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

Percentage Accurate: 71.3% → 85.3%

Time: 3.4s

Alternatives: 9

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: 71.3% 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: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-291}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\sqrt{x + y}}{\sqrt{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.3e-291)
   (* 2.0 (sqrt (* (+ z y) x)))
   (* (* 2.0 (/ (sqrt (+ x y)) (sqrt z))) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.3e-291) {
		tmp = 2.0 * sqrt(((z + y) * x));
	} else {
		tmp = (2.0 * (sqrt((x + y)) / sqrt(z))) * 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 <= 4.3d-291) then
        tmp = 2.0d0 * sqrt(((z + y) * x))
    else
        tmp = (2.0d0 * (sqrt((x + y)) / sqrt(z))) * 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 <= 4.3e-291) {
		tmp = 2.0 * Math.sqrt(((z + y) * x));
	} else {
		tmp = (2.0 * (Math.sqrt((x + y)) / Math.sqrt(z))) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 4.3e-291:
		tmp = 2.0 * math.sqrt(((z + y) * x))
	else:
		tmp = (2.0 * (math.sqrt((x + y)) / math.sqrt(z))) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.3e-291)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + y) * x)));
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(Float64(x + y)) / sqrt(z))) * 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 <= 4.3e-291)
		tmp = 2.0 * sqrt(((z + y) * x));
	else
		tmp = (2.0 * (sqrt((x + y)) / sqrt(z))) * 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, 4.3e-291], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[N[(x + y), $MachinePrecision]], $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.30000000000000035e-291

   1. Initial program 69.1%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 47.6

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

   5. Applied rewrites 47.6%

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

   if 4.30000000000000035e-291 < y

   1. Initial program 71.1%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-sqrt.f64 42.6

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

   7. Applied rewrites 42.6%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 43.0

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

   10. Applied rewrites 43.0%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. sqrt-div N/A

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

       5. lower-/.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-sqrt.f64 48.4

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

   12. Applied rewrites 48.4%

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

Alternative 2: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z + x, y, z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{\frac{x + y}{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.4e-13)
   (* 2.0 (sqrt (fma (+ z x) y (* z x))))
   (* (* 2.0 (sqrt (/ (+ x y) z))) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.4e-13) {
		tmp = 2.0 * sqrt(fma((z + x), y, (z * x)));
	} else {
		tmp = (2.0 * sqrt(((x + y) / z))) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6.4e-13)
		tmp = Float64(2.0 * sqrt(fma(Float64(z + x), y, Float64(z * x))));
	else
		tmp = Float64(Float64(2.0 * sqrt(Float64(Float64(x + y) / z))) * 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, 6.4e-13], N[(2.0 * N[Sqrt[N[(N[(z + x), $MachinePrecision] * y + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[N[(N[(x + y), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.39999999999999999e-13

   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 y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   if 6.39999999999999999e-13 < y

   1. Initial program 62.7%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 39.8

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

   8. Applied rewrites 39.8%

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

Alternative 3: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z + x, y, z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.55e+16)
   (* 2.0 (sqrt (fma (+ z x) y (* z x))))
   (* (* 2.0 (sqrt (/ y z))) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.55e+16) {
		tmp = 2.0 * sqrt(fma((z + x), y, (z * x)));
	} else {
		tmp = (2.0 * sqrt((y / z))) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.55e+16)
		tmp = Float64(2.0 * sqrt(fma(Float64(z + x), y, Float64(z * x))));
	else
		tmp = Float64(Float64(2.0 * sqrt(Float64(y / z))) * 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, 1.55e+16], N[(2.0 * N[Sqrt[N[(N[(z + x), $MachinePrecision] * y + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55e16

   1. Initial program 73.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if 1.55e16 < y

   1. Initial program 59.6%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

      3. lower-/.f64 35.8

         \[\leadsto \left(2 \cdot \sqrt{\frac{y}{z}}\right) \cdot z \]

   8. Applied rewrites 35.8%

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

Alternative 4: 71.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y, z \cdot x\right)}\\ \end{array} \end{array} \]

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-296)
   (* 2.0 (sqrt (* (+ z y) x)))
   (* 2.0 (sqrt (fma z y (* z x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-296) {
		tmp = 2.0 * sqrt(((z + y) * x));
	} else {
		tmp = 2.0 * sqrt(fma(z, y, (z * x)));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-296)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + y) * x)));
	else
		tmp = Float64(2.0 * sqrt(fma(z, y, Float64(z * x))));
	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, -5e-296], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * y + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.0000000000000003e-296

   1. Initial program 68.3%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 45.7

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

   5. Applied rewrites 45.7%

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

   if -5.0000000000000003e-296 < y

   1. Initial program 71.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.0%

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

Alternative 5: 71.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

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-296) (* 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-296) {
		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-296)) 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-296) {
		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-296:
		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-296)
		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-296)
		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-296], 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 < -5.0000000000000003e-296

   1. Initial program 68.3%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 45.7

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

   5. Applied rewrites 45.7%

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

   if -5.0000000000000003e-296 < y

   1. Initial program 71.6%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 46.0

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

   5. Applied rewrites 46.0%

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

Alternative 6: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e-285) (* 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 <= -2.3e-285) {
		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 <= (-2.3d-285)) 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 <= -2.3e-285) {
		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 <= -2.3e-285:
		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 <= -2.3e-285)
		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 <= -2.3e-285)
		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, -2.3e-285], 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 < -2.29999999999999996e-285

   1. Initial program 67.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 32.9

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

   5. Applied rewrites 32.9%

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

   if -2.29999999999999996e-285 < 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 46.8

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

   5. Applied rewrites 46.8%

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

Alternative 7: 71.3% accurate, 1.2× speedup

Math

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

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 (* z x)))))

C

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

Julia

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

Derivation

1. Initial program 70.2%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 70.2

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

5. Applied rewrites 70.2%

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

Alternative 8: 69.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-311}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

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-311) (* 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-311) {
		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-311)) 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-311) {
		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-311:
		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-311)
		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-311)
		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-311], 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 < -1.9999999999999e-311

   1. Initial program 68.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.9

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

   5. Applied rewrites 31.9%

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

   if -1.9999999999999e-311 < y

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 21.3

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

   5. Applied rewrites 21.3%

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

Alternative 9: 35.6% 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 70.2%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 29.4

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

5. Applied rewrites 29.4%

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

Developer Target 1: 83.4% 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 2025080 
(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)))))