2nthrt (problem 3.4.6)

Percentage Accurate: 54.2% → 85.7%

Time: 29.5s

Alternatives: 13

Speedup: 1.4×

• could not determine a ground truth

  1. x = -2.722504979829982e+73
  2. n = -4.2660075644445295e+220

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: 54.2% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -2e-94)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (pow n -1.0) 2e-15)
     (/ (log (/ (+ 1.0 x) x)) n)
     (- (exp (/ (log1p x) n)) (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -2e-94) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if (pow(n, -1.0) <= 2e-15) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -2e-94) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if (Math.pow(n, -1.0) <= 2e-15) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, Math.pow(n, -1.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -2e-94:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif math.pow(n, -1.0) <= 2e-15:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, math.pow(n, -1.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -2e-94)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif ((n ^ -1.0) <= 2e-15)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e-94], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 2e-15], 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[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-94

   1. Initial program 78.5%

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -1.9999999999999999e-94 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000002e-15

   1. Initial program 31.4%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 83.6%

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

   if 2.0000000000000002e-15 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 47.1%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 90.0

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

   4. Applied rewrites 90.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (pow (+ x 1.0) (pow n -1.0)) t_0)))
   (if (<= t_1 -2e-7)
     (- 1.0 t_0)
     (if (<= t_1 2e-7)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (fma (fma (/ (+ -0.5 (/ 0.5 n)) n) x (pow n -1.0)) x 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = pow((x + 1.0), pow(n, -1.0)) - t_0;
	double tmp;
	if (t_1 <= -2e-7) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 2e-7) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(fma(((-0.5 + (0.5 / n)) / n), x, pow(n, -1.0)), x, 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0)
	tmp = 0.0
	if (t_1 <= -2e-7)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 2e-7)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(-0.5 + Float64(0.5 / n)) / n), x, (n ^ -1.0)), x, 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-7], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 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))) < -1.9999999999999999e-7

   1. Initial program 98.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 98.9%

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

   if -1.9999999999999999e-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))) < 1.9999999999999999e-7

   1. Initial program 42.1%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 78.6%

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

   if 1.9999999999999999e-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)))

   1. Initial program 49.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 73.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.85e-8)
   (/
    (-
     (+
      (log1p x)
      (/
       (fma
        0.16666666666666666
        (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)
        (* (- (pow (log1p x) 2.0) (pow (log x) 2.0)) 0.5))
       n))
     (log x))
    n)
   (/ (/ (exp (/ (log x) n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = ((log1p(x) + (fma(0.16666666666666666, ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n), ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) * 0.5)) / n)) - log(x)) / n;
	} else {
		tmp = (exp((log(x) / n)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.85e-8)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(fma(0.16666666666666666, Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n), Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) * 0.5)) / n)) - log(x)) / n);
	else
		tmp = Float64(Float64(exp(Float64(log(x) / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.85e-8], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85e-8

   1. Initial program 41.9%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 78.8%

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

   if 1.85e-8 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in x around 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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   7. Applied rewrites 97.8%

      \[\leadsto \frac{\frac{e^{\frac{\log x}{n}}}{n}}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) \cdot 0.5\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{\log x}{n}}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (pow (+ x 1.0) (pow n -1.0)) t_0)))
   (if (<= t_1 -2e-7)
     (- 1.0 t_0)
     (if (<= t_1 2e-7) (/ (log (/ (+ 1.0 x) x)) n) (- (+ (/ x n) 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = pow((x + 1.0), pow(n, -1.0)) - t_0;
	double tmp;
	if (t_1 <= -2e-7) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 2e-7) {
		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 ** (n ** (-1.0d0))
    t_1 = ((x + 1.0d0) ** (n ** (-1.0d0))) - t_0
    if (t_1 <= (-2d-7)) then
        tmp = 1.0d0 - t_0
    else if (t_1 <= 2d-7) 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, Math.pow(n, -1.0));
	double t_1 = Math.pow((x + 1.0), Math.pow(n, -1.0)) - t_0;
	double tmp;
	if (t_1 <= -2e-7) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 2e-7) {
		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, math.pow(n, -1.0))
	t_1 = math.pow((x + 1.0), math.pow(n, -1.0)) - t_0
	tmp = 0
	if t_1 <= -2e-7:
		tmp = 1.0 - t_0
	elif t_1 <= 2e-7:
		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 ^ (n ^ -1.0)
	t_1 = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0)
	tmp = 0.0
	if (t_1 <= -2e-7)
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 2e-7)
		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 ^ (n ^ -1.0);
	t_1 = ((x + 1.0) ^ (n ^ -1.0)) - t_0;
	tmp = 0.0;
	if (t_1 <= -2e-7)
		tmp = 1.0 - t_0;
	elseif (t_1 <= 2e-7)
		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[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-7], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], 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))) < -1.9999999999999999e-7

