Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 6.3s

Alternatives: 9

Speedup: 1.2×

• Could not converge on a ground truth

  1. k = 6.708709046906742e+184
  2. n = -1.2262771804906428e+86

Specification

Math

\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\frac{{\left(6.283185307179586 \cdot n\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}} \cdot \sqrt{6.283185307179586 \cdot n} \]

FPCore

(FPCore (k n)
 :precision binary64
 (*
  (/ (pow (* 6.283185307179586 n) (* -0.5 k)) (sqrt k))
  (sqrt (* 6.283185307179586 n))))

C

double code(double k, double n) {
	return (pow((6.283185307179586 * n), (-0.5 * k)) / sqrt(k)) * sqrt((6.283185307179586 * 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(k, n)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = (((6.283185307179586d0 * n) ** ((-0.5d0) * k)) / sqrt(k)) * sqrt((6.283185307179586d0 * n))
end function

Java

public static double code(double k, double n) {
	return (Math.pow((6.283185307179586 * n), (-0.5 * k)) / Math.sqrt(k)) * Math.sqrt((6.283185307179586 * n));
}

Python

def code(k, n):
	return (math.pow((6.283185307179586 * n), (-0.5 * k)) / math.sqrt(k)) * math.sqrt((6.283185307179586 * n))

Julia

function code(k, n)
	return Float64(Float64((Float64(6.283185307179586 * n) ^ Float64(-0.5 * k)) / sqrt(k)) * sqrt(Float64(6.283185307179586 * n)))
end

MATLAB

function tmp = code(k, n)
	tmp = (((6.283185307179586 * n) ^ (-0.5 * k)) / sqrt(k)) * sqrt((6.283185307179586 * n));
end

Wolfram

code[k_, n_] := N[(N[(N[Power[N[(6.283185307179586 * n), $MachinePrecision], N[(-0.5 * k), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(6.283185307179586 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(\frac{1 - k}{2}\right)}} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\color{blue}{1 - k}}{2}\right)} \]

   4. sub-flip N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\color{blue}{1 + \left(\mathsf{neg}\left(k\right)\right)}}{2}\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\color{blue}{\left(\mathsf{neg}\left(k\right)\right) + 1}}{2}\right)} \]

   6. div-add N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(k\right)}{2} + \frac{1}{2}\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\mathsf{neg}\left(k\right)}{2} + \color{blue}{\frac{1}{2}}\right)} \]

   8. pow-add N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\mathsf{neg}\left(k\right)}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\frac{1}{2}}\right)} \]

   9. lower-unsound-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\mathsf{neg}\left(k\right)}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\frac{1}{2}}\right)} \]

3. Applied rewrites 99.5%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\left({\left(n \cdot \left(\pi + \pi\right)\right)}^{\left(k \cdot -0.5\right)} \cdot {\left(n \cdot \left(\pi + \pi\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. exp-to-pow N/A

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

   5. lift-log.f64 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. pow-exp N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{{\left(\left(\pi + \pi\right) \cdot n\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}} \cdot \sqrt{\left(\pi + \pi\right) \cdot n}} \]

6. Evaluated real constant 99.5%

   \[\leadsto \frac{{\left(\color{blue}{6.283185307179586} \cdot n\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}} \cdot \sqrt{\left(\pi + \pi\right) \cdot n} \]

7. Evaluated real constant 99.5%

   \[\leadsto \frac{{\left(6.283185307179586 \cdot n\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}} \cdot \sqrt{\color{blue}{6.283185307179586} \cdot n} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 1.2× speedup

Math

\[\frac{{\left(n \cdot 6.283185307179586\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* n 6.283185307179586) (fma k -0.5 0.5)) (sqrt k)))

C

double code(double k, double n) {
	return pow((n * 6.283185307179586), fma(k, -0.5, 0.5)) / sqrt(k);
}

Julia

function code(k, n)
	return Float64((Float64(n * 6.283185307179586) ^ fma(k, -0.5, 0.5)) / sqrt(k))
end

Wolfram

code[k_, n_] := N[(N[Power[N[(n * 6.283185307179586), $MachinePrecision], N[(k * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(\frac{1 - k}{2}\right)}} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\color{blue}{1 - k}}{2}\right)} \]

   4. sub-flip N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\color{blue}{1 + \left(\mathsf{neg}\left(k\right)\right)}}{2}\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\color{blue}{\left(\mathsf{neg}\left(k\right)\right) + 1}}{2}\right)} \]

   6. div-add N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(k\right)}{2} + \frac{1}{2}\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\mathsf{neg}\left(k\right)}{2} + \color{blue}{\frac{1}{2}}\right)} \]

   8. pow-add N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\mathsf{neg}\left(k\right)}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\frac{1}{2}}\right)} \]

   9. lower-unsound-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\mathsf{neg}\left(k\right)}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\frac{1}{2}}\right)} \]

3. Applied rewrites 99.5%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\left({\left(n \cdot \left(\pi + \pi\right)\right)}^{\left(k \cdot -0.5\right)} \cdot {\left(n \cdot \left(\pi + \pi\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. exp-to-pow N/A

