Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5

Percentage Accurate: 70.0% → 92.7%

Time: 6.6s

Alternatives: 10

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))

C

double code(double x, double y, double z) {
	return 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
}

Python

def code(x, y, z):
	return 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))

Julia

function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
end

Wolfram

code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))

C

double code(double x, double y, double z) {
	return 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
}

Python

def code(x, y, z):
	return 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))

Julia

function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
end

Wolfram

code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+64}:\\ \;\;\;\;2 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 740000000:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, x, z \cdot \left(x + y\right)\right)}\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log z + \log \left(x + y\right)\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.8e+64)
   (* 2.0 (pow (pow (exp 0.25) (- (log (- (- y) z)) (log (/ -1.0 x)))) 2.0))
   (if (<= y 740000000.0)
     (* 2.0 (sqrt (fma y x (* z (+ x y)))))
     (if (<= y 1.46e+200)
       (*
        (fma
         (sqrt (/ (/ 1.0 (pow z 3.0)) (+ y x)))
         (* y x)
         (* (sqrt (/ (+ y x) z)) 2.0))
        z)
       (* 2.0 (pow (pow (exp 0.25) (+ (log z) (log (+ x y)))) 2.0))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.8e+64) {
		tmp = 2.0 * pow(pow(exp(0.25), (log((-y - z)) - log((-1.0 / x)))), 2.0);
	} else if (y <= 740000000.0) {
		tmp = 2.0 * sqrt(fma(y, x, (z * (x + y))));
	} else if (y <= 1.46e+200) {
		tmp = fma(sqrt(((1.0 / pow(z, 3.0)) / (y + x))), (y * x), (sqrt(((y + x) / z)) * 2.0)) * z;
	} else {
		tmp = 2.0 * pow(pow(exp(0.25), (log(z) + log((x + y)))), 2.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.8e+64)
		tmp = Float64(2.0 * ((exp(0.25) ^ Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x)))) ^ 2.0));
	elseif (y <= 740000000.0)
		tmp = Float64(2.0 * sqrt(fma(y, x, Float64(z * Float64(x + y)))));
	elseif (y <= 1.46e+200)
		tmp = Float64(fma(sqrt(Float64(Float64(1.0 / (z ^ 3.0)) / Float64(y + x))), Float64(y * x), Float64(sqrt(Float64(Float64(y + x) / z)) * 2.0)) * z);
	else
		tmp = Float64(2.0 * ((exp(0.25) ^ Float64(log(z) + log(Float64(x + y)))) ^ 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -2.8e+64], N[(2.0 * N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 740000000.0], N[(2.0 * N[Sqrt[N[(y * x + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.46e+200], N[(N[(N[Sqrt[N[(N[(1.0 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y * x), $MachinePrecision] + N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(2.0 * N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.80000000000000024e64

   1. Initial program 51.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

      2. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

      3. associate-+l+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot z + y \cdot z\right) + x \cdot y}} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x \cdot z + \color{blue}{y \cdot z}\right) + x \cdot y} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(\color{blue}{x \cdot z} + y \cdot z\right) + x \cdot y} \]

      7. distribute-rgt-out N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)} + x \cdot y} \]

      8. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, x \cdot y\right)}} \]

      9. lower-+.f64 51.6

         \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, \color{blue}{x + y}, x \cdot y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{x \cdot y}\right)} \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

      12. lower-*.f64 51.6

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

   4. Applied rewrites 51.6%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, y \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\sqrt{\mathsf{fma}\left(z, x + y, y \cdot x\right)}} \]

      2. pow1/2 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-+l+ N/A

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

      10. distribute-lft-out N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. metadata-eval N/A

          \[\leadsto 2 \cdot {\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}} \]

      18. pow-prod-up N/A

          \[\leadsto 2 \cdot \color{blue}{\left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}\right)} \]

      19. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\color{blue}{{\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}} \cdot {\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}\right) \]

      20. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}} \cdot \color{blue}{{\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}}\right) \]