   1. Initial program 98.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 98.9%

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

   if -1.9999999999999999e-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))) < 1.9999999999999999e-7

   1. Initial program 42.1%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 78.6%

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

   if 1.9999999999999999e-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)))

   1. Initial program 49.4%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 48.9

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

   5. Applied rewrites 48.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{elif}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left({n}^{-1}\right)}\\ t_1 := {\left(x + 1\right)}^{\left({n}^{-1}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-7} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (pow n -1.0))) (t_1 (- (pow (+ x 1.0) (pow n -1.0)) t_0)))
   (if (or (<= t_1 -2e-7) (not (<= t_1 2e-7)))
     (- 1.0 t_0)
     (/ (log (/ (+ 1.0 x) x)) n))))

C

double code(double x, double n) {
	double t_0 = pow(x, pow(n, -1.0));
	double t_1 = pow((x + 1.0), pow(n, -1.0)) - t_0;
	double tmp;
	if ((t_1 <= -2e-7) || !(t_1 <= 2e-7)) {
		tmp = 1.0 - t_0;
	} else {
		tmp = log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (n ** (-1.0d0))
    t_1 = ((x + 1.0d0) ** (n ** (-1.0d0))) - t_0
    if ((t_1 <= (-2d-7)) .or. (.not. (t_1 <= 2d-7))) then
        tmp = 1.0d0 - t_0
    else
        tmp = log(((1.0d0 + x) / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, Math.pow(n, -1.0));
	double t_1 = Math.pow((x + 1.0), Math.pow(n, -1.0)) - t_0;
	double tmp;
	if ((t_1 <= -2e-7) || !(t_1 <= 2e-7)) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, math.pow(n, -1.0))
	t_1 = math.pow((x + 1.0), math.pow(n, -1.0)) - t_0
	tmp = 0
	if (t_1 <= -2e-7) or not (t_1 <= 2e-7):
		tmp = 1.0 - t_0
	else:
		tmp = math.log(((1.0 + x) / x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ (n ^ -1.0)
	t_1 = Float64((Float64(x + 1.0) ^ (n ^ -1.0)) - t_0)
	tmp = 0.0
	if ((t_1 <= -2e-7) || !(t_1 <= 2e-7))
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (n ^ -1.0);
	t_1 = ((x + 1.0) ^ (n ^ -1.0)) - t_0;
	tmp = 0.0;
	if ((t_1 <= -2e-7) || ~((t_1 <= 2e-7)))
		tmp = 1.0 - t_0;
	else
		tmp = log(((1.0 + x) / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-7], N[Not[LessEqual[t$95$1, 2e-7]], $MachinePrecision]], N[(1.0 - t$95$0), $MachinePrecision], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]

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))) < -1.9999999999999999e-7 or 1.9999999999999999e-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)))

   1. Initial program 73.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 72.7%

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

   if -1.9999999999999999e-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))) < 1.9999999999999999e-7

