2nthrt (problem 3.4.6)

Percentage Accurate: 53.5% → 85.8%

Time: 17.0s

Alternatives: 14

Speedup: 1.9×

• could not determine a ground truth

  1. x = 1.6202709024647961e+239
  2. n = 3.6204846158308346e-246

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: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;-\frac{\mathsf{fma}\left(-1, \log \left(1 + x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-84)
   (/ (exp (- (/ (- (log x)) n))) (* n x))
   (if (<= (/ 1.0 n) 0.02)
     (-
      (/
       (fma
        -1.0
        (+
         (log (+ 1.0 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) <= -2e-84) {
		tmp = exp(-(-log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 0.02) {
		tmp = -(fma(-1.0, (log((1.0 + 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) <= -2e-84)
		tmp = Float64(exp(Float64(-Float64(Float64(-log(x)) / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(-Float64(fma(-1.0, Float64(log(Float64(1.0 + 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], -2e-84], N[(N[Exp[(-N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision])], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], (-N[(N[(-1.0 * N[(N[Log[N[(1.0 + x), $MachinePrecision]], $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) < -2.0000000000000001e-84

   1. Initial program 82.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

   if -2.0000000000000001e-84 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 30.9%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 79.2%

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

      1. lift-log1p.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 79.2

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

   6. Applied rewrites 79.2%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 55.7%

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

   2. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-log1p.f64 99.2

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

   4. Applied rewrites 99.2%

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

Alternative 2: 78.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ x 1.0) (/ 1.0 n)) t_0)))
   (if (<= t_1 (- INFINITY))
     (- 1.0 t_0)
     (if (<= t_1 0.2)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ (log1p x) n)) 1.0)))))

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 <= 0.2) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.2) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 1.0 - t_0
	elif t_1 <= 0.2:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - 1.0
	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 <= 0.2)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, 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, 0.2], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (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. Taylor expanded in x around 0

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

   4. Applied rewrites 100.0%

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

   if -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))) < 0.20000000000000001

   1. Initial program 43.6%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 79.2

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

   4. Applied rewrites 79.2%

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

   if 0.20000000000000001 < (-.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 56.1%

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

   2. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-log1p.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in n around inf

      \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.2%

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

Alternative 3: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;-\frac{-1 \cdot \frac{0.5 \cdot \frac{1 + \left(-\log x\right)}{n} - 0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

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 0.2)
       (/ (- (log1p x) (log x)) n)
       (-
        (/
         (* -1.0 (/ (- (* 0.5 (/ (+ 1.0 (- (log x))) n)) 0.5) (* x x)))
         n))))))

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 <= 0.2) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = -((-1.0 * (((0.5 * ((1.0 + -log(x)) / n)) - 0.5) / (x * x))) / n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 - t_0;
	} else if (t_1 <= 0.2) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = -((-1.0 * (((0.5 * ((1.0 + -Math.log(x)) / n)) - 0.5) / (x * x))) / n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.pow((x + 1.0), (1.0 / n)) - t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 1.0 - t_0
	elif t_1 <= 0.2:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = -((-1.0 * (((0.5 * ((1.0 + -math.log(x)) / n)) - 0.5) / (x * x))) / n)
	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 <= 0.2)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(-Float64(Float64(-1.0 * Float64(Float64(Float64(0.5 * Float64(Float64(1.0 + Float64(-log(x))) / n)) - 0.5) / Float64(x * x))) / n));
	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, 0.2], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], (-N[(N[(-1.0 * N[(N[(N[(0.5 * N[(N[(1.0 + (-N[Log[x], $MachinePrecision])), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $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. Taylor expanded in x around 0

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

   4. Applied rewrites 100.0%

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

   if -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))) < 0.20000000000000001

   1. Initial program 43.6%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 79.2

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

   4. Applied rewrites 79.2%

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

   if 0.20000000000000001 < (-.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 56.1%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 0.7%

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

   5. Taylor expanded in x around inf

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto -\frac{0 \cdot \log \left(\frac{1}{x}\right) + \left(-1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + -1 \cdot \frac{\frac{1}{2} \cdot \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}\right) - \frac{1}{2}}{{x}^{2}}\right)}{n} \]

      4. log-pow-rev N/A

         \[\leadsto -\frac{\log \left({\left(\frac{1}{x}\right)}^{0}\right) + \left(-1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + -1 \cdot \frac{\frac{1}{2} \cdot \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}\right) - \frac{1}{2}}{{x}^{2}}\right)}{n} \]