      21. pow2 N/A

          \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}\right)}^{2}} \]

      22. lower-pow.f64 51.7

          \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{0.25}\right)}^{2}} \]

   6. Applied rewrites 51.7%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y + z, x, y \cdot z\right)\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in x around -inf

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

   9. Applied rewrites 36.2%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(e^{0.25}\right)}^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}}^{2} \]

   if -2.80000000000000024e64 < y < 7.4e8

   1. Initial program 86.9%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

      2. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

      3. associate-+l+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y} + \left(x \cdot z + y \cdot z\right)} \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot x} + \left(x \cdot z + y \cdot z\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(y, x, x \cdot z + y \cdot z\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, x \cdot z + \color{blue}{y \cdot z}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, \color{blue}{x \cdot z} + y \cdot z\right)} \]

      9. distribute-rgt-out N/A

         \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, \color{blue}{z \cdot \left(x + y\right)}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, \color{blue}{z \cdot \left(x + y\right)}\right)} \]

      11. lower-+.f64 86.9

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, z \cdot \color{blue}{\left(x + y\right)}\right)} \]

   4. Applied rewrites 86.9%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(y, x, z \cdot \left(x + y\right)\right)}} \]

   if 7.4e8 < y < 1.46e200

   1. Initial program 62.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   if 1.46e200 < y

   1. Initial program 43.7%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

      2. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

      3. associate-+l+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot z + y \cdot z\right) + x \cdot y}} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x \cdot z + \color{blue}{y \cdot z}\right) + x \cdot y} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(\color{blue}{x \cdot z} + y \cdot z\right) + x \cdot y} \]

      7. distribute-rgt-out N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)} + x \cdot y} \]

      8. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, x \cdot y\right)}} \]

      9. lower-+.f64 44.1

         \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, \color{blue}{x + y}, x \cdot y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{x \cdot y}\right)} \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

      12. lower-*.f64 44.1

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

   4. Applied rewrites 44.1%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, y \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\sqrt{\mathsf{fma}\left(z, x + y, y \cdot x\right)}} \]

      2. pow1/2 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-+l+ N/A

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

      10. distribute-lft-out N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. metadata-eval N/A

          \[\leadsto 2 \cdot {\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}} \]

      18. pow-prod-up N/A

          \[\leadsto 2 \cdot \color{blue}{\left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}\right)} \]

      19. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\color{blue}{{\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}} \cdot {\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}\right) \]

      20. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}} \cdot \color{blue}{{\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}}\right) \]

      21. pow2 N/A

          \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}\right)}^{2}} \]

      22. lower-pow.f64 44.5

          \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{0.25}\right)}^{2}} \]

   6. Applied rewrites 44.5%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y + z, x, y \cdot z\right)\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 57.1%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(e^{0.25}\right)}^{\left(\log z + \log \left(x + y\right)\right)}\right)}}^{2} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 740000000:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, y \cdot x\right)}\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log z + \log \left(x + y\right)\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 740000000.0)
   (* 2.0 (sqrt (fma z (+ x y) (* y x))))
   (if (<= y 1.46e+200)
     (*
      (fma
       (sqrt (/ (/ 1.0 (pow z 3.0)) (+ y x)))
       (* y x)
       (* (sqrt (/ (+ y x) z)) 2.0))
      z)
     (* 2.0 (pow (pow (exp 0.25) (+ (log z) (log (+ x y)))) 2.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 740000000.0) {
		tmp = 2.0 * sqrt(fma(z, (x + y), (y * x)));
	} else if (y <= 1.46e+200) {
		tmp = fma(sqrt(((1.0 / pow(z, 3.0)) / (y + x))), (y * x), (sqrt(((y + x) / z)) * 2.0)) * z;
	} else {
		tmp = 2.0 * pow(pow(exp(0.25), (log(z) + log((x + y)))), 2.0);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 740000000.0)
		tmp = Float64(2.0 * sqrt(fma(z, Float64(x + y), Float64(y * x))));
	elseif (y <= 1.46e+200)
		tmp = Float64(fma(sqrt(Float64(Float64(1.0 / (z ^ 3.0)) / Float64(y + x))), Float64(y * x), Float64(sqrt(Float64(Float64(y + x) / z)) * 2.0)) * z);
	else
		tmp = Float64(2.0 * ((exp(0.25) ^ Float64(log(z) + log(Float64(x + y)))) ^ 2.0));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 740000000.0], N[(2.0 * N[Sqrt[N[(z * N[(x + y), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.46e+200], N[(N[(N[Sqrt[N[(N[(1.0 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y * x), $MachinePrecision] + N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(2.0 * N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.4e8