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

   5. lift-log.f64 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. pow-exp N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{{\left(\left(\pi + \pi\right) \cdot n\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}} \cdot \sqrt{\left(\pi + \pi\right) \cdot n}} \]

6. Evaluated real constant 99.5%

   \[\leadsto \frac{{\left(\color{blue}{6.283185307179586} \cdot n\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}} \cdot \sqrt{\left(\pi + \pi\right) \cdot n} \]

7. Evaluated real constant 99.5%

   \[\leadsto \frac{{\left(6.283185307179586 \cdot n\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}} \cdot \sqrt{\color{blue}{6.283185307179586} \cdot n} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(\frac{884279719003555}{140737488355328} \cdot n\right)}^{\left(\frac{-1}{2} \cdot k\right)}}{\sqrt{k}} \cdot \sqrt{\frac{884279719003555}{140737488355328} \cdot n}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(\frac{884279719003555}{140737488355328} \cdot n\right)}^{\left(\frac{-1}{2} \cdot k\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{884279719003555}{140737488355328} \cdot n} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{{\left(\frac{884279719003555}{140737488355328} \cdot n\right)}^{\left(\frac{-1}{2} \cdot k\right)} \cdot \sqrt{\frac{884279719003555}{140737488355328} \cdot n}}{\sqrt{k}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(\frac{884279719003555}{140737488355328} \cdot n\right)}^{\left(\frac{-1}{2} \cdot k\right)} \cdot \sqrt{\frac{884279719003555}{140737488355328} \cdot n}}{\sqrt{k}}} \]

9. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{{\left(n \cdot 6.283185307179586\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}}} \]
10. Add Preprocessing

Alternative 3: 73.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;k \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{6.283185307179586}{k}}\\ \mathbf{elif}\;k \leq 2.25 \cdot 10^{+182}:\\ \;\;\;\;\frac{n \cdot \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{n \cdot n}}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.9e-5)
   (* (sqrt n) (sqrt (/ 6.283185307179586 k)))
   (if (<= k 2.25e+182)
     (/ (* n (sqrt (* (* n (+ PI PI)) (/ 1.0 (* n n))))) (sqrt k))
     (* (sqrt (* n PI)) (sqrt (sqrt (* (/ 2.0 k) (/ 2.0 k))))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.9e-5) {
		tmp = sqrt(n) * sqrt((6.283185307179586 / k));
	} else if (k <= 2.25e+182) {
		tmp = (n * sqrt(((n * (((double) M_PI) + ((double) M_PI))) * (1.0 / (n * n))))) / sqrt(k);
	} else {
		tmp = sqrt((n * ((double) M_PI))) * sqrt(sqrt(((2.0 / k) * (2.0 / k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5.9e-5) {
		tmp = Math.sqrt(n) * Math.sqrt((6.283185307179586 / k));
	} else if (k <= 2.25e+182) {
		tmp = (n * Math.sqrt(((n * (Math.PI + Math.PI)) * (1.0 / (n * n))))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((n * Math.PI)) * Math.sqrt(Math.sqrt(((2.0 / k) * (2.0 / k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5.9e-5:
		tmp = math.sqrt(n) * math.sqrt((6.283185307179586 / k))
	elif k <= 2.25e+182:
		tmp = (n * math.sqrt(((n * (math.pi + math.pi)) * (1.0 / (n * n))))) / math.sqrt(k)
	else:
		tmp = math.sqrt((n * math.pi)) * math.sqrt(math.sqrt(((2.0 / k) * (2.0 / k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.9e-5)
		tmp = Float64(sqrt(n) * sqrt(Float64(6.283185307179586 / k)));
	elseif (k <= 2.25e+182)
		tmp = Float64(Float64(n * sqrt(Float64(Float64(n * Float64(pi + pi)) * Float64(1.0 / Float64(n * n))))) / sqrt(k));
	else
		tmp = Float64(sqrt(Float64(n * pi)) * sqrt(sqrt(Float64(Float64(2.0 / k) * Float64(2.0 / k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5.9e-5)
		tmp = sqrt(n) * sqrt((6.283185307179586 / k));
	elseif (k <= 2.25e+182)
		tmp = (n * sqrt(((n * (pi + pi)) * (1.0 / (n * n))))) / sqrt(k);
	else
		tmp = sqrt((n * pi)) * sqrt(sqrt(((2.0 / k) * (2.0 / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.9e-5], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(6.283185307179586 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.25e+182], N[(N[(n * N[Sqrt[N[(N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(n * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[N[(N[(2.0 / k), $MachinePrecision] * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.8999999999999998e-5

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. count-2-rev N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      12. pow-exp N/A

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

      13. *-commutative N/A

          \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      16. exp-to-pow N/A

          \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      17. unpow1/2 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

   7. Evaluated real constant 37.2%

      \[\leadsto \sqrt{\frac{6.283185307179586 \cdot n}{k}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