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\frac{0 + \left(-1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + -1 \cdot \frac{\frac{1}{2} \cdot \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}\right) - \frac{1}{2}}{{x}^{2}}\right)}{n} \]

      7. lower-+.f64 N/A

         \[\leadsto -\frac{0 + \left(-1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + -1 \cdot \frac{\frac{1}{2} \cdot \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}\right) - \frac{1}{2}}{{x}^{2}}\right)}{n} \]

   7. Applied rewrites 17.8%

      \[\leadsto -\frac{0 + \mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(\log x, -1, 1\right)}{n} \cdot 0.5 - 0.5}{x \cdot x}, -1, -\frac{\mathsf{fma}\left(\frac{-\log x}{n}, -1, 1\right)}{x}\right)}{n} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.5%

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

Alternative 4: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;-\frac{\mathsf{fma}\left(-1, \mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n}, \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-84)
   (/ (exp (- (/ (- (log x)) n))) (* n x))
   (if (<= (/ 1.0 n) 0.02)
     (-
      (/
       (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) <= -2e-84) {
		tmp = exp(-(-log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 0.02) {
		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) <= -2e-84)
		tmp = Float64(exp(Float64(-Float64(Float64(-log(x)) / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.02)
		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], -2e-84], N[(N[Exp[(-N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision])], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], (-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) < -2.0000000000000001e-84

   1. Initial program 82.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

   if -2.0000000000000001e-84 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 30.9%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 79.2%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 55.7%

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

   2. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-log1p.f64 99.2

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

   4. Applied rewrites 99.2%

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

Alternative 5: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-84)
   (/ (exp (- (/ (- (log x)) n))) (* n x))
   (if (<= (/ 1.0 n) 0.02)
     (/ (- (log1p x) (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) <= -2e-84) {
		tmp = exp(-(-log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 0.02) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = Math.exp(-(-Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 0.02) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-84:
		tmp = math.exp(-(-math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 0.02:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-84)
		tmp = Float64(exp(Float64(-Float64(Float64(-log(x)) / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 0.02)
		tmp = Float64(Float64(log1p(x) - 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], -2e-84], N[(N[Exp[(-N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision])], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.02], N[(N[(N[Log[1 + x], $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) < -2.0000000000000001e-84

   1. Initial program 82.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

   if -2.0000000000000001e-84 < (/.f64 #s(literal 1 binary64) n) < 0.0200000000000000004

   1. Initial program 30.9%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 78.9

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

   4. Applied rewrites 78.9%

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

   if 0.0200000000000000004 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 55.7%

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

   2. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-log1p.f64 99.2

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

   4. Applied rewrites 99.2%

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

Alternative 6: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\frac{e^{-\frac{-\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 1:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-84)
   (/ (exp (- (/ (- (log x)) n))) (* n x))
   (if (<= (/ 1.0 n) 1.0)
     (/ (- (log1p x) (log x)) n)
     (- (exp (/ x n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = exp(-(-log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1.0) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-84) {
		tmp = Math.exp(-(-Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1.0) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-84:
		tmp = math.exp(-(-math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 1.0:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-84)
		tmp = Float64(exp(Float64(-Float64(Float64(-log(x)) / n))) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1.0)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-84], N[(N[Exp[(-N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision])], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1.0], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / 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) < -2.0000000000000001e-84

   1. Initial program 82.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

   if -2.0000000000000001e-84 < (/.f64 #s(literal 1 binary64) n) < 1

   1. Initial program 31.1%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 78.7

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

   4. Applied rewrites 78.7%

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

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

   1. Initial program 55.5%

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

   2. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-log1p.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.4%

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

Alternative 7: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{if}\;n \leq -7.5 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -280000000:\\ \;\;\;\;\frac{1 + -1 \cdot \frac{-\log x}{n}}{n \cdot x}\\ \mathbf{elif}\;n \leq 1.1:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log1p x) (log x)) n)))
   (if (<= n -7.5e+84)
     t_0
     (if (<= n -280000000.0)
       (/ (+ 1.0 (* -1.0 (/ (- (log x)) n))) (* n x))
       (if (<= n 1.1) (- (exp (/ x n)) (pow x (/ 1.0 n))) t_0)))))

