2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 86.7%

Time: 45.4s

Alternatives: 14

Speedup: 1.9×

• could not determine a ground truth

  1. x = -7.6410374963508475e+56
  2. n = 2.627839169970152e+162

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-12}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-17)
   (/ (exp (/ (log x) n)) (* x n))
   (if (<= (/ 1.0 n) 1e-12)
     (-
      (/
       (fma
        -1.0
        (+ (log1p x) (/ (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))) n))
        (log x))
       n))
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-17) {
		tmp = exp((log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-12) {
		tmp = -(fma(-1.0, (log1p(x) + ((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) / n)), log(x)) / n);
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-17)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-12)
		tmp = Float64(-Float64(fma(-1.0, Float64(log1p(x) + Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) / n)), log(x)) / n));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-17], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-12], (-N[(N[(-1.0 * N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-17

   1. Initial program 96.4%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   if -4.9999999999999999e-17 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-13

   1. Initial program 29.0%

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

   3. Taylor expanded in n around -inf

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

   5. Applied rewrites 80.4%

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

   if 9.9999999999999998e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.6%

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

   3. Taylor expanded in n around 0

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

   5. Applied rewrites 94.0%

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

Alternative 2: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{1}{n}\right), 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 1e-9)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (fma x (fma x (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 1.0 n)) 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 1e-9) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) - (0.5 / n)), (1.0 / n)), 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 1e-9)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(1.0 / n)), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-9], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 1.00000000000000006e-9

   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

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.5%

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

   if 1.00000000000000006e-9 < (-.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 48.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.8%

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

Alternative 3: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1 + \mathsf{fma}\left(-0.5, x, 0.5 \cdot \frac{x}{n}\right)}{n}, 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 1e-9)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (fma x (/ (+ 1.0 (fma -0.5 x (* 0.5 (/ x n)))) n) 1.0) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 1e-9) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(x, ((1.0 + fma(-0.5, x, (0.5 * (x / n)))) / n), 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 1e-9)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(fma(x, Float64(Float64(1.0 + fma(-0.5, x, Float64(0.5 * Float64(x / n)))) / n), 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-9], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(x * N[(N[(1.0 + N[(-0.5 * x + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 1.00000000000000006e-9

   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

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.5%

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

   if 1.00000000000000006e-9 < (-.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 48.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 64.9%

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

Alternative 4: 77.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \frac{\frac{0.5}{n \cdot n} - \frac{0.5}{n}}{x}, \frac{1}{n}\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 0.02)
       (/ (log (/ (+ x 1.0) x)) n)
       (/ (fma 1.0 (/ (- (/ 0.5 (* n n)) (/ 0.5 n)) x) (/ 1.0 n)) x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.02) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(1.0, (((0.5 / (n * n)) - (0.5 / n)) / x), (1.0 / n)) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(1.0 - t_0);
	elseif (t_1 <= 0.02)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(fma(1.0, Float64(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)) / x), Float64(1.0 / n)) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 * N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 42.4%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 81.2%

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

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

   1. Initial program 46.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 50.5%

      \[\leadsto \frac{\mathsf{fma}\left(1, \frac{\frac{0.5}{n \cdot n} - \frac{0.5}{n}}{x}, \frac{e^{-\frac{\log x}{-n}}}{n}\right)}{x} \]

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 50.5%

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

Alternative 5: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-17)
   (/ (exp (/ (log x) n)) (* x n))
   (if (<= (/ 1.0 n) 1e-12)
     (/ (log (/ (+ x 1.0) x)) n)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-17) {
		tmp = exp((log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-12) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-17) {
		tmp = Math.exp((Math.log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 1e-12) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-17:
		tmp = math.exp((math.log(x) / n)) / (x * n)
	elif (1.0 / n) <= 1e-12:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-17)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 1e-12)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-17], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-12], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-17

   1. Initial program 96.4%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   if -4.9999999999999999e-17 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-13

   1. Initial program 29.0%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 80.2%

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

   if 9.9999999999999998e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 45.6%

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

   3. Taylor expanded in n around 0

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

   5. Applied rewrites 94.0%

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

Alternative 6: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{1}{n}\right), 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-17)
   (/ (exp (/ (log x) n)) (* x n))
   (if (<= (/ 1.0 n) 2e-11)
     (/ (log (/ (+ x 1.0) x)) n)
     (-
      (fma x (fma x (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 1.0 n)) 1.0)
      (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-17) {
		tmp = exp((log(x) / n)) / (x * n);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = fma(x, fma(x, ((0.5 / (n * n)) - (0.5 / n)), (1.0 / n)), 1.0) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-17)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(fma(x, fma(x, Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(1.0 / n)), 1.0) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-17], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(x * N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-17