      3. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{884279719003555}{140737488355328} \cdot n\right) \cdot \frac{1}{k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{884279719003555}{140737488355328} \cdot n\right) \cdot \frac{1}{k}} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot \frac{884279719003555}{140737488355328}\right) \cdot \frac{1}{k}} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{n \cdot \left(\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}\right)} \]

      7. sqrt-prod N/A

         \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

      8. lower-unsound-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

      9. lower-unsound-sqrt.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

      10. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}} \]

      11. mult-flip-rev N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\frac{\frac{884279719003555}{140737488355328}}{k}} \]

      12. lower-/.f64 49.0%

          \[\leadsto \sqrt{n} \cdot \sqrt{\frac{6.283185307179586}{k}} \]

   9. Applied rewrites 49.0%

      \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{6.283185307179586}{k}}} \]

   if 5.8999999999999998e-5 < k < 2.2500000000000001e182

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\color{blue}{\sqrt{k}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{k}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{k}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{k}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{k}} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\sqrt{k}} \]

      7. lower-sqrt.f64 48.9%

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\sqrt{k}} \]

   7. Applied rewrites 48.9%

      \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\color{blue}{\sqrt{k}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\sqrt{k}} \]

      2. count-2-rev N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\pi}{n} + \frac{\pi}{n}}}{\sqrt{k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\pi}{n} + \frac{\pi}{n}}}{\sqrt{k}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\pi}{n} + \frac{\pi}{n}}}{\sqrt{k}} \]

      5. common-denominator N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\pi \cdot n + \pi \cdot n}{n \cdot n}}}{\sqrt{k}} \]

      6. distribute-rgt-in N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{n \cdot n}}}{\sqrt{k}} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{n \cdot n}}}{\sqrt{k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{n \cdot n}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{n \cdot n}}}{\sqrt{k}} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{n \cdot \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      11. mult-flip N/A

          \[\leadsto \frac{n \cdot \sqrt{\left(\left(\pi + \pi\right) \cdot n\right) \cdot \frac{1}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{n \cdot \sqrt{\left(\left(\pi + \pi\right) \cdot n\right) \cdot \frac{1}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{n \cdot \sqrt{\left(\left(\pi + \pi\right) \cdot n\right) \cdot \frac{1}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{n \cdot \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{n \cdot \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{n \cdot \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      17. sqr-neg-rev N/A

          \[\leadsto \frac{n \cdot \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{n \cdot n}}}{\sqrt{k}} \]

      18. lower-*.f64 49.4%

          \[\leadsto \frac{n \cdot \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{n \cdot n}}}{\sqrt{k}} \]

   9. Applied rewrites 49.4%

      \[\leadsto \frac{n \cdot \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{n \cdot n}}}{\sqrt{k}} \]

   if 2.2500000000000001e182 < k

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. count-2-rev N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      12. pow-exp N/A

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

      13. *-commutative N/A

          \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      16. exp-to-pow N/A

          \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      17. unpow1/2 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

      3. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(\pi + \pi\right) \cdot n\right) \cdot \frac{1}{k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\pi + \pi\right) \cdot n\right) \cdot \frac{1}{k}} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{k}} \]

      6. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{k}} \]

      7. count-2-rev N/A

         \[\leadsto \sqrt{\left(n \cdot \left(2 \cdot \pi\right)\right) \cdot \frac{1}{k}} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{k}} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(n \cdot \pi\right) \cdot 2\right) \cdot \frac{1}{k}} \]

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \frac{1}{k}} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot \left(2 \cdot \frac{1}{k}\right)} \]

      12. sqrt-prod N/A

          \[\leadsto \sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{k}}} \]

      13. lower-unsound-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{k}}} \]

      14. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \sqrt{\color{blue}{2 \cdot \frac{1}{k}}} \]

      15. lift-PI.f64 N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{2 \cdot \frac{1}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\color{blue}{2} \cdot \frac{1}{k}} \]

      17. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{2 \cdot \frac{1}{k}} \]

      18. mult-flip-rev N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\frac{2}{k}} \]

      19. lower-/.f64 48.9%

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\frac{2}{k}} \]

   8. Applied rewrites 48.9%

      \[\leadsto \sqrt{n \cdot \pi} \cdot \color{blue}{\sqrt{\frac{2}{k}}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k}} \cdot \sqrt{\frac{2}{k}}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]

      4. lower-*.f64 37.2%

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]

