2nthrt (problem 3.4.6)

Percentage Accurate: 54.1% → 90.5%

Time: 20.1s

Alternatives: 17

Speedup: 2.9×

• Could not converge on a ground truth

  1. x = -2.666172354659113e-6
  2. n = 6.125532548748677e-272

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 6.2 \cdot 10^{-26}:\\ \;\;\;\;-\mathsf{expm1}\left(t\_0\right)\\ \mathbf{elif}\;x \leq 1500:\\ \;\;\;\;\left(n \cdot \log \left(\frac{1 + x}{x}\right)\right) \cdot \frac{1}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t\_0}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= x 6.2e-26)
     (- (expm1 t_0))
     (if (<= x 1500.0)
       (* (* n (log (/ (+ 1.0 x) x))) (/ 1.0 (* n n)))
       (/ (exp t_0) (* n x))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if (x <= 6.2e-26) {
		tmp = -expm1(t_0);
	} else if (x <= 1500.0) {
		tmp = (n * log(((1.0 + x) / x))) * (1.0 / (n * n));
	} else {
		tmp = exp(t_0) / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if (x <= 6.2e-26) {
		tmp = -Math.expm1(t_0);
	} else if (x <= 1500.0) {
		tmp = (n * Math.log(((1.0 + x) / x))) * (1.0 / (n * n));
	} else {
		tmp = Math.exp(t_0) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if x <= 6.2e-26:
		tmp = -math.expm1(t_0)
	elif x <= 1500.0:
		tmp = (n * math.log(((1.0 + x) / x))) * (1.0 / (n * n))
	else:
		tmp = math.exp(t_0) / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 6.2e-26)
		tmp = Float64(-expm1(t_0));
	elseif (x <= 1500.0)
		tmp = Float64(Float64(n * log(Float64(Float64(1.0 + x) / x))) * Float64(1.0 / Float64(n * n)));
	else
		tmp = Float64(exp(t_0) / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 6.2e-26], (-N[(Exp[t$95$0] - 1), $MachinePrecision]), If[LessEqual[x, 1500.0], N[(N[(n * N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[t$95$0], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.19999999999999966e-26

   1. Initial program 44.3%

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

   2. Taylor expanded in x around 0

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

      1. sub-negate N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      5. lower-log.f64 88.3

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

   4. Applied rewrites 88.3%

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 6.19999999999999966e-26 < x < 1500

   1. Initial program 40.6%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in n around inf

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 49.7

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

   10. Applied rewrites 49.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. pow2 N/A

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

       4. mult-flip N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 50.6

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

   12. Applied rewrites 50.6%

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

   if 1500 < x

   1. Initial program 67.8%

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

   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 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in x around 0

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 97.8

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

   7. Applied rewrites 97.8%

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

Alternative 2: 80.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-6}:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;-\frac{\frac{n \cdot \log \left(\frac{x}{1 + x}\right)}{n}}{n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{n \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.45e-6)
   (- (expm1 (/ (log x) n)))
   (if (<= x 1.4e+77)
     (- (/ (/ (* n (log (/ x (+ 1.0 x)))) n) n))
     (if (<= x 5.8e+194)
       (/ (+ (/ 1.0 x) (/ (log x) (* n x))) n)
       (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.45e-6) {
		tmp = -expm1((log(x) / n));
	} else if (x <= 1.4e+77) {
		tmp = -(((n * log((x / (1.0 + x)))) / n) / n);
	} else if (x <= 5.8e+194) {
		tmp = ((1.0 / x) + (log(x) / (n * x))) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.45e-6) {
		tmp = -Math.expm1((Math.log(x) / n));
	} else if (x <= 1.4e+77) {
		tmp = -(((n * Math.log((x / (1.0 + x)))) / n) / n);
	} else if (x <= 5.8e+194) {
		tmp = ((1.0 / x) + (Math.log(x) / (n * x))) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.45e-6:
		tmp = -math.expm1((math.log(x) / n))
	elif x <= 1.4e+77:
		tmp = -(((n * math.log((x / (1.0 + x)))) / n) / n)
	elif x <= 5.8e+194:
		tmp = ((1.0 / x) + (math.log(x) / (n * x))) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.45e-6)
		tmp = Float64(-expm1(Float64(log(x) / n)));
	elseif (x <= 1.4e+77)
		tmp = Float64(-Float64(Float64(Float64(n * log(Float64(x / Float64(1.0 + x)))) / n) / n));
	elseif (x <= 5.8e+194)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(log(x) / Float64(n * x))) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.45e-6], (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), If[LessEqual[x, 1.4e+77], (-N[(N[(N[(n * N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[x, 5.8e+194], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.44999999999999984e-6

