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

Percentage Accurate: 70.5% → 83.8%

Time: 8.7s

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

real(8) function code(x, y, z)
    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: 70.5% 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

real(8) function code(x, y, z)
    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: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{z}{y}, x, x\right) \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{z} \cdot 2, \sqrt{y}, \frac{z + y}{\sqrt{y} \cdot \sqrt{z}} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6e-286)
   (* (sqrt (* (fma (/ z y) x x) y)) 2.0)
   (fma (* (sqrt z) 2.0) (sqrt y) (* (/ (+ z y) (* (sqrt y) (sqrt z))) x))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e-286) {
		tmp = sqrt((fma((z / y), x, x) * y)) * 2.0;
	} else {
		tmp = fma((sqrt(z) * 2.0), sqrt(y), (((z + y) / (sqrt(y) * sqrt(z))) * x));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6e-286)
		tmp = Float64(sqrt(Float64(fma(Float64(z / y), x, x) * y)) * 2.0);
	else
		tmp = fma(Float64(sqrt(z) * 2.0), sqrt(y), Float64(Float64(Float64(z + y) / Float64(sqrt(y) * sqrt(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, 6e-286], N[(N[Sqrt[N[(N[(N[(z / y), $MachinePrecision] * x + x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[z], $MachinePrecision] * 2.0), $MachinePrecision] * N[Sqrt[y], $MachinePrecision] + N[(N[(N[(z + y), $MachinePrecision] / N[(N[Sqrt[y], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.0000000000000001e-286

   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 y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 34.0%

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

   if 6.0000000000000001e-286 < y

   1. Initial program 70.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 22.5

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

   5. Applied rewrites 22.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{z \cdot y}} \cdot \left(z + y\right), x, \sqrt{z \cdot y} \cdot 2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 33.0%

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

   9. Applied rewrites 33.6%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{z}{y}, x, x\right) \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{z} \cdot 2, \sqrt{y}, \frac{z + y}{\sqrt{y} \cdot \sqrt{z}} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x + y, z, x \cdot y\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{z} \cdot 2, \sqrt{y}, \sqrt{\frac{z}{y}} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.6e+49)
   (* (sqrt (fma (+ x y) z (* x y))) 2.0)
   (fma (* (sqrt z) 2.0) (sqrt y) (* (sqrt (/ z y)) x))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e+49) {
		tmp = sqrt(fma((x + y), z, (x * y))) * 2.0;
	} else {
		tmp = fma((sqrt(z) * 2.0), sqrt(y), (sqrt((z / y)) * x));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.6e+49)
		tmp = Float64(sqrt(fma(Float64(x + y), z, Float64(x * y))) * 2.0);
	else
		tmp = fma(Float64(sqrt(z) * 2.0), sqrt(y), Float64(sqrt(Float64(z / y)) * 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, 1.6e+49], N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[z], $MachinePrecision] * 2.0), $MachinePrecision] * N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[N[(z / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.60000000000000007e49

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.9

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 76.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

   if 1.60000000000000007e49 < y

   1. Initial program 52.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 25.3

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

   5. Applied rewrites 25.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 24.9%

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

   10. Applied rewrites 45.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x + y, z, x \cdot y\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{z} \cdot 2, \sqrt{y}, \sqrt{\frac{z}{y}} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x + y, z, x \cdot y\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{z}{y}} \cdot 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5e-15)
   (* (sqrt (fma (+ x y) z (* x y))) 2.0)
   (* (* (sqrt (/ z y)) 2.0) y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e-15) {
		tmp = sqrt(fma((x + y), z, (x * y))) * 2.0;
	} else {
		tmp = (sqrt((z / y)) * 2.0) * y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5e-15)
		tmp = Float64(sqrt(fma(Float64(x + y), z, Float64(x * y))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(z / 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, 1.5e-15], N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(z / y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e-15

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.3

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 76.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 76.3

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

   4. Applied rewrites 76.3%

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

   if 1.5e-15 < y

   1. Initial program 58.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 29.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 39.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 39.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x + y, z, x \cdot y\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{z}{y}} \cdot 2\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, x, z \cdot y\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-281)
   (* (sqrt (* (+ z y) x)) 2.0)
   (* (sqrt (fma z x (* z y))) 2.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-281) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = sqrt(fma(z, x, (z * y))) * 2.0;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-281)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	else
		tmp = Float64(sqrt(fma(z, x, Float64(z * 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, -1e-281], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Sqrt[N[(z * x + N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e-281