   1. Initial program 42.1%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 78.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq -2 \cdot 10^{-7} \lor \neg \left({\left(x + 1\right)}^{\left({n}^{-1}\right)} - {x}^{\left({n}^{-1}\right)} \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;1 - {x}^{\left({n}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -2e-94)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (pow n -1.0) 1e-13)
     (/ (log (/ (+ 1.0 x) x)) n)
     (-
      (fma (fma (/ (+ -0.5 (/ 0.5 n)) n) x (pow n -1.0)) x 1.0)
      (pow x (pow n -1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -2e-94) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if (pow(n, -1.0) <= 1e-13) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = fma(fma(((-0.5 + (0.5 / n)) / n), x, pow(n, -1.0)), x, 1.0) - pow(x, pow(n, -1.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -2e-94)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif ((n ^ -1.0) <= 1e-13)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(fma(fma(Float64(Float64(-0.5 + Float64(0.5 / n)) / n), x, (n ^ -1.0)), x, 1.0) - (x ^ (n ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -2e-94], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], 1e-13], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * x + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-94

   1. Initial program 78.5%

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -1.9999999999999999e-94 < (/.f64 #s(literal 1 binary64) n) < 1e-13

   1. Initial program 31.4%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 82.8

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 83.1%

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

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

   1. Initial program 47.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 68.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;{n}^{-1} \leq 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5 + \frac{0.5}{n}}{n}, x, {n}^{-1}\right), x, 1\right) - {x}^{\left({n}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1.5 \cdot 10^{+230}:\\ \;\;\;\;{\left(n \cdot x\right)}^{-1}\\ \mathbf{elif}\;{n}^{-1} \leq -10:\\ \;\;\;\;\frac{n \cdot x}{\left(n \cdot x\right) \cdot \left(n \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (pow n -1.0) -1.5e+230)
   (pow (* n x) -1.0)
   (if (<= (pow n -1.0) -10.0)
     (/ (* n x) (* (* n x) (* n x)))
     (/ (pow n -1.0) x))))

C

double code(double x, double n) {
	double tmp;
	if (pow(n, -1.0) <= -1.5e+230) {
		tmp = pow((n * x), -1.0);
	} else if (pow(n, -1.0) <= -10.0) {
		tmp = (n * x) / ((n * x) * (n * x));
	} else {
		tmp = pow(n, -1.0) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n ** (-1.0d0)) <= (-1.5d+230)) then
        tmp = (n * x) ** (-1.0d0)
    else if ((n ** (-1.0d0)) <= (-10.0d0)) then
        tmp = (n * x) / ((n * x) * (n * x))
    else
        tmp = (n ** (-1.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (Math.pow(n, -1.0) <= -1.5e+230) {
		tmp = Math.pow((n * x), -1.0);
	} else if (Math.pow(n, -1.0) <= -10.0) {
		tmp = (n * x) / ((n * x) * (n * x));
	} else {
		tmp = Math.pow(n, -1.0) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if math.pow(n, -1.0) <= -1.5e+230:
		tmp = math.pow((n * x), -1.0)
	elif math.pow(n, -1.0) <= -10.0:
		tmp = (n * x) / ((n * x) * (n * x))
	else:
		tmp = math.pow(n, -1.0) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n ^ -1.0) <= -1.5e+230)
		tmp = Float64(n * x) ^ -1.0;
	elseif ((n ^ -1.0) <= -10.0)
		tmp = Float64(Float64(n * x) / Float64(Float64(n * x) * Float64(n * x)));
	else
		tmp = Float64((n ^ -1.0) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n ^ -1.0) <= -1.5e+230)
		tmp = (n * x) ^ -1.0;
	elseif ((n ^ -1.0) <= -10.0)
		tmp = (n * x) / ((n * x) * (n * x));
	else
		tmp = (n ^ -1.0) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -1.5e+230], N[Power[N[(n * x), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[Power[n, -1.0], $MachinePrecision], -10.0], N[(N[(n * x), $MachinePrecision] / N[(N[(n * x), $MachinePrecision] * N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[n, -1.0], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.50000000000000004e230

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 49.4%

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

   10. Applied rewrites 49.4%

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

   if -1.50000000000000004e230 < (/.f64 #s(literal 1 binary64) n) < -10

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 41.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cosh \left(\frac{\log x}{n}\right), n \cdot x, \left(n \cdot x\right) \cdot \sinh \left(\frac{\log x}{n}\right)\right)}{\color{blue}{\left(n \cdot x\right) \cdot \left(n \cdot x\right)}} \]