C

double code(double x, double n) {
	double t_0 = (log1p(x) - log(x)) / n;
	double tmp;
	if (n <= -7.5e+84) {
		tmp = t_0;
	} else if (n <= -280000000.0) {
		tmp = (1.0 + (-1.0 * (-log(x) / n))) / (n * x);
	} else if (n <= 1.1) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (Math.log1p(x) - Math.log(x)) / n;
	double tmp;
	if (n <= -7.5e+84) {
		tmp = t_0;
	} else if (n <= -280000000.0) {
		tmp = (1.0 + (-1.0 * (-Math.log(x) / n))) / (n * x);
	} else if (n <= 1.1) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (math.log1p(x) - math.log(x)) / n
	tmp = 0
	if n <= -7.5e+84:
		tmp = t_0
	elif n <= -280000000.0:
		tmp = (1.0 + (-1.0 * (-math.log(x) / n))) / (n * x)
	elif n <= 1.1:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(log1p(x) - log(x)) / n)
	tmp = 0.0
	if (n <= -7.5e+84)
		tmp = t_0;
	elseif (n <= -280000000.0)
		tmp = Float64(Float64(1.0 + Float64(-1.0 * Float64(Float64(-log(x)) / n))) / Float64(n * x));
	elseif (n <= 1.1)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -7.5e+84], t$95$0, If[LessEqual[n, -280000000.0], N[(N[(1.0 + N[(-1.0 * N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.1], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.5000000000000001e84 or 1.1000000000000001 < n

   1. Initial program 31.1%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-log.f64 78.8

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

   4. Applied rewrites 78.8%

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

   if -7.5000000000000001e84 < n < -2.8e8

   1. Initial program 12.8%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 59.1%

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

   6. Applied rewrites 59.0%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. neg-log N/A

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

      6. lift-log.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-*.f64 50.3

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

   9. Applied rewrites 50.3%

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

   if -2.8e8 < n < 1.1000000000000001

   1. Initial program 84.7%

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

   2. Taylor expanded in n around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-log1p.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.5%

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

Alternative 8: 57.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\frac{x}{n} \cdot \frac{x}{n} - 1}{\frac{x}{n} - 1} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{1 - 0.5 \cdot {x}^{-1}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (- (/ (- (* (/ x n) (/ x n)) 1.0) (- (/ x n) 1.0)) (pow x (/ 1.0 n)))
   (if (<= x 4.2e+143) (/ (/ (- 1.0 (* 0.5 (pow x -1.0))) n) x) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - pow(x, (1.0 / n));
	} else if (x <= 4.2e+143) {
		tmp = ((1.0 - (0.5 * pow(x, -1.0))) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = ((((x / n) * (x / n)) - 1.0d0) / ((x / n) - 1.0d0)) - (x ** (1.0d0 / n))
    else if (x <= 4.2d+143) then
        tmp = ((1.0d0 - (0.5d0 * (x ** (-1.0d0)))) / n) / x
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - Math.pow(x, (1.0 / n));
	} else if (x <= 4.2e+143) {
		tmp = ((1.0 - (0.5 * Math.pow(x, -1.0))) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - math.pow(x, (1.0 / n))
	elif x <= 4.2e+143:
		tmp = ((1.0 - (0.5 * math.pow(x, -1.0))) / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(Float64(Float64(Float64(x / n) * Float64(x / n)) - 1.0) / Float64(Float64(x / n) - 1.0)) - (x ^ Float64(1.0 / n)));
	elseif (x <= 4.2e+143)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 * (x ^ -1.0))) / n) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = ((((x / n) * (x / n)) - 1.0) / ((x / n) - 1.0)) - (x ^ (1.0 / n));
	elseif (x <= 4.2e+143)
		tmp = ((1.0 - (0.5 * (x ^ -1.0))) / n) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(N[(N[(N[(x / n), $MachinePrecision] * N[(x / n), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[(x / n), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+143], N[(N[(N[(1.0 - N[(0.5 * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 43.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 42.6

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

   4. Applied rewrites 42.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-/.f64 45.4

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

   6. Applied rewrites 45.4%

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

   if 1 < x < 4.19999999999999975e143

   1. Initial program 50.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 64.2

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

   7. Applied rewrites 64.2%

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

   if 4.19999999999999975e143 < x

   1. Initial program 82.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 82.1%

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

Alternative 9: 55.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{1 - 0.5 \cdot {x}^{-1}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
   (if (<= x 4.2e+143) (/ (/ (- 1.0 (* 0.5 (pow x -1.0))) n) x) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else if (x <= 4.2e+143) {
		tmp = ((1.0 - (0.5 * pow(x, -1.0))) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else if (x <= 4.2d+143) then
        tmp = ((1.0d0 - (0.5d0 * (x ** (-1.0d0)))) / n) / x
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else if (x <= 4.2e+143) {
		tmp = ((1.0 - (0.5 * Math.pow(x, -1.0))) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	elif x <= 4.2e+143:
		tmp = ((1.0 - (0.5 * math.pow(x, -1.0))) / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	elseif (x <= 4.2e+143)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 * (x ^ -1.0))) / n) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	elseif (x <= 4.2e+143)
		tmp = ((1.0 - (0.5 * (x ^ -1.0))) / n) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+143], N[(N[(N[(1.0 - N[(0.5 * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 43.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 42.6