   10. Applied rewrites 37.2%

       \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;k \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{6.283185307179586}{k}}\\ \mathbf{elif}\;k \leq 2.25 \cdot 10^{+182}:\\ \;\;\;\;\frac{n \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{n \cdot n}}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.9e-5)
   (* (sqrt n) (sqrt (/ 6.283185307179586 k)))
   (if (<= k 2.25e+182)
     (/ (* n (sqrt (/ (* n (+ PI PI)) (* n n)))) (sqrt k))
     (* (sqrt (* n PI)) (sqrt (sqrt (* (/ 2.0 k) (/ 2.0 k))))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.9e-5) {
		tmp = sqrt(n) * sqrt((6.283185307179586 / k));
	} else if (k <= 2.25e+182) {
		tmp = (n * sqrt(((n * (((double) M_PI) + ((double) M_PI))) / (n * n)))) / sqrt(k);
	} else {
		tmp = sqrt((n * ((double) M_PI))) * sqrt(sqrt(((2.0 / k) * (2.0 / k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5.9e-5) {
		tmp = Math.sqrt(n) * Math.sqrt((6.283185307179586 / k));
	} else if (k <= 2.25e+182) {
		tmp = (n * Math.sqrt(((n * (Math.PI + Math.PI)) / (n * n)))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((n * Math.PI)) * Math.sqrt(Math.sqrt(((2.0 / k) * (2.0 / k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5.9e-5:
		tmp = math.sqrt(n) * math.sqrt((6.283185307179586 / k))
	elif k <= 2.25e+182:
		tmp = (n * math.sqrt(((n * (math.pi + math.pi)) / (n * n)))) / math.sqrt(k)
	else:
		tmp = math.sqrt((n * math.pi)) * math.sqrt(math.sqrt(((2.0 / k) * (2.0 / k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.9e-5)
		tmp = Float64(sqrt(n) * sqrt(Float64(6.283185307179586 / k)));
	elseif (k <= 2.25e+182)
		tmp = Float64(Float64(n * sqrt(Float64(Float64(n * Float64(pi + pi)) / Float64(n * n)))) / sqrt(k));
	else
		tmp = Float64(sqrt(Float64(n * pi)) * sqrt(sqrt(Float64(Float64(2.0 / k) * Float64(2.0 / k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5.9e-5)
		tmp = sqrt(n) * sqrt((6.283185307179586 / k));
	elseif (k <= 2.25e+182)
		tmp = (n * sqrt(((n * (pi + pi)) / (n * n)))) / sqrt(k);
	else
		tmp = sqrt((n * pi)) * sqrt(sqrt(((2.0 / k) * (2.0 / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.9e-5], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(6.283185307179586 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.25e+182], N[(N[(n * N[Sqrt[N[(N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(n * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[N[(N[(2.0 / k), $MachinePrecision] * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.8999999999999998e-5

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. count-2-rev N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      12. pow-exp N/A

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

      13. *-commutative N/A

          \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      16. exp-to-pow N/A

          \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      17. unpow1/2 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

   7. Evaluated real constant 37.2%

      \[\leadsto \sqrt{\frac{6.283185307179586 \cdot n}{k}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

      3. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{884279719003555}{140737488355328} \cdot n\right) \cdot \frac{1}{k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{884279719003555}{140737488355328} \cdot n\right) \cdot \frac{1}{k}} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot \frac{884279719003555}{140737488355328}\right) \cdot \frac{1}{k}} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{n \cdot \left(\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}\right)} \]

      7. sqrt-prod N/A

         \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

      8. lower-unsound-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

      9. lower-unsound-sqrt.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

      10. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}} \]

      11. mult-flip-rev N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\frac{\frac{884279719003555}{140737488355328}}{k}} \]

      12. lower-/.f64 49.0%

          \[\leadsto \sqrt{n} \cdot \sqrt{\frac{6.283185307179586}{k}} \]

   9. Applied rewrites 49.0%

      \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{6.283185307179586}{k}}} \]

   if 5.8999999999999998e-5 < k < 2.2500000000000001e182

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\color{blue}{\sqrt{k}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{k}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{k}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{k}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{k}} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\sqrt{k}} \]

      7. lower-sqrt.f64 48.9%

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\sqrt{k}} \]

   7. Applied rewrites 48.9%

      \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\color{blue}{\sqrt{k}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\sqrt{k}} \]

      2. count-2-rev N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\pi}{n} + \frac{\pi}{n}}}{\sqrt{k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\pi}{n} + \frac{\pi}{n}}}{\sqrt{k}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\pi}{n} + \frac{\pi}{n}}}{\sqrt{k}} \]

      5. common-denominator N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\pi \cdot n + \pi \cdot n}{n \cdot n}}}{\sqrt{k}} \]

      6. distribute-rgt-in N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{n \cdot n}}}{\sqrt{k}} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{n \cdot n}}}{\sqrt{k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{n \cdot n}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{n \cdot \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{n \cdot n}}}{\sqrt{k}} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{n \cdot \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{n \cdot \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{n \cdot \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      13. *-commutative N/A

          \[\leadsto \frac{n \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{n \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{\left(\mathsf{neg}\left(n\right)\right) \cdot \left(\mathsf{neg}\left(n\right)\right)}}}{\sqrt{k}} \]

      15. sqr-neg-rev N/A

          \[\leadsto \frac{n \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{n \cdot n}}}{\sqrt{k}} \]

      16. lower-*.f64 49.2%

          \[\leadsto \frac{n \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{n \cdot n}}}{\sqrt{k}} \]

   9. Applied rewrites 49.2%

      \[\leadsto \frac{n \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{n \cdot n}}}{\sqrt{k}} \]

   if 2.2500000000000001e182 < k

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. count-2-rev N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      12. pow-exp N/A

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

      13. *-commutative N/A

          \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      16. exp-to-pow N/A

          \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      17. unpow1/2 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