   1. Initial program 96.4%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   if -4.9999999999999999e-17 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999988e-11

   1. Initial program 28.8%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 79.7%

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

   7. Applied rewrites 79.7%

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

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

   1. Initial program 46.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.2%

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

Alternative 7: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.86)
   (/ (- x (log x)) n)
   (if (<= x 6.2e+82)
     (/ (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) x)) n)
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.86) {
		tmp = (x - log(x)) / n;
	} else if (x <= 6.2e+82) {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.86d0) then
        tmp = (x - log(x)) / n
    else if (x <= 6.2d+82) then
        tmp = -((-(((0.3333333333333333d0 / x) - 0.5d0) / x) - 1.0d0) / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.86) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 6.2e+82) {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.86:
		tmp = (x - math.log(x)) / n
	elif x <= 6.2e+82:
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.86)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 6.2e+82)
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.86)
		tmp = (x - log(x)) / n;
	elseif (x <= 6.2e+82)
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.86], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 6.2e+82], N[((-N[(N[((-N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.859999999999999987

   1. Initial program 39.7%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.8%

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

   if 0.859999999999999987 < x < 6.20000000000000065e82

   1. Initial program 33.6%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 30.3%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 70.9%

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

   if 6.20000000000000065e82 < x

   1. Initial program 82.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 82.3%

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

Alternative 8: 60.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \frac{\frac{0.5}{n \cdot n} - \frac{0.5}{n}}{x}, \frac{1}{n}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7.2e-10)
   (/ (- (log x)) n)
   (if (<= x 5.7e+82)
     (/ (fma 1.0 (/ (- (/ 0.5 (* n n)) (/ 0.5 n)) x) (/ 1.0 n)) x)
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7.2e-10) {
		tmp = -log(x) / n;
	} else if (x <= 5.7e+82) {
		tmp = fma(1.0, (((0.5 / (n * n)) - (0.5 / n)) / x), (1.0 / n)) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7.2e-10)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 5.7e+82)
		tmp = Float64(fma(1.0, Float64(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)) / x), Float64(1.0 / n)) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7.2e-10], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 5.7e+82], N[(N[(1.0 * N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.2e-10

   1. Initial program 39.2%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.9%

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

   if 7.2e-10 < x < 5.70000000000000016e82

   1. Initial program 36.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 68.9%

      \[\leadsto \frac{\mathsf{fma}\left(1, \frac{\frac{0.5}{n \cdot n} - \frac{0.5}{n}}{x}, \frac{e^{-\frac{\log x}{-n}}}{n}\right)}{x} \]

   9. Taylor expanded in n around inf

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

   11. Applied rewrites 67.1%

       \[\leadsto \frac{\mathsf{fma}\left(1, \frac{\frac{0.5}{n \cdot n} - \frac{0.5}{n}}{x}, \frac{1}{n}\right)}{x} \]

   if 5.70000000000000016e82 < x

   1. Initial program 82.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 82.3%

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

Alternative 9: 49.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x} - 0.5}{x}\right) - 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.2e+82)
   (/ (- (/ (- (- (/ (- (/ 0.3333333333333333 x) 0.5) x)) 1.0) x)) n)
   (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.2e+82) {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 6.2d+82) then
        tmp = -((-(((0.3333333333333333d0 / x) - 0.5d0) / x) - 1.0d0) / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6.2e+82) {
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6.2e+82:
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.2e+82)
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) - 1.0) / x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6.2e+82)
		tmp = -((-(((0.3333333333333333 / x) - 0.5) / x) - 1.0) / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.2e+82], N[((-N[(N[((-N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.20000000000000065e82

   1. Initial program 38.8%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 38.7%

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

   if 6.20000000000000065e82 < x

   1. Initial program 82.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 82.3%

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

Alternative 10: 49.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{-\frac{\left(-\frac{\frac{0.3333333333333333}{x}}{x}\right) - 1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.2e+82)
   (/ (- (/ (- (- (/ (/ 0.3333333333333333 x) x)) 1.0) x)) n)
   (- 1.0 1.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.2e+82) {
		tmp = -((-((0.3333333333333333 / x) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 6.2d+82) then
        tmp = -((-((0.3333333333333333d0 / x) / x) - 1.0d0) / x) / n
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6.2e+82) {
		tmp = -((-((0.3333333333333333 / x) / x) - 1.0) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6.2e+82:
		tmp = -((-((0.3333333333333333 / x) / x) - 1.0) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.2e+82)
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(0.3333333333333333 / x) / x)) - 1.0) / x)) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6.2e+82)
		tmp = -((-((0.3333333333333333 / x) / x) - 1.0) / x) / n;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.2e+82], N[((-N[(N[((-N[(N[(0.3333333333333333 / x), $MachinePrecision] / x), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.20000000000000065e82