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

   4. Applied rewrites 42.6%

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

   if 1 < x < 4.19999999999999975e143

   1. Initial program 50.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 64.2

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

   7. Applied rewrites 64.2%

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

   if 4.19999999999999975e143 < x

   1. Initial program 82.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 82.1%

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

Alternative 10: 55.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{1 - 0.5 \cdot {x}^{-1}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (- (+ (/ x n) 1.0) (pow x (/ 1.0 n)))
   (if (<= x 4.2e+143) (/ (- 1.0 (* 0.5 (pow x -1.0))) (* n x)) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = ((x / n) + 1.0) - pow(x, (1.0 / n));
	} else if (x <= 4.2e+143) {
		tmp = (1.0 - (0.5 * pow(x, -1.0))) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = ((x / n) + 1.0d0) - (x ** (1.0d0 / n))
    else if (x <= 4.2d+143) then
        tmp = (1.0d0 - (0.5d0 * (x ** (-1.0d0)))) / (n * x)
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = ((x / n) + 1.0) - Math.pow(x, (1.0 / n));
	} else if (x <= 4.2e+143) {
		tmp = (1.0 - (0.5 * Math.pow(x, -1.0))) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = ((x / n) + 1.0) - math.pow(x, (1.0 / n))
	elif x <= 4.2e+143:
		tmp = (1.0 - (0.5 * math.pow(x, -1.0))) / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - (x ^ Float64(1.0 / n)));
	elseif (x <= 4.2e+143)
		tmp = Float64(Float64(1.0 - Float64(0.5 * (x ^ -1.0))) / Float64(n * x));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = ((x / n) + 1.0) - (x ^ (1.0 / n));
	elseif (x <= 4.2e+143)
		tmp = (1.0 - (0.5 * (x ^ -1.0))) / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+143], N[(N[(1.0 - N[(0.5 * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 43.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 42.6

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

   4. Applied rewrites 42.6%

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

   if 1 < x < 4.19999999999999975e143

   1. Initial program 50.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

   if 4.19999999999999975e143 < x

   1. Initial program 82.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 82.1%

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

Alternative 11: 55.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{1 - 0.5 \cdot {x}^{-1}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 4.2e+143) (/ (- 1.0 (* 0.5 (pow x -1.0))) (* n x)) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 4.2e+143) {
		tmp = (1.0 - (0.5 * pow(x, -1.0))) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 4.2d+143) then
        tmp = (1.0d0 - (0.5d0 * (x ** (-1.0d0)))) / (n * x)
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 4.2e+143) {
		tmp = (1.0 - (0.5 * Math.pow(x, -1.0))) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 4.2e+143:
		tmp = (1.0 - (0.5 * math.pow(x, -1.0))) / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 4.2e+143)
		tmp = Float64(Float64(1.0 - Float64(0.5 * (x ^ -1.0))) / Float64(n * x));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 4.2e+143)
		tmp = (1.0 - (0.5 * (x ^ -1.0))) / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+143], N[(N[(1.0 - N[(0.5 * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 43.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 42.2%

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

   if 1 < x < 4.19999999999999975e143

   1. Initial program 50.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

   if 4.19999999999999975e143 < x

   1. Initial program 82.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 82.1%

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

Alternative 12: 55.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00078:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+143}:\\ \;\;\;\;-\frac{\frac{-1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.00078)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 4.2e+143) (- (/ (/ -1.0 x) n)) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.00078) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 4.2e+143) {
		tmp = -((-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 <= 0.00078d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 4.2d+143) then
        tmp = -(((-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 <= 0.00078) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 4.2e+143) {
		tmp = -((-1.0 / x) / n);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.00078:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 4.2e+143:
		tmp = -((-1.0 / x) / n)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.00078)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 4.2e+143)
		tmp = Float64(-Float64(Float64(-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 <= 0.00078)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 4.2e+143)
		tmp = -((-1.0 / x) / n);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.00078], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+143], (-N[(N[(-1.0 / x), $MachinePrecision] / n), $MachinePrecision]), N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.79999999999999986e-4