      3. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(\pi + \pi\right) \cdot n\right) \cdot \frac{1}{k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\pi + \pi\right) \cdot n\right) \cdot \frac{1}{k}} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{k}} \]

      6. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{k}} \]

      7. count-2-rev N/A

         \[\leadsto \sqrt{\left(n \cdot \left(2 \cdot \pi\right)\right) \cdot \frac{1}{k}} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{k}} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(n \cdot \pi\right) \cdot 2\right) \cdot \frac{1}{k}} \]

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \frac{1}{k}} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot \left(2 \cdot \frac{1}{k}\right)} \]

      12. sqrt-prod N/A

          \[\leadsto \sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{k}}} \]

      13. lower-unsound-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{k}}} \]

      14. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \sqrt{\color{blue}{2 \cdot \frac{1}{k}}} \]

      15. lift-PI.f64 N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{2 \cdot \frac{1}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\color{blue}{2} \cdot \frac{1}{k}} \]

      17. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{2 \cdot \frac{1}{k}} \]

      18. mult-flip-rev N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\frac{2}{k}} \]

      19. lower-/.f64 48.9%

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\frac{2}{k}} \]

   8. Applied rewrites 48.9%

      \[\leadsto \sqrt{n \cdot \pi} \cdot \color{blue}{\sqrt{\frac{2}{k}}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k}} \cdot \sqrt{\frac{2}{k}}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]

      4. lower-*.f64 37.2%

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]

   10. Applied rewrites 37.2%

       \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 64.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{-31}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{6.283185307179586}{k}}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+164}:\\ \;\;\;\;n \cdot \sqrt{\frac{6.283185307179586}{k \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1e-31)
   (* (sqrt n) (sqrt (/ 6.283185307179586 k)))
   (if (<= k 6e+164)
     (* n (sqrt (/ 6.283185307179586 (* k n))))
     (* (sqrt (* n PI)) (sqrt (sqrt (* (/ 2.0 k) (/ 2.0 k))))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1e-31) {
		tmp = sqrt(n) * sqrt((6.283185307179586 / k));
	} else if (k <= 6e+164) {
		tmp = n * sqrt((6.283185307179586 / (k * n)));
	} else {
		tmp = sqrt((n * ((double) M_PI))) * sqrt(sqrt(((2.0 / k) * (2.0 / k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1e-31) {
		tmp = Math.sqrt(n) * Math.sqrt((6.283185307179586 / k));
	} else if (k <= 6e+164) {
		tmp = n * Math.sqrt((6.283185307179586 / (k * n)));
	} else {
		tmp = Math.sqrt((n * Math.PI)) * Math.sqrt(Math.sqrt(((2.0 / k) * (2.0 / k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1e-31:
		tmp = math.sqrt(n) * math.sqrt((6.283185307179586 / k))
	elif k <= 6e+164:
		tmp = n * math.sqrt((6.283185307179586 / (k * n)))
	else:
		tmp = math.sqrt((n * math.pi)) * math.sqrt(math.sqrt(((2.0 / k) * (2.0 / k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1e-31)
		tmp = Float64(sqrt(n) * sqrt(Float64(6.283185307179586 / k)));
	elseif (k <= 6e+164)
		tmp = Float64(n * sqrt(Float64(6.283185307179586 / Float64(k * n))));
	else
		tmp = Float64(sqrt(Float64(n * pi)) * sqrt(sqrt(Float64(Float64(2.0 / k) * Float64(2.0 / k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1e-31)
		tmp = sqrt(n) * sqrt((6.283185307179586 / k));
	elseif (k <= 6e+164)
		tmp = n * sqrt((6.283185307179586 / (k * n)));
	else
		tmp = sqrt((n * pi)) * sqrt(sqrt(((2.0 / k) * (2.0 / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1e-31], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(6.283185307179586 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6e+164], N[(n * N[Sqrt[N[(6.283185307179586 / N[(k * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(n * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[N[(N[(2.0 / k), $MachinePrecision] * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.0000000000000001e-31

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. count-2-rev N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      12. pow-exp N/A

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

      13. *-commutative N/A

          \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      16. exp-to-pow N/A

          \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      17. unpow1/2 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

   7. Evaluated real constant 37.2%

      \[\leadsto \sqrt{\frac{6.283185307179586 \cdot n}{k}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

      3. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{884279719003555}{140737488355328} \cdot n\right) \cdot \frac{1}{k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{884279719003555}{140737488355328} \cdot n\right) \cdot \frac{1}{k}} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot \frac{884279719003555}{140737488355328}\right) \cdot \frac{1}{k}} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{n \cdot \left(\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}\right)} \]

      7. sqrt-prod N/A

         \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

      8. lower-unsound-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

      9. lower-unsound-sqrt.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

      10. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}} \]

      11. mult-flip-rev N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\frac{\frac{884279719003555}{140737488355328}}{k}} \]

      12. lower-/.f64 49.0%

          \[\leadsto \sqrt{n} \cdot \sqrt{\frac{6.283185307179586}{k}} \]