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 37.6%

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

   if -10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 33.4%

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

   3. Taylor expanded in x around 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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 43.5%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{n}^{-1} \leq -1.5 \cdot 10^{+230}:\\ \;\;\;\;{\left(n \cdot x\right)}^{-1}\\ \mathbf{elif}\;{n}^{-1} \leq -10:\\ \;\;\;\;\frac{n \cdot x}{\left(n \cdot x\right) \cdot \left(n \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{n}^{-1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} + {n}^{-1}\right) - \frac{\frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.85e-8)
   (/ (- x (log x)) n)
   (if (<= x 4.4e+101)
     (/
      (- (+ (/ 0.3333333333333333 (* (* x x) n)) (pow n -1.0)) (/ (/ 0.5 n) x))
      x)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.4e+101) {
		tmp = (((0.3333333333333333 / ((x * x) * n)) + pow(n, -1.0)) - ((0.5 / n) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.85d-8) then
        tmp = (x - log(x)) / n
    else if (x <= 4.4d+101) then
        tmp = (((0.3333333333333333d0 / ((x * x) * n)) + (n ** (-1.0d0))) - ((0.5d0 / n) / x)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.4e+101) {
		tmp = (((0.3333333333333333 / ((x * x) * n)) + Math.pow(n, -1.0)) - ((0.5 / n) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.85e-8:
		tmp = (x - math.log(x)) / n
	elif x <= 4.4e+101:
		tmp = (((0.3333333333333333 / ((x * x) * n)) + math.pow(n, -1.0)) - ((0.5 / n) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.85e-8)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.4e+101)
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) + (n ^ -1.0)) - Float64(Float64(0.5 / n) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.85e-8)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.4e+101)
		tmp = (((0.3333333333333333 / ((x * x) * n)) + (n ^ -1.0)) - ((0.5 / n) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.85e-8], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.4e+101], N[(N[(N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 1.85e-8

   1. Initial program 41.9%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.8%

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

   if 1.85e-8 < x < 4.4000000000000001e101

   1. Initial program 56.8%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 45.6

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

   5. Applied rewrites 45.6%

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

   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

   8. Applied rewrites 65.0%

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

   if 4.4000000000000001e101 < x

   1. Initial program 70.9%

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

   3. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \log \left(1 + x\right) + -1 \cdot \frac{\left(-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{24} \cdot {\log \left(1 + x\right)}^{4} - \frac{1}{24} \cdot {\log x}^{4}}{n} + \frac{-1}{6} \cdot {\log \left(1 + x\right)}^{3}\right) - \frac{-1}{6} \cdot {\log x}^{3}}{n} + \frac{1}{2} \cdot {\log \left(1 + x\right)}^{2}\right) - \frac{1}{2} \cdot {\log x}^{2}}{n}\right) - -1 \cdot \log x}{n}} \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\frac{\left(\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.5, \frac{\mathsf{fma}\left(-0.041666666666666664, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}}{n}, \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot -0.16666666666666666\right)}{-n}\right)}{n}\right) + \log x}{-n}} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.0%

      \[\leadsto \frac{\left(-\mathsf{log1p}\left(x\right)\right) - \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, \frac{\mathsf{fma}\left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{4} - {\log x}^{4}}{n}, -0.041666666666666664, -0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)\right)}{-n}\right)}{n}}{-n} + \color{blue}{\frac{-\log x}{n}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 71.0%

       \[\leadsto 0 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} + {n}^{-1}\right) - \frac{\frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} + {n}^{-1}\right) - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.85e-8)
   (/ (- x (log x)) n)
   (/
    (- (+ (/ 0.3333333333333333 (* (* x x) n)) (pow n -1.0)) (/ (/ 0.5 n) x))
    x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (((0.3333333333333333 / ((x * x) * n)) + pow(n, -1.0)) - ((0.5 / n) / x)) / 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 (x <= 1.85d-8) then
        tmp = (x - log(x)) / n
    else
        tmp = (((0.3333333333333333d0 / ((x * x) * n)) + (n ** (-1.0d0))) - ((0.5d0 / n) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (((0.3333333333333333 / ((x * x) * n)) + Math.pow(n, -1.0)) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.85e-8:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (((0.3333333333333333 / ((x * x) * n)) + math.pow(n, -1.0)) - ((0.5 / n) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.85e-8)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) + (n ^ -1.0)) - Float64(Float64(0.5 / n) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.85e-8)
		tmp = (x - log(x)) / n;
	else
		tmp = (((0.3333333333333333 / ((x * x) * n)) + (n ^ -1.0)) - ((0.5 / n) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.85e-8], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85e-8