   1. Initial program 43.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 42.4%

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

   if 7.79999999999999986e-4 < x < 4.19999999999999975e143

   1. Initial program 50.5%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in x around inf

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto -\frac{0 \cdot \log \left(\frac{1}{x}\right) + -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}}{n} \]

      4. log-pow-rev N/A

         \[\leadsto -\frac{\log \left({\left(\frac{1}{x}\right)}^{0}\right) + -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}}{n} \]

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\frac{0 + -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}}{n} \]

      7. lower-+.f64 N/A

         \[\leadsto -\frac{0 + -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}}{n} \]

      8. mul-1-neg N/A

         \[\leadsto -\frac{0 + \left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}\right)\right)}{n} \]

      9. lower-neg.f64 N/A

         \[\leadsto -\frac{0 + \left(-\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}\right)}{n} \]

      10. lower-/.f64 N/A

          \[\leadsto -\frac{0 + \left(-\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}\right)}{n} \]

   7. Applied rewrites 61.9%

      \[\leadsto -\frac{0 + \left(-\frac{\mathsf{fma}\left(\frac{-\log x}{n}, -1, 1\right)}{x}\right)}{n} \]

   8. Taylor expanded in n around inf

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

      1. lower-/.f64 61.6

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

   10. Applied rewrites 61.6%

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

   if 4.19999999999999975e143 < x

   1. Initial program 82.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 82.1%

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

Alternative 13: 45.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{\frac{-1}{x}}{n}\\ \mathbf{if}\;n \leq -0.99:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -9 \cdot 10^{-226}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (/ (/ -1.0 x) n))))
   (if (<= n -0.99) t_0 (if (<= n -9e-226) (- 1.0 1.0) t_0))))

C

double code(double x, double n) {
	double t_0 = -((-1.0 / x) / n);
	double tmp;
	if (n <= -0.99) {
		tmp = t_0;
	} else if (n <= -9e-226) {
		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 <= (-0.99d0)) then
        tmp = t_0
    else if (n <= (-9d-226)) 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 <= -0.99) {
		tmp = t_0;
	} else if (n <= -9e-226) {
		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 <= -0.99:
		tmp = t_0
	elif n <= -9e-226:
		tmp = 1.0 - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-Float64(Float64(-1.0 / x) / n))
	tmp = 0.0
	if (n <= -0.99)
		tmp = t_0;
	elseif (n <= -9e-226)
		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 <= -0.99)
		tmp = t_0;
	elseif (n <= -9e-226)
		tmp = 1.0 - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = (-N[(N[(-1.0 / x), $MachinePrecision] / n), $MachinePrecision])}, If[LessEqual[n, -0.99], t$95$0, If[LessEqual[n, -9e-226], N[(1.0 - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -0.98999999999999999 or -9.00000000000000023e-226 < n

   1. Initial program 40.7%

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

   2. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 64.3%

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

   5. Taylor expanded in x around inf

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto -\frac{0 \cdot \log \left(\frac{1}{x}\right) + -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}}{n} \]

      4. log-pow-rev N/A

         \[\leadsto -\frac{\log \left({\left(\frac{1}{x}\right)}^{0}\right) + -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}}{n} \]

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\frac{0 + -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}}{n} \]

      7. lower-+.f64 N/A

         \[\leadsto -\frac{0 + -1 \cdot \frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}}{n} \]

      8. mul-1-neg N/A

         \[\leadsto -\frac{0 + \left(\mathsf{neg}\left(\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}\right)\right)}{n} \]

      9. lower-neg.f64 N/A

         \[\leadsto -\frac{0 + \left(-\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}\right)}{n} \]

      10. lower-/.f64 N/A

          \[\leadsto -\frac{0 + \left(-\frac{1 + -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}{x}\right)}{n} \]

   7. Applied rewrites 40.8%

      \[\leadsto -\frac{0 + \left(-\frac{\mathsf{fma}\left(\frac{-\log x}{n}, -1, 1\right)}{x}\right)}{n} \]

   8. Taylor expanded in n around inf

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

      1. lower-/.f64 44.7

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

   10. Applied rewrites 44.7%

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

   if -0.98999999999999999 < n < -9.00000000000000023e-226

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 50.4%

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

Alternative 14: 30.6% accurate, 57.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 38.6%

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

5. Taylor expanded in n around inf

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

7. Applied rewrites 30.6%

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

Reproduce

herbie shell --seed 2025095 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))