   1. Initial program 38.8%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 38.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 38.7%

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

   if 6.20000000000000065e82 < x

   1. Initial program 82.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 82.3%

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

Alternative 11: 46.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -9 \cdot 10^{-197}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -5.0)
   (/ (/ 1.0 n) x)
   (if (<= n -9e-197) (- 1.0 1.0) (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -5.0) {
		tmp = (1.0 / n) / x;
	} else if (n <= -9e-197) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.0d0)) then
        tmp = (1.0d0 / n) / x
    else if (n <= (-9d-197)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -5.0) {
		tmp = (1.0 / n) / x;
	} else if (n <= -9e-197) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -5.0:
		tmp = (1.0 / n) / x
	elif n <= -9e-197:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -5.0)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (n <= -9e-197)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -5.0)
		tmp = (1.0 / n) / x;
	elseif (n <= -9e-197)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -5.0], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -9e-197], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5

   1. Initial program 26.5%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 43.7%

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

   10. Applied rewrites 44.9%

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

   if -5 < n < -9.0000000000000002e-197

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 54.3%

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

   if -9.0000000000000002e-197 < n

   1. Initial program 46.6%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.6%

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

Alternative 12: 46.2% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -9 \cdot 10^{-197}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -5.0)
   (/ (/ 1.0 n) x)
   (if (<= n -9e-197) (- 1.0 1.0) (/ 1.0 (* x n)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -5.0) {
		tmp = (1.0 / n) / x;
	} else if (n <= -9e-197) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.0d0)) then
        tmp = (1.0d0 / n) / x
    else if (n <= (-9d-197)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 1.0d0 / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -5.0) {
		tmp = (1.0 / n) / x;
	} else if (n <= -9e-197) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -5.0:
		tmp = (1.0 / n) / x
	elif n <= -9e-197:
		tmp = 1.0 - 1.0
	else:
		tmp = 1.0 / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -5.0)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (n <= -9e-197)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(1.0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -5.0)
		tmp = (1.0 / n) / x;
	elseif (n <= -9e-197)
		tmp = 1.0 - 1.0;
	else
		tmp = 1.0 / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -5.0], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -9e-197], N[(1.0 - 1.0), $MachinePrecision], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5

   1. Initial program 26.5%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 43.7%

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

   10. Applied rewrites 44.9%

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

   if -5 < n < -9.0000000000000002e-197

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 54.3%

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

   if -9.0000000000000002e-197 < n

   1. Initial program 46.6%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 47.5%

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

Alternative 13: 45.9% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot n}\\ \mathbf{if}\;n \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -9 \cdot 10^{-197}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x n))))
   (if (<= n -5.0) t_0 (if (<= n -9e-197) (- 1.0 1.0) t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (x * n);
	double tmp;
	if (n <= -5.0) {
		tmp = t_0;
	} else if (n <= -9e-197) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 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) :: tmp
    t_0 = 1.0d0 / (x * n)
    if (n <= (-5.0d0)) then
        tmp = t_0
    else if (n <= (-9d-197)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (x * n);
	double tmp;
	if (n <= -5.0) {
		tmp = t_0;
	} else if (n <= -9e-197) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (x * n)
	tmp = 0
	if n <= -5.0:
		tmp = t_0
	elif n <= -9e-197:
		tmp = 1.0 - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(x * n))
	tmp = 0.0
	if (n <= -5.0)
		tmp = t_0;
	elseif (n <= -9e-197)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (x * n);
	tmp = 0.0;
	if (n <= -5.0)
		tmp = t_0;
	elseif (n <= -9e-197)
		tmp = 1.0 - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.0], t$95$0, If[LessEqual[n, -9e-197], N[(1.0 - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -5 or -9.0000000000000002e-197 < n

   1. Initial program 38.4%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.9%

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

   if -5 < n < -9.0000000000000002e-197

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 54.3%

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

Alternative 14: 31.6% accurate, 57.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 37.8%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 30.7%

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

Reproduce

herbie shell --seed 2025036 -o generate:simplify -o generate:proofs
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))