   1. Initial program 41.9%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.8%

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

   if 1.85e-8 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 62.5

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

   5. Applied rewrites 62.5%

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

   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

   8. Applied rewrites 62.0%

      \[\leadsto \frac{\left(\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} + \frac{1}{n}\right) - \frac{\frac{0.5}{n}}{x}}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} + {n}^{-1}\right) - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} + {n}^{-1}\right) - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.85e-8)
   (/ (- (log x)) n)
   (/
    (- (+ (/ 0.3333333333333333 (* (* x x) n)) (pow n -1.0)) (/ (/ 0.5 n) x))
    x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = -log(x) / n;
	} else {
		tmp = (((0.3333333333333333 / ((x * x) * n)) + pow(n, -1.0)) - ((0.5 / n) / x)) / 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 (x <= 1.85d-8) then
        tmp = -log(x) / n
    else
        tmp = (((0.3333333333333333d0 / ((x * x) * n)) + (n ** (-1.0d0))) - ((0.5d0 / n) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.85e-8) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (((0.3333333333333333 / ((x * x) * n)) + Math.pow(n, -1.0)) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.85e-8:
		tmp = -math.log(x) / n
	else:
		tmp = (((0.3333333333333333 / ((x * x) * n)) + math.pow(n, -1.0)) - ((0.5 / n) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.85e-8)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(Float64(x * x) * n)) + (n ^ -1.0)) - Float64(Float64(0.5 / n) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.85e-8)
		tmp = -log(x) / n;
	else
		tmp = (((0.3333333333333333 / ((x * x) * n)) + (n ^ -1.0)) - ((0.5 / n) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.85e-8], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 / N[(N[(x * x), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] + N[Power[n, -1.0], $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85e-8

   1. Initial program 41.9%

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

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.7%

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

   if 1.85e-8 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 62.5

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

   5. Applied rewrites 62.5%

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

   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

   8. Applied rewrites 62.0%

      \[\leadsto \frac{\left(\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} + \frac{1}{n}\right) - \frac{\frac{0.5}{n}}{x}}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{0.3333333333333333}{\left(x \cdot x\right) \cdot n} + {n}^{-1}\right) - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in x around 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

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

   2. log-rec N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r/ N/A

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

   5. mul-1-neg N/A

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

   6. distribute-lft-neg-in N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. lower-*.f64 57.2

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

5. Applied rewrites 57.2%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 38.6%

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

9. Final simplification 38.6%

   \[\leadsto \frac{{n}^{-1}}{x} \]
10. Add Preprocessing

Alternative 12: 40.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double x, double n) {
	return pow((n * x), -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 = (n * x) ** (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in x around 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

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

   2. log-rec N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r/ N/A

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

   5. mul-1-neg N/A

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

   6. distribute-lft-neg-in N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. lower-*.f64 57.2

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

5. Applied rewrites 57.2%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 38.6%

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

10. Applied rewrites 38.3%

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

11. Final simplification 38.3%

    \[\leadsto {\left(n \cdot x\right)}^{-1} \]
12. Add Preprocessing

Alternative 13: 46.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} - -1}{x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (- (/ (- (/ 0.3333333333333333 x) 0.5) x) -1.0) x) n))

C

double code(double x, double n) {
	return (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (((((0.3333333333333333d0 / x) - 0.5d0) / x) - (-1.0d0)) / x) / n
end function

Java

public static double code(double x, double n) {
	return (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n;
}

Python

def code(x, n):
	return (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (((((0.3333333333333333 / x) - 0.5) / x) - -1.0) / x) / n;
end

Wolfram

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

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-log.f64 56.3

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

5. Applied rewrites 56.3%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 47.0%

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

9. Final simplification 47.0%

   \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{x} - 0.5}{x} - -1}{x}}{n} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))