   9. Applied rewrites 49.0%

      \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{6.283185307179586}{k}}} \]

   if 1.0000000000000001e-31 < k < 6e164

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. count-2-rev N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      12. pow-exp N/A

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

      13. *-commutative N/A

          \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      16. exp-to-pow N/A

          \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      17. unpow1/2 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

   7. Evaluated real constant 37.2%

      \[\leadsto \sqrt{\frac{6.283185307179586 \cdot n}{k}} \]

   8. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\sqrt{\frac{\frac{884279719003555}{140737488355328}}{k \cdot n}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{\frac{\frac{884279719003555}{140737488355328}}{k \cdot n}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto n \cdot \sqrt{\frac{\frac{884279719003555}{140737488355328}}{k \cdot n}} \]

      3. lower-/.f64 N/A

         \[\leadsto n \cdot \sqrt{\frac{\frac{884279719003555}{140737488355328}}{k \cdot n}} \]

      4. lower-*.f64 49.2%

         \[\leadsto n \cdot \sqrt{\frac{6.283185307179586}{k \cdot n}} \]

   10. Applied rewrites 49.2%

       \[\leadsto n \cdot \color{blue}{\sqrt{\frac{6.283185307179586}{k \cdot n}}} \]

   if 6e164 < k

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. count-2-rev N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      12. pow-exp N/A

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

      13. *-commutative N/A

          \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      16. exp-to-pow N/A

          \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      17. unpow1/2 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

      3. mult-flip N/A

         \[\leadsto \sqrt{\left(\left(\pi + \pi\right) \cdot n\right) \cdot \frac{1}{k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\pi + \pi\right) \cdot n\right) \cdot \frac{1}{k}} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{k}} \]

      6. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{k}} \]

      7. count-2-rev N/A

         \[\leadsto \sqrt{\left(n \cdot \left(2 \cdot \pi\right)\right) \cdot \frac{1}{k}} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{k}} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(n \cdot \pi\right) \cdot 2\right) \cdot \frac{1}{k}} \]

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \frac{1}{k}} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot \left(2 \cdot \frac{1}{k}\right)} \]

      12. sqrt-prod N/A

          \[\leadsto \sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{k}}} \]

      13. lower-unsound-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{k}}} \]

      14. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \sqrt{\color{blue}{2 \cdot \frac{1}{k}}} \]

      15. lift-PI.f64 N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{2 \cdot \frac{1}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\color{blue}{2} \cdot \frac{1}{k}} \]

      17. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{2 \cdot \frac{1}{k}} \]

      18. mult-flip-rev N/A

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\frac{2}{k}} \]

      19. lower-/.f64 48.9%

          \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\frac{2}{k}} \]

   8. Applied rewrites 48.9%

      \[\leadsto \sqrt{n \cdot \pi} \cdot \color{blue}{\sqrt{\frac{2}{k}}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k}} \cdot \sqrt{\frac{2}{k}}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]

      4. lower-*.f64 37.2%

         \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]

   10. Applied rewrites 37.2%

       \[\leadsto \sqrt{n \cdot \pi} \cdot \sqrt{\sqrt{\frac{2}{k} \cdot \frac{2}{k}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 60.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{\frac{n}{k} \cdot 6.283185307179586}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{\frac{6.283185307179586}{k \cdot n}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 5.2e+48)
   (sqrt (* (/ n k) 6.283185307179586))
   (* n (sqrt (/ 6.283185307179586 (* k n))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 5.2e+48) {
		tmp = sqrt(((n / k) * 6.283185307179586));
	} else {
		tmp = n * sqrt((6.283185307179586 / (k * 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(k, n)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 5.2d+48) then
        tmp = sqrt(((n / k) * 6.283185307179586d0))
    else
        tmp = n * sqrt((6.283185307179586d0 / (k * n)))
    end if
    code = tmp
end function

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 5.2e+48) {
		tmp = Math.sqrt(((n / k) * 6.283185307179586));
	} else {
		tmp = n * Math.sqrt((6.283185307179586 / (k * n)));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 5.2e+48:
		tmp = math.sqrt(((n / k) * 6.283185307179586))
	else:
		tmp = n * math.sqrt((6.283185307179586 / (k * n)))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 5.2e+48)
		tmp = sqrt(Float64(Float64(n / k) * 6.283185307179586));
	else
		tmp = Float64(n * sqrt(Float64(6.283185307179586 / Float64(k * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 5.2e+48)
		tmp = sqrt(((n / k) * 6.283185307179586));
	else
		tmp = n * sqrt((6.283185307179586 / (k * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 5.2e+48], N[Sqrt[N[(N[(n / k), $MachinePrecision] * 6.283185307179586), $MachinePrecision]], $MachinePrecision], N[(n * N[Sqrt[N[(6.283185307179586 / N[(k * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.1999999999999999e48

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. count-2-rev N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      12. pow-exp N/A

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

      13. *-commutative N/A

          \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      16. exp-to-pow N/A

          \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      17. unpow1/2 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

   7. Evaluated real constant 37.2%

      \[\leadsto \sqrt{\frac{6.283185307179586 \cdot n}{k}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