   1. Initial program 71.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 45.9

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

   5. Applied rewrites 45.9%

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

   if -1e-281 < y

   1. Initial program 70.4%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 43.9

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

   5. Applied rewrites 43.9%

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

   7. Applied rewrites 43.8%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, x, z \cdot y\right)} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-281) (* (sqrt (* (+ z y) x)) 2.0) (* (sqrt (* (+ x y) z)) 2.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-281) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = sqrt(((x + y) * z)) * 2.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1d-281)) then
        tmp = sqrt(((z + y) * x)) * 2.0d0
    else
        tmp = sqrt(((x + y) * z)) * 2.0d0
    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 <= -1e-281) {
		tmp = Math.sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = Math.sqrt(((x + y) * z)) * 2.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1e-281:
		tmp = math.sqrt(((z + y) * x)) * 2.0
	else:
		tmp = math.sqrt(((x + y) * z)) * 2.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-281)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	else
		tmp = Float64(sqrt(Float64(Float64(x + y) * z)) * 2.0);
	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 <= -1e-281)
		tmp = sqrt(((z + y) * x)) * 2.0;
	else
		tmp = sqrt(((x + y) * z)) * 2.0;
	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, -1e-281], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e-281

   1. Initial program 71.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 45.9

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

   5. Applied rewrites 45.9%

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

   if -1e-281 < y

   1. Initial program 70.4%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 43.9

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

   5. Applied rewrites 43.9%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot y} \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.2e-279) (* (sqrt (* (+ z y) x)) 2.0) (* (sqrt (* z y)) 2.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.2e-279) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = sqrt((z * y)) * 2.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.2d-279) then
        tmp = sqrt(((z + y) * x)) * 2.0d0
    else
        tmp = sqrt((z * y)) * 2.0d0
    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 <= 2.2e-279) {
		tmp = Math.sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = Math.sqrt((z * y)) * 2.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.2e-279:
		tmp = math.sqrt(((z + y) * x)) * 2.0
	else:
		tmp = math.sqrt((z * y)) * 2.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.2e-279)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	else
		tmp = Float64(sqrt(Float64(z * y)) * 2.0);
	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 <= 2.2e-279)
		tmp = sqrt(((z + y) * x)) * 2.0;
	else
		tmp = sqrt((z * y)) * 2.0;
	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, 2.2e-279], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Sqrt[N[(z * y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.2e-279

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 47.5

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

   5. Applied rewrites 47.5%

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

   if 2.2e-279 < y

   1. Initial program 70.8%

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

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

      2. lower-*.f64 23.1

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

   5. Applied rewrites 23.1%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot y} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 70.9

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-rgt-out N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 71.0

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 71.0

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

4. Applied rewrites 71.0%

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

5. Final simplification 71.0%

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

Alternative 8: 68.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{x \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot y} \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.85e-285) (* (sqrt (* x y)) 2.0) (* (sqrt (* z y)) 2.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.85e-285) {
		tmp = sqrt((x * y)) * 2.0;
	} else {
		tmp = sqrt((z * y)) * 2.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.85d-285)) then
        tmp = sqrt((x * y)) * 2.0d0
    else
        tmp = sqrt((z * y)) * 2.0d0
    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 <= -1.85e-285) {
		tmp = Math.sqrt((x * y)) * 2.0;
	} else {
		tmp = Math.sqrt((z * y)) * 2.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.85e-285:
		tmp = math.sqrt((x * y)) * 2.0
	else:
		tmp = math.sqrt((z * y)) * 2.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.85e-285)
		tmp = Float64(sqrt(Float64(x * y)) * 2.0);
	else
		tmp = Float64(sqrt(Float64(z * y)) * 2.0);
	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 <= -1.85e-285)
		tmp = sqrt((x * y)) * 2.0;
	else
		tmp = sqrt((z * y)) * 2.0;
	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, -1.85e-285], N[(N[Sqrt[N[(x * y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Sqrt[N[(z * y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8499999999999999e-285

   1. Initial program 71.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.7

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

   5. Applied rewrites 20.7%

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

   if -1.8499999999999999e-285 < y

   1. Initial program 70.4%

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

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

      2. lower-*.f64 21.5

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

   5. Applied rewrites 21.5%

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

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{x \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot y} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt((x * y)) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

3. Taylor expanded in z around 0

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

   1. lower-sqrt.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 25.1

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

5. Applied rewrites 25.1%

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

6. Final simplification 25.1%

   \[\leadsto \sqrt{x \cdot y} \cdot 2 \]
7. Add Preprocessing

Developer Target 1: 82.4% 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

real(8) function code(x, y, z)
    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 2024327 
(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)))))