   1. Initial program 43.9%

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

   2. Taylor expanded in x around 0

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

      1. sub-negate N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      5. lower-log.f64 87.2

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

   4. Applied rewrites 87.2%

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 2.44999999999999984e-6 < x < 1.4e77

   1. Initial program 45.3%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in n around inf

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 47.1

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

   10. Applied rewrites 47.1%

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

   if 1.4e77 < x < 5.8000000000000001e194

   1. Initial program 63.1%

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

   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 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if 5.8000000000000001e194 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.4%

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

Alternative 3: 80.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-6}:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{n \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.45e-6)
   (- (expm1 (/ (log x) n)))
   (if (<= x 1.4e+77)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= x 5.8e+194)
       (/ (+ (/ 1.0 x) (/ (log x) (* n x))) n)
       (- 1.0 1.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.45e-6) {
		tmp = -expm1((log(x) / n));
	} else if (x <= 1.4e+77) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (x <= 5.8e+194) {
		tmp = ((1.0 / x) + (log(x) / (n * x))) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.45e-6) {
		tmp = -Math.expm1((Math.log(x) / n));
	} else if (x <= 1.4e+77) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (x <= 5.8e+194) {
		tmp = ((1.0 / x) + (Math.log(x) / (n * x))) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.45e-6:
		tmp = -math.expm1((math.log(x) / n))
	elif x <= 1.4e+77:
		tmp = math.log(((1.0 + x) / x)) / n
	elif x <= 5.8e+194:
		tmp = ((1.0 / x) + (math.log(x) / (n * x))) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.45e-6)
		tmp = Float64(-expm1(Float64(log(x) / n)));
	elseif (x <= 1.4e+77)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (x <= 5.8e+194)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(log(x) / Float64(n * x))) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.45e-6], (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), If[LessEqual[x, 1.4e+77], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.8e+194], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.44999999999999984e-6

   1. Initial program 43.9%

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

   2. Taylor expanded in x around 0

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

      1. sub-negate N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      5. lower-log.f64 87.2

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

   4. Applied rewrites 87.2%

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 2.44999999999999984e-6 < x < 1.4e77

   1. Initial program 45.3%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 46.9

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

   4. Applied rewrites 46.9%

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

   if 1.4e77 < x < 5.8000000000000001e194

   1. Initial program 63.1%

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

   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 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if 5.8000000000000001e194 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.4%

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

Alternative 4: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 2.45 \cdot 10^{-6}:\\ \;\;\;\;-\mathsf{expm1}\left(t\_0\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{1 + t\_0}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= x 2.45e-6)
     (- (expm1 t_0))
     (if (<= x 1.4e+77)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= x 5.8e+194) (/ (+ 1.0 t_0) (* n x)) (- 1.0 1.0))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if (x <= 2.45e-6) {
		tmp = -expm1(t_0);
	} else if (x <= 1.4e+77) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if (x <= 5.8e+194) {
		tmp = (1.0 + t_0) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if (x <= 2.45e-6) {
		tmp = -Math.expm1(t_0);
	} else if (x <= 1.4e+77) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if (x <= 5.8e+194) {
		tmp = (1.0 + t_0) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if x <= 2.45e-6:
		tmp = -math.expm1(t_0)
	elif x <= 1.4e+77:
		tmp = math.log(((1.0 + x) / x)) / n
	elif x <= 5.8e+194:
		tmp = (1.0 + t_0) / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 2.45e-6)
		tmp = Float64(-expm1(t_0));
	elseif (x <= 1.4e+77)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (x <= 5.8e+194)
		tmp = Float64(Float64(1.0 + t_0) / Float64(n * x));
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 2.45e-6], (-N[(Exp[t$95$0] - 1), $MachinePrecision]), If[LessEqual[x, 1.4e+77], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.8e+194], N[(N[(1.0 + t$95$0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.44999999999999984e-6

   1. Initial program 43.9%

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

   2. Taylor expanded in x around 0

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

      1. sub-negate N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      5. lower-log.f64 87.2

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

   4. Applied rewrites 87.2%

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 2.44999999999999984e-6 < x < 1.4e77

