0.2d0)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.2:
		tmp = 1.0 - 1.0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.2)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.2)
		tmp = 1.0 - 1.0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.2], N[(1.0 - 1.0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.20000000000000001

   1. Initial program 99.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 49.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 52.6%

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

   if -0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 41.3

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 45.1%

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

Alternative 16: 31.4% accurate, 57.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 38.3%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 31.4%

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

Reproduce

herbie shell --seed 2025088 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))