   1. Initial program 78.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

      2. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

      3. associate-+l+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot z + y \cdot z\right) + x \cdot y}} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x \cdot z + \color{blue}{y \cdot z}\right) + x \cdot y} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(\color{blue}{x \cdot z} + y \cdot z\right) + x \cdot y} \]

      7. distribute-rgt-out N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)} + x \cdot y} \]

      8. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, x \cdot y\right)}} \]

      9. lower-+.f64 78.0

         \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, \color{blue}{x + y}, x \cdot y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{x \cdot y}\right)} \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

      12. lower-*.f64 78.0

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

   4. Applied rewrites 78.0%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, y \cdot x\right)}} \]

   if 7.4e8 < y < 1.46e200

   1. Initial program 62.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]

   if 1.46e200 < y

   1. Initial program 43.7%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

      2. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

      3. associate-+l+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot z + y \cdot z\right) + x \cdot y}} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x \cdot z + \color{blue}{y \cdot z}\right) + x \cdot y} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(\color{blue}{x \cdot z} + y \cdot z\right) + x \cdot y} \]

      7. distribute-rgt-out N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)} + x \cdot y} \]

      8. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, x \cdot y\right)}} \]

      9. lower-+.f64 44.1

         \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, \color{blue}{x + y}, x \cdot y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{x \cdot y}\right)} \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

      12. lower-*.f64 44.1

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

   4. Applied rewrites 44.1%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, y \cdot x\right)}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\sqrt{\mathsf{fma}\left(z, x + y, y \cdot x\right)}} \]

      2. pow1/2 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-+l+ N/A

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

      10. distribute-lft-out N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. metadata-eval N/A

          \[\leadsto 2 \cdot {\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}} \]

      18. pow-prod-up N/A

          \[\leadsto 2 \cdot \color{blue}{\left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}\right)} \]

      19. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\color{blue}{{\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}} \cdot {\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}\right) \]

      20. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}} \cdot \color{blue}{{\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}}\right) \]

      21. pow2 N/A

          \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{\frac{1}{4}}\right)}^{2}} \]

      22. lower-pow.f64 44.5

          \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(z + y, x, z \cdot y\right)\right)}^{0.25}\right)}^{2}} \]

   6. Applied rewrites 44.5%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y + z, x, y \cdot z\right)\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in z around inf

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

   9. Applied rewrites 57.1%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(e^{0.25}\right)}^{\left(\log z + \log \left(x + y\right)\right)}\right)}}^{2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 740000000:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 740000000.0)
   (* 2.0 (sqrt (fma z (+ x y) (* y x))))
   (*
    (fma
     (sqrt (/ (/ 1.0 (pow z 3.0)) (+ y x)))
     (* y x)
     (* (sqrt (/ (+ y x) z)) 2.0))
    z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 740000000.0) {
		tmp = 2.0 * sqrt(fma(z, (x + y), (y * x)));
	} else {
		tmp = fma(sqrt(((1.0 / pow(z, 3.0)) / (y + x))), (y * x), (sqrt(((y + x) / z)) * 2.0)) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 740000000.0)
		tmp = Float64(2.0 * sqrt(fma(z, Float64(x + y), Float64(y * x))));
	else
		tmp = Float64(fma(sqrt(Float64(Float64(1.0 / (z ^ 3.0)) / Float64(y + x))), Float64(y * x), Float64(sqrt(Float64(Float64(y + x) / z)) * 2.0)) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 740000000.0], N[(2.0 * N[Sqrt[N[(z * N[(x + y), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(1.0 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y * x), $MachinePrecision] + N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.4e8