   1. Initial program 45.3%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 46.9

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

   4. Applied rewrites 46.9%

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

   if 1.4e77 < x < 5.8000000000000001e194

   1. Initial program 63.1%

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

   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 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in n around inf

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 65.4

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

   7. Applied rewrites 65.4%

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

   if 5.8000000000000001e194 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.4%

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

Alternative 5: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{\left(1 + \frac{0.3333333333333333}{x \cdot x}\right) - 0.5 \cdot \frac{1}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.35)
   (- (expm1 (/ (log x) n)))
   (if (<= x 5.8e+194)
     (/ (/ (- (+ 1.0 (/ 0.3333333333333333 (* x x))) (* 0.5 (/ 1.0 x))) x) n)
     (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.35) {
		tmp = -expm1((log(x) / n));
	} else if (x <= 5.8e+194) {
		tmp = (((1.0 + (0.3333333333333333 / (x * x))) - (0.5 * (1.0 / x))) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.35) {
		tmp = -Math.expm1((Math.log(x) / n));
	} else if (x <= 5.8e+194) {
		tmp = (((1.0 + (0.3333333333333333 / (x * x))) - (0.5 * (1.0 / x))) / x) / n;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.35:
		tmp = -math.expm1((math.log(x) / n))
	elif x <= 5.8e+194:
		tmp = (((1.0 + (0.3333333333333333 / (x * x))) - (0.5 * (1.0 / x))) / x) / n
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.35)
		tmp = Float64(-expm1(Float64(log(x) / n)));
	elseif (x <= 5.8e+194)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(0.3333333333333333 / Float64(x * x))) - Float64(0.5 * Float64(1.0 / x))) / x) / n);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.35], (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), If[LessEqual[x, 5.8e+194], N[(N[(N[(N[(1.0 + N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.34999999999999998

   1. Initial program 44.1%

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

   2. Taylor expanded in x around 0

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

      1. sub-negate N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      5. lower-log.f64 86.6

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

   4. Applied rewrites 86.6%

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 0.34999999999999998 < x < 5.8000000000000001e194

   1. Initial program 55.4%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 56.0

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 64.7

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

   7. Applied rewrites 64.7%

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

   if 5.8000000000000001e194 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.4%

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

Alternative 6: 79.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;-\mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.92)
   (- (expm1 (/ (log x) n)))
   (if (<= x 5.8e+194) (/ (/ (- 1.0 (/ 0.5 x)) n) x) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.92) {
		tmp = -expm1((log(x) / n));
	} else if (x <= 5.8e+194) {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.92) {
		tmp = -Math.expm1((Math.log(x) / n));
	} else if (x <= 5.8e+194) {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.92:
		tmp = -math.expm1((math.log(x) / n))
	elif x <= 5.8e+194:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.92)
		tmp = Float64(-expm1(Float64(log(x) / n)));
	elseif (x <= 5.8e+194)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	else
		tmp = Float64(1.0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.92], (-N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), If[LessEqual[x, 5.8e+194], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.92000000000000004

   1. Initial program 44.0%

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

   2. Taylor expanded in x around 0

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

      1. sub-negate N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

      5. lower-log.f64 86.5

         \[\leadsto -\mathsf{expm1}\left(\frac{\log x}{n}\right) \]

   4. Applied rewrites 86.5%

      \[\leadsto \color{blue}{-\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 0.92000000000000004 < x < 5.8000000000000001e194

   1. Initial program 55.4%

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

   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 81.4%

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

   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. lower-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 64.5

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

   10. Applied rewrites 64.5%

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

   if 5.8000000000000001e194 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.4%

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

Alternative 7: 71.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (/ n x) (* n n))
     (if (<= t_0 0.0)
       (/ (log (/ (+ 1.0 x) x)) n)
       (/ (* n (- (log x))) (* n n))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (n / x) / (n * n);
	} else if (t_0 <= 0.0) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = (n * -log(x)) / (n * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (n / x) / (n * n);
	} else if (t_0 <= 0.0) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = (n * -Math.log(x)) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (n / x) / (n * n)
	elif t_0 <= 0.0:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = (n * -math.log(x)) / (n * n)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(n * (-N[Log[x], $MachinePrecision])), $MachinePrecision] / N[(n * n), $MachinePrecision]), $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 n around -inf