      3. associate-/l* N/A

         \[\leadsto \sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{n}{k}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{n}{k} \cdot \frac{884279719003555}{140737488355328}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{n}{k} \cdot \frac{884279719003555}{140737488355328}} \]

      6. lower-/.f64 37.3%

         \[\leadsto \sqrt{\frac{n}{k} \cdot 6.283185307179586} \]

   9. Applied rewrites 37.3%

      \[\leadsto \sqrt{\frac{n}{k} \cdot 6.283185307179586} \]

   if 5.1999999999999999e48 < n

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lower-PI.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lower-sqrt.f64 49.0%

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      4. count-2-rev N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      11. lift-log.f64 N/A

          \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

      12. pow-exp N/A

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

      13. *-commutative N/A

          \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

      16. exp-to-pow N/A

          \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

      17. unpow1/2 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

   7. Evaluated real constant 37.2%

      \[\leadsto \sqrt{\frac{6.283185307179586 \cdot n}{k}} \]

   8. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\sqrt{\frac{\frac{884279719003555}{140737488355328}}{k \cdot n}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto n \cdot \sqrt{\frac{\frac{884279719003555}{140737488355328}}{k \cdot n}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto n \cdot \sqrt{\frac{\frac{884279719003555}{140737488355328}}{k \cdot n}} \]

      3. lower-/.f64 N/A

         \[\leadsto n \cdot \sqrt{\frac{\frac{884279719003555}{140737488355328}}{k \cdot n}} \]

      4. lower-*.f64 49.2%

         \[\leadsto n \cdot \sqrt{\frac{6.283185307179586}{k \cdot n}} \]

   10. Applied rewrites 49.2%

       \[\leadsto n \cdot \color{blue}{\sqrt{\frac{6.283185307179586}{k \cdot n}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 49.0% accurate, 3.4× speedup

Math

\[\sqrt{n} \cdot \sqrt{\frac{6.283185307179586}{k}} \]

FPCore

(FPCore (k n) :precision binary64 (* (sqrt n) (sqrt (/ 6.283185307179586 k))))

C

double code(double k, double n) {
	return sqrt(n) * sqrt((6.283185307179586 / k));
}

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(k, n)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt(n) * sqrt((6.283185307179586d0 / k))
end function

Java

public static double code(double k, double n) {
	return Math.sqrt(n) * Math.sqrt((6.283185307179586 / k));
}

Python

def code(k, n):
	return math.sqrt(n) * math.sqrt((6.283185307179586 / k))

Julia

function code(k, n)
	return Float64(sqrt(n) * sqrt(Float64(6.283185307179586 / k)))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt(n) * sqrt((6.283185307179586 / k));
end

Wolfram

code[k_, n_] := N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(6.283185307179586 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   5. lower-PI.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   6. lower-sqrt.f64 49.0%

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

4. Applied rewrites 49.0%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   4. count-2-rev N/A

      \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   7. distribute-lft-in N/A

      \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   10. rem-exp-log N/A

       \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   11. lift-log.f64 N/A

       \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   12. pow-exp N/A

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

   13. *-commutative N/A

       \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

   14. *-commutative N/A

       \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

   15. lift-log.f64 N/A

       \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

   16. exp-to-pow N/A

       \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

   17. unpow1/2 N/A

       \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

   18. lift-sqrt.f64 N/A

       \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

6. Applied rewrites 37.2%

   \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

7. Evaluated real constant 37.2%

   \[\leadsto \sqrt{\frac{6.283185307179586 \cdot n}{k}} \]
8. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

   2. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

   3. mult-flip N/A

      \[\leadsto \sqrt{\left(\frac{884279719003555}{140737488355328} \cdot n\right) \cdot \frac{1}{k}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\frac{884279719003555}{140737488355328} \cdot n\right) \cdot \frac{1}{k}} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\left(n \cdot \frac{884279719003555}{140737488355328}\right) \cdot \frac{1}{k}} \]

   6. associate-*l* N/A

      \[\leadsto \sqrt{n \cdot \left(\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}\right)} \]

   7. sqrt-prod N/A

      \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

   8. lower-unsound-*.f64 N/A

      \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

   9. lower-unsound-sqrt.f64 N/A

      \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}}} \]

   10. lower-unsound-sqrt.f64 N/A

       \[\leadsto \sqrt{n} \cdot \sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}} \]

   11. mult-flip-rev N/A

       \[\leadsto \sqrt{n} \cdot \sqrt{\frac{\frac{884279719003555}{140737488355328}}{k}} \]

   12. lower-/.f64 49.0%

       \[\leadsto \sqrt{n} \cdot \sqrt{\frac{6.283185307179586}{k}} \]

9. Applied rewrites 49.0%

   \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{6.283185307179586}{k}}} \]
10. Add Preprocessing

Alternative 8: 37.3% accurate, 4.0× speedup

Math

\[\sqrt{\frac{n}{k} \cdot 6.283185307179586} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* (/ n k) 6.283185307179586)))