   1. Initial program 78.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

      2. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

      3. associate-+l+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot z + y \cdot z\right) + x \cdot y}} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x \cdot z + \color{blue}{y \cdot z}\right) + x \cdot y} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(\color{blue}{x \cdot z} + y \cdot z\right) + x \cdot y} \]

      7. distribute-rgt-out N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)} + x \cdot y} \]

      8. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, x \cdot y\right)}} \]

      9. lower-+.f64 78.0

         \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, \color{blue}{x + y}, x \cdot y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{x \cdot y}\right)} \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

      12. lower-*.f64 78.0

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

   4. Applied rewrites 78.0%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, y \cdot x\right)}} \]

   if 7.4e8 < y

   1. Initial program 57.8%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 33.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{\frac{1}{{y}^{3}}}{z + x}}, z \cdot x, \sqrt{\frac{z + x}{y}} \cdot 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5.7e-5)
   (* 2.0 (sqrt (fma z (+ x y) (* y x))))
   (*
    (fma
     (sqrt (/ (/ 1.0 (pow y 3.0)) (+ z x)))
     (* z x)
     (* (sqrt (/ (+ z x) y)) 2.0))
    y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.7e-5) {
		tmp = 2.0 * sqrt(fma(z, (x + y), (y * x)));
	} else {
		tmp = fma(sqrt(((1.0 / pow(y, 3.0)) / (z + x))), (z * x), (sqrt(((z + x) / y)) * 2.0)) * y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 5.7e-5)
		tmp = Float64(2.0 * sqrt(fma(z, Float64(x + y), Float64(y * x))));
	else
		tmp = Float64(fma(sqrt(Float64(Float64(1.0 / (y ^ 3.0)) / Float64(z + x))), Float64(z * x), Float64(sqrt(Float64(Float64(z + x) / y)) * 2.0)) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 5.7e-5], N[(2.0 * N[Sqrt[N[(z * N[(x + y), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(1.0 / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] / N[(z + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(z * x), $MachinePrecision] + N[(N[Sqrt[N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.7000000000000003e-5

   1. Initial program 77.9%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

      2. lift-+.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

      3. associate-+l+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot z + y \cdot z\right) + x \cdot y}} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(x \cdot z + \color{blue}{y \cdot z}\right) + x \cdot y} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\left(\color{blue}{x \cdot z} + y \cdot z\right) + x \cdot y} \]

      7. distribute-rgt-out N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)} + x \cdot y} \]

      8. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, x \cdot y\right)}} \]

      9. lower-+.f64 77.9

         \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, \color{blue}{x + y}, x \cdot y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{x \cdot y}\right)} \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

      12. lower-*.f64 77.9

          \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

   4. Applied rewrites 77.9%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, y \cdot x\right)}} \]

   if 5.7000000000000003e-5 < y

   1. Initial program 58.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{\frac{1}{{y}^{3}}}{z + x}}, z \cdot x, \sqrt{\frac{z + x}{y}} \cdot 2\right) \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-303}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-303) (* 2.0 (sqrt (* (+ z y) x))) (* 2.0 (sqrt (* (+ y x) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-303) {
		tmp = 2.0 * sqrt(((z + y) * x));
	} else {
		tmp = 2.0 * sqrt(((y + x) * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5d-303) then
        tmp = 2.0d0 * sqrt(((z + y) * x))
    else
        tmp = 2.0d0 * sqrt(((y + x) * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-303) {
		tmp = 2.0 * Math.sqrt(((z + y) * x));
	} else {
		tmp = 2.0 * Math.sqrt(((y + x) * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 5e-303:
		tmp = 2.0 * math.sqrt(((z + y) * x))
	else:
		tmp = 2.0 * math.sqrt(((y + x) * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-303)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + y) * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + x) * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e-303)
		tmp = 2.0 * sqrt(((z + y) * x));
	else
		tmp = 2.0 * sqrt(((y + x) * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 5e-303], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999998e-303