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 54.0%

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

   8. Taylor expanded in n around inf

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 63.4

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

   13. Applied rewrites 63.4%

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

   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))) < 0.0

   1. Initial program 44.0%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 79.8

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

   4. Applied rewrites 79.8%

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

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

   1. Initial program 56.1%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 4.2%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in n around inf

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 39.8

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

   10. Applied rewrites 39.8%

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

   11. Taylor expanded in x around 0

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

       1. log-pow-rev N/A

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

       2. inv-pow N/A

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

       3. neg-log N/A

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

       4. lift-neg.f64 N/A

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

       5. lift-log.f64 39.7

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

   13. Applied rewrites 39.7%

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

Alternative 8: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \frac{1}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))
   (if (<= t_0 (- INFINITY))
     (/ (/ n x) (* n n))
     (if (<= t_0 2e-6)
       (/ (log (/ (+ 1.0 x) x)) n)
       (/ (* n (/ 1.0 x)) (* n n))))))

C

double code(double x, double n) {
	double t_0 = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (n / x) / (n * n);
	} else if (t_0 <= 2e-6) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = (n * (1.0 / x)) / (n * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (n / x) / (n * n);
	} else if (t_0 <= 2e-6) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = (n * (1.0 / x)) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (n / x) / (n * n)
	elif t_0 <= 2e-6:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = (n * (1.0 / x)) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(n / x) / Float64(n * n));
	elseif (t_0 <= 2e-6)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(Float64(n * Float64(1.0 / x)) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (n / x) / (n * n);
	elseif (t_0 <= 2e-6)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = (n * (1.0 / x)) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

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

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 54.0%

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

   8. Taylor expanded in n around inf

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 49.8

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

   10. Applied rewrites 49.8%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 63.4

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

   13. Applied rewrites 63.4%

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

   if -inf.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 1.99999999999999991e-6

   1. Initial program 44.1%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 79.7

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

   4. Applied rewrites 79.7%

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

   if 1.99999999999999991e-6 < (-.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 56.2%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 1.5%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in n around inf

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 38.9

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

   10. Applied rewrites 38.9%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 37.1

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

   13. Applied rewrites 37.1%

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

Alternative 9: 60.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\frac{x + \left(-\log x\right)}{n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.96)
   (/ (+ x (- (log x))) n)
   (if (<= x 5.8e+194) (/ (/ (- 1.0 (/ 0.5 x)) n) x) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.96) {
		tmp = (x + -log(x)) / n;
	} else if (x <= 5.8e+194) {
		tmp = ((1.0 - (0.5 / x)) / 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 <= 0.96d0) then
        tmp = (x + -log(x)) / n
    else if (x <= 5.8d+194) then
        tmp = ((1.0d0 - (0.5d0 / x)) / 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 <= 0.96) {
		tmp = (x + -Math.log(x)) / n;
	} else if (x <= 5.8e+194) {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.96:
		tmp = (x + -math.log(x)) / n
	elif x <= 5.8e+194:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.96)
		tmp = Float64(Float64(x + Float64(-log(x))) / n);
	elseif (x <= 5.8e+194)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / 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 <= 0.96)
		tmp = (x + -log(x)) / n;
	elseif (x <= 5.8e+194)
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.96], N[(N[(x + (-N[Log[x], $MachinePrecision])), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.8e+194], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.95999999999999996

   1. Initial program 44.0%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 50.9

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

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. lower-+.f64 N/A

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

      4. neg-log N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-log.f64 50.4

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

   7. Applied rewrites 50.4%

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

   if 0.95999999999999996 < x < 5.8000000000000001e194

   1. Initial program 55.5%

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

   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 81.4%

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

   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. lower-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 64.5

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

   10. Applied rewrites 64.5%

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

   if 5.8000000000000001e194 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.4%

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

Alternative 10: 59.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.68)
   (/ (- (log x)) n)
   (if (<= x 5.8e+194) (/ (/ (- 1.0 (/ 0.5 x)) n) x) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = -log(x) / n;
	} else if (x <= 5.8e+194) {
		tmp = ((1.0 - (0.5 / x)) / 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 <= 0.68d0) then
        tmp = -log(x) / n
    else if (x <= 5.8d+194) then
        tmp = ((1.0d0 - (0.5d0 / x)) / 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 <= 0.68) {
		tmp = -Math.log(x) / n;
	} else if (x <= 5.8e+194) {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.68:
		tmp = -math.log(x) / n
	elif x <= 5.8e+194:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 5.8e+194)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / 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 <= 0.68)
		tmp = -log(x) / n;
	elseif (x <= 5.8e+194)
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.68], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 5.8e+194], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.680000000000000049