C

double code(double k, double n) {
	return sqrt(((n / k) * 6.283185307179586));
}

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(k, n)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt(((n / k) * 6.283185307179586d0))
end function

Java

public static double code(double k, double n) {
	return Math.sqrt(((n / k) * 6.283185307179586));
}

Python

def code(k, n):
	return math.sqrt(((n / k) * 6.283185307179586))

Julia

function code(k, n)
	return sqrt(Float64(Float64(n / k) * 6.283185307179586))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt(((n / k) * 6.283185307179586));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(n / k), $MachinePrecision] * 6.283185307179586), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   5. lower-PI.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   6. lower-sqrt.f64 49.0%

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

4. Applied rewrites 49.0%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   4. count-2-rev N/A

      \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   7. distribute-lft-in N/A

      \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   10. rem-exp-log N/A

       \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   11. lift-log.f64 N/A

       \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   12. pow-exp N/A

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

   13. *-commutative N/A

       \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

   14. *-commutative N/A

       \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

   15. lift-log.f64 N/A

       \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

   16. exp-to-pow N/A

       \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

   17. unpow1/2 N/A

       \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

   18. lift-sqrt.f64 N/A

       \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

6. Applied rewrites 37.2%

   \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

7. Evaluated real constant 37.2%

   \[\leadsto \sqrt{\frac{6.283185307179586 \cdot n}{k}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

   3. associate-/l* N/A

      \[\leadsto \sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{n}{k}} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{n}{k} \cdot \frac{884279719003555}{140737488355328}} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{n}{k} \cdot \frac{884279719003555}{140737488355328}} \]

   6. lower-/.f64 37.3%

      \[\leadsto \sqrt{\frac{n}{k} \cdot 6.283185307179586} \]

9. Applied rewrites 37.3%

   \[\leadsto \sqrt{\frac{n}{k} \cdot 6.283185307179586} \]
10. Add Preprocessing

Alternative 9: 37.2% accurate, 4.0× speedup

Math

\[\sqrt{n \cdot \frac{6.283185307179586}{k}} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* n (/ 6.283185307179586 k))))

C

double code(double k, double n) {
	return sqrt((n * (6.283185307179586 / k)));
}

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(k, n)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt((n * (6.283185307179586d0 / k)))
end function

Java

public static double code(double k, double n) {
	return Math.sqrt((n * (6.283185307179586 / k)));
}

Python

def code(k, n):
	return math.sqrt((n * (6.283185307179586 / k)))

Julia

function code(k, n)
	return sqrt(Float64(n * Float64(6.283185307179586 / k)))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((n * (6.283185307179586 / k)));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(n * N[(6.283185307179586 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   5. lower-PI.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   6. lower-sqrt.f64 49.0%

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

4. Applied rewrites 49.0%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\color{blue}{\sqrt{k}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   4. count-2-rev N/A

      \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \pi + n \cdot \pi\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   7. distribute-lft-in N/A

      \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   10. rem-exp-log N/A

       \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   11. lift-log.f64 N/A

       \[\leadsto \frac{{\left(e^{\log \left(n \cdot \left(\pi + \pi\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{k}} \]

   12. pow-exp N/A

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

   13. *-commutative N/A

       \[\leadsto \frac{e^{\frac{1}{2} \cdot \log \left(n \cdot \left(\pi + \pi\right)\right)}}{\sqrt{k}} \]

   14. *-commutative N/A

       \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

   15. lift-log.f64 N/A

       \[\leadsto \frac{e^{\log \left(n \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}}}{\sqrt{k}} \]

   16. exp-to-pow N/A

       \[\leadsto \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{\color{blue}{k}}} \]

   17. unpow1/2 N/A

       \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{\color{blue}{k}}} \]

   18. lift-sqrt.f64 N/A

       \[\leadsto \frac{\sqrt{n \cdot \left(\pi + \pi\right)}}{\sqrt{k}} \]

6. Applied rewrites 37.2%

   \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]

7. Evaluated real constant 37.2%

   \[\leadsto \sqrt{\frac{6.283185307179586 \cdot n}{k}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{\frac{884279719003555}{140737488355328} \cdot n}{k}} \]

   2. mult-flip N/A

      \[\leadsto \sqrt{\left(\frac{884279719003555}{140737488355328} \cdot n\right) \cdot \frac{1}{k}} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\frac{884279719003555}{140737488355328} \cdot n\right) \cdot \frac{1}{k}} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(n \cdot \frac{884279719003555}{140737488355328}\right) \cdot \frac{1}{k}} \]

   5. associate-*l* N/A

      \[\leadsto \sqrt{n \cdot \left(\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{n \cdot \left(\frac{884279719003555}{140737488355328} \cdot \frac{1}{k}\right)} \]

   7. mult-flip-rev N/A

      \[\leadsto \sqrt{n \cdot \frac{\frac{884279719003555}{140737488355328}}{k}} \]

   8. lower-/.f64 37.2%

      \[\leadsto \sqrt{n \cdot \frac{6.283185307179586}{k}} \]

9. Applied rewrites 37.2%

   \[\leadsto \sqrt{n \cdot \frac{6.283185307179586}{k}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))