   1. Initial program 73.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)}} \]
   4. Step-by-step derivation

   5. Applied rewrites 48.5%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(z + y\right) \cdot x}} \]

   if 4.9999999999999998e-303 < y

   1. Initial program 72.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)}} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.0%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(y + x\right) \cdot z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-303}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e-303) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* (+ y x) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e-303) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt(((y + x) * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.5d-303) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt(((y + x) * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e-303) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt(((y + x) * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 3.5e-303:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt(((y + x) * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 3.5e-303)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + x) * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.5e-303)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt(((y + x) * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 3.5e-303], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5e-303

   1. Initial program 73.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y}} \]
   4. Step-by-step derivation

   5. Applied rewrites 19.0%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot x}} \]

   if 3.5e-303 < y

   1. Initial program 72.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)}} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.0%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(y + x\right) \cdot z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, y \cdot x\right)} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* 2.0 (sqrt (fma z (+ x y) (* y x)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 2.0 * sqrt(fma(z, (x + y), (y * x)));
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(2.0 * sqrt(fma(z, Float64(x + y), Float64(y * x))))
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(z * N[(x + y), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

   2. lift-+.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

   3. associate-+l+ N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]

   4. +-commutative N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot z + y \cdot z\right) + x \cdot y}} \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\left(x \cdot z + \color{blue}{y \cdot z}\right) + x \cdot y} \]

   6. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\left(\color{blue}{x \cdot z} + y \cdot z\right) + x \cdot y} \]

   7. distribute-rgt-out N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)} + x \cdot y} \]

   8. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, x \cdot y\right)}} \]

   9. lower-+.f64 72.7

      \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, \color{blue}{x + y}, x \cdot y\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{x \cdot y}\right)} \]

   11. *-commutative N/A

       \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

   12. lower-*.f64 72.7

       \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, \color{blue}{y \cdot x}\right)} \]

4. Applied rewrites 72.7%

   \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, y \cdot x\right)}} \]
5. Add Preprocessing

Alternative 8: 70.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ 2 \cdot \sqrt{\mathsf{fma}\left(y, x, z \cdot \left(x + y\right)\right)} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* 2.0 (sqrt (fma y x (* z (+ x y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 2.0 * sqrt(fma(y, x, (z * (x + y))));
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(2.0 * sqrt(fma(y, x, Float64(z * Float64(x + y)))))
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(y * x + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \]

   2. lift-+.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]

   3. associate-+l+ N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y} + \left(x \cdot z + y \cdot z\right)} \]

   5. *-commutative N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot x} + \left(x \cdot z + y \cdot z\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(y, x, x \cdot z + y \cdot z\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, x \cdot z + \color{blue}{y \cdot z}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, \color{blue}{x \cdot z} + y \cdot z\right)} \]

   9. distribute-rgt-out N/A

      \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, \color{blue}{z \cdot \left(x + y\right)}\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, \color{blue}{z \cdot \left(x + y\right)}\right)} \]

   11. lower-+.f64 72.9

       \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(y, x, z \cdot \color{blue}{\left(x + y\right)}\right)} \]

4. Applied rewrites 72.9%

   \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(y, x, z \cdot \left(x + y\right)\right)}} \]
5. Add Preprocessing

Alternative 9: 67.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* z y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((z * y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2d-310)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((z * y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((z * y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2e-310:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((z * y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-310)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((z * y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -2e-310], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 73.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y}} \]
   4. Step-by-step derivation