   1. Initial program 44.0%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 50.9

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

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 49.9

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

   7. Applied rewrites 49.9%

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

   if 0.680000000000000049 < x < 5.8000000000000001e194

   1. Initial program 55.4%

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

   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 81.3%

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

   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. lower-/.f64 64.4

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 64.4

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

   10. Applied rewrites 64.4%

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

   if 5.8000000000000001e194 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.4%

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

Alternative 11: 59.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{1 - \frac{0.5}{x}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.68)
   (/ (- (log x)) n)
   (if (<= x 5.8e+194) (/ (- 1.0 (/ 0.5 x)) (* n x)) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.68) {
		tmp = -log(x) / n;
	} else if (x <= 5.8e+194) {
		tmp = (1.0 - (0.5 / x)) / (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 <= 0.68d0) then
        tmp = -log(x) / n
    else if (x <= 5.8d+194) then
        tmp = (1.0d0 - (0.5d0 / x)) / (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 <= 0.68) {
		tmp = -Math.log(x) / n;
	} else if (x <= 5.8e+194) {
		tmp = (1.0 - (0.5 / x)) / (n * x);
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.68:
		tmp = -math.log(x) / n
	elif x <= 5.8e+194:
		tmp = (1.0 - (0.5 / x)) / (n * x)
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 5.8e+194)
		tmp = Float64(Float64(1.0 - Float64(0.5 / x)) / 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 <= 0.68)
		tmp = -log(x) / n;
	elseif (x <= 5.8e+194)
		tmp = (1.0 - (0.5 / x)) / (n * x);
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.68], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 5.8e+194], N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.680000000000000049

   1. Initial program 44.0%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 50.9

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

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 49.9

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

   7. Applied rewrites 49.9%

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

   if 0.680000000000000049 < x < 5.8000000000000001e194

   1. Initial program 55.4%

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

   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 81.3%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 66.6%

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

   8. Taylor expanded in n around inf

      \[\leadsto \frac{\mathsf{fma}\left(1, \frac{\frac{\frac{1}{2}}{n \cdot n} - \frac{\frac{1}{2}}{n}}{x}, \frac{1}{n}\right)}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.3%

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

   11. Taylor expanded in n around inf

       \[\leadsto \frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{\color{blue}{n \cdot x}} \]
   12. 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. mult-flip-rev N/A

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

       4. lower-/.f64 N/A

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

       5. lift-*.f64 63.2

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

   13. Applied rewrites 63.2%

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

   if 5.8000000000000001e194 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.4%

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

Alternative 12: 59.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.35)
   (/ (- (log x)) n)
   (if (<= x 5.8e+194) (/ (/ 1.0 n) x) (- 1.0 1.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.35) {
		tmp = -log(x) / n;
	} else if (x <= 5.8e+194) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.35d0) then
        tmp = -log(x) / n
    else if (x <= 5.8d+194) then
        tmp = (1.0d0 / n) / x
    else
        tmp = 1.0d0 - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.35) {
		tmp = -Math.log(x) / n;
	} else if (x <= 5.8e+194) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 1.0 - 1.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.35:
		tmp = -math.log(x) / n
	elif x <= 5.8e+194:
		tmp = (1.0 / n) / x
	else:
		tmp = 1.0 - 1.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.35)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 5.8e+194)
		tmp = Float64(Float64(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 <= 0.35)
		tmp = -log(x) / n;
	elseif (x <= 5.8e+194)
		tmp = (1.0 / n) / x;
	else
		tmp = 1.0 - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.35], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 5.8e+194], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.34999999999999998

   1. Initial program 44.1%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in x around 0

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

      1. log-pow-rev N/A

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

      2. inv-pow N/A

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

      3. neg-log N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 50.0

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

   7. Applied rewrites 50.0%

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

   if 0.34999999999999998 < x < 5.8000000000000001e194

   1. Initial program 55.4%

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

   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 81.1%

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

   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. lower-/.f64 64.3

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 63.3%

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

   if 5.8000000000000001e194 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.4%

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

Alternative 13: 47.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+288}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;1 - 1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ n x) (* n n))))
   (if (<= (/ 1.0 n) -2e+288)
     (- 1.0 1.0)
     (if (<= (/ 1.0 n) -1e+83)
       t_0
       (if (<= (/ 1.0 n) -2e+15)
         (- 1.0 1.0)
         (if (<= (/ 1.0 n) 4e-9) (/ (/ 1.0 n) x) t_0))))))

C

double code(double x, double n) {
	double t_0 = (n / x) / (n * n);
	double tmp;
	if ((1.0 / n) <= -2e+288) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= -1e+83) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e+15) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= 4e-9) {
		tmp = (1.0 / n) / x;
	} 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 = (n / x) / (n * n)
    if ((1.0d0 / n) <= (-2d+288)) then
        tmp = 1.0d0 - 1.0d0
    else if ((1.0d0 / n) <= (-1d+83)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-2d+15)) then
        tmp = 1.0d0 - 1.0d0
    else if ((1.0d0 / n) <= 4d-9) then
        tmp = (1.0d0 / n) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (n / x) / (n * n);
	double tmp;
	if ((1.0 / n) <= -2e+288) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= -1e+83) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e+15) {
		tmp = 1.0 - 1.0;
	} else if ((1.0 / n) <= 4e-9) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (n / x) / (n * n)
	tmp = 0
	if (1.0 / n) <= -2e+288:
		tmp = 1.0 - 1.0
	elif (1.0 / n) <= -1e+83:
		tmp = t_0
	elif (1.0 / n) <= -2e+15:
		tmp = 1.0 - 1.0
	elif (1.0 / n) <= 4e-9:
		tmp = (1.0 / n) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(n / x) / Float64(n * n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+288)
		tmp = Float64(1.0 - 1.0);
	elseif (Float64(1.0 / n) <= -1e+83)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -2e+15)
		tmp = Float64(1.0 - 1.0);
	elseif (Float64(1.0 / n) <= 4e-9)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (n / x) / (n * n);
	tmp = 0.0;
	if ((1.0 / n) <= -2e+288)
		tmp = 1.0 - 1.0;
	elseif ((1.0 / n) <= -1e+83)
		tmp = t_0;
	elseif ((1.0 / n) <= -2e+15)
		tmp = 1.0 - 1.0;
	elseif ((1.0 / n) <= 4e-9)
		tmp = (1.0 / n) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+288], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+83], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+15], N[(1.0 - 1.0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-9], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e288 or -1.00000000000000003e83 < (/.f64 #s(literal 1 binary64) n) < -2e15

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 48.5%

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

   if -2e288 < (/.f64 #s(literal 1 binary64) n) < -1.00000000000000003e83 or 4.00000000000000025e-9 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 80.3%

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

   2. Taylor expanded in n around -inf

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 29.8%

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

   8. Taylor expanded in n around inf

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 44.2

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

   10. Applied rewrites 44.2%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 41.7

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

   13. Applied rewrites 41.7%

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

   if -2e15 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000025e-9

   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 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 37.3%

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

   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. lower-/.f64 1.9

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

   7. Applied rewrites 1.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 21.7%

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

Alternative 14: 46.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e+15) (- 1.0 1.0) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+15) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d+15)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+15) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e+15:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+15)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e+15)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+15], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e15

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 50.1%

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

   if -2e15 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.6%

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

   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 40.7%

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

   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. lower-/.f64 38.5

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

   7. Applied rewrites 38.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 45.2%

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

Alternative 15: 46.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e+15) (- 1.0 1.0) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+15) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d+15)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+15) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e+15:
		tmp = 1.0 - 1.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+15)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e+15)
		tmp = 1.0 - 1.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+15], N[(1.0 - 1.0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e15

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 50.1%

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

   if -2e15 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.6%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 45.1

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

   7. Applied rewrites 45.1%

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

Alternative 16: 46.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+15}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e+15) (- 1.0 1.0) (/ 1.0 (* n x))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, n)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d+15)) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e+15) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e+15:
		tmp = 1.0 - 1.0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e+15)
		tmp = 1.0 - 1.0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e15

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 50.1%

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

   if -2e15 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 36.6%

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

   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. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 61.2

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 44.5

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

   7. Applied rewrites 44.5%

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

Alternative 17: 30.9% accurate, 12.4× 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 54.1%

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

2. Taylor expanded in n around inf

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

4. Applied rewrites 17.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 30.9%

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

Reproduce

herbie shell --seed 2025127 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))