   5. Applied rewrites 19.1%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot x}} \]

   if -1.999999999999994e-310 < y

   1. Initial program 72.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot z}} \]
   4. Step-by-step derivation

   5. Applied rewrites 28.6%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 35.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ 2 \cdot \sqrt{y \cdot x} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* 2.0 (sqrt (* y x))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 2.0 * sqrt((y * x));
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt((y * x))
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt((y * x));
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 2.0 * math.sqrt((y * x))

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(y * x)))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((y * x));
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto 2 \cdot \sqrt{\color{blue}{x \cdot y}} \]
4. Step-by-step derivation

5. Applied rewrites 21.6%

   \[\leadsto 2 \cdot \sqrt{\color{blue}{y \cdot x}} \]
6. Add Preprocessing

Developer Target 1: 83.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\\ \mathbf{if}\;z < 7.636950090573675 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z)))
          (* (pow z 0.25) (pow y 0.25)))))
   (if (< z 7.636950090573675e+176)
     (* 2.0 (sqrt (+ (* (+ x y) z) (* x y))))
     (* (* t_0 t_0) 2.0))))

C

double code(double x, double y, double z) {
	double t_0 = (0.25 * ((pow(y, -0.75) * (pow(z, -0.75) * x)) * (y + z))) + (pow(z, 0.25) * pow(y, 0.25));
	double tmp;
	if (z < 7.636950090573675e+176) {
		tmp = 2.0 * sqrt((((x + y) * z) + (x * y)));
	} else {
		tmp = (t_0 * t_0) * 2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.25d0 * (((y ** (-0.75d0)) * ((z ** (-0.75d0)) * x)) * (y + z))) + ((z ** 0.25d0) * (y ** 0.25d0))
    if (z < 7.636950090573675d+176) then
        tmp = 2.0d0 * sqrt((((x + y) * z) + (x * y)))
    else
        tmp = (t_0 * t_0) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.25 * ((Math.pow(y, -0.75) * (Math.pow(z, -0.75) * x)) * (y + z))) + (Math.pow(z, 0.25) * Math.pow(y, 0.25));
	double tmp;
	if (z < 7.636950090573675e+176) {
		tmp = 2.0 * Math.sqrt((((x + y) * z) + (x * y)));
	} else {
		tmp = (t_0 * t_0) * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.25 * ((math.pow(y, -0.75) * (math.pow(z, -0.75) * x)) * (y + z))) + (math.pow(z, 0.25) * math.pow(y, 0.25))
	tmp = 0
	if z < 7.636950090573675e+176:
		tmp = 2.0 * math.sqrt((((x + y) * z) + (x * y)))
	else:
		tmp = (t_0 * t_0) * 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.25 * Float64(Float64((y ^ -0.75) * Float64((z ^ -0.75) * x)) * Float64(y + z))) + Float64((z ^ 0.25) * (y ^ 0.25)))
	tmp = 0.0
	if (z < 7.636950090573675e+176)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x + y) * z) + Float64(x * y))));
	else
		tmp = Float64(Float64(t_0 * t_0) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.25 * (((y ^ -0.75) * ((z ^ -0.75) * x)) * (y + z))) + ((z ^ 0.25) * (y ^ 0.25));
	tmp = 0.0;
	if (z < 7.636950090573675e+176)
		tmp = 2.0 * sqrt((((x + y) * z) + (x * y)));
	else
		tmp = (t_0 * t_0) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.25 * N[(N[(N[Power[y, -0.75], $MachinePrecision] * N[(N[Power[z, -0.75], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[z, 0.25], $MachinePrecision] * N[Power[y, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, 7.636950090573675e+176], N[(2.0 * N[Sqrt[N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 2.0), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025022 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 763695009057367500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 2 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 1/4 (* (* (pow y -3/4) (* (pow z -3/4) x)) (+ y z))) (* (pow z 1/4) (pow y 1/4))) (+ (* 1/4 (* (* (pow y -3/4) (* (pow z -3/4) x)) (+ y z))) (* (pow z 1/4) (pow y 1/4)))) 2)))

  (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))