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

Percentage Accurate: 70.3% → 96.2%

Time: 8.8s

Alternatives: 12

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: 70.3% 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

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: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1700000000:\\ \;\;\;\;\left(\left(\sqrt{\frac{z + x}{y}} \cdot -1\right) \cdot 2\right) \cdot y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(\frac{z}{x} + 1, y, z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1700000000.0)
   (* (* (* (sqrt (/ (+ z x) y)) -1.0) 2.0) y)
   (if (<= y 2.9e-294)
     (* 2.0 (sqrt (* (fma (+ (/ z x) 1.0) y z) x)))
     (* (/ (* (sqrt y) 2.0) (sqrt z)) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1700000000.0) {
		tmp = ((sqrt(((z + x) / y)) * -1.0) * 2.0) * y;
	} else if (y <= 2.9e-294) {
		tmp = 2.0 * sqrt((fma(((z / x) + 1.0), y, z) * x));
	} else {
		tmp = ((sqrt(y) * 2.0) / sqrt(z)) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1700000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(z + x) / y)) * -1.0) * 2.0) * y);
	elseif (y <= 2.9e-294)
		tmp = Float64(2.0 * sqrt(Float64(fma(Float64(Float64(z / x) + 1.0), y, z) * x)));
	else
		tmp = Float64(Float64(Float64(sqrt(y) * 2.0) / sqrt(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, -1700000000.0], N[(N[(N[(N[Sqrt[N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision] * 2.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.9e-294], N[(2.0 * N[Sqrt[N[(N[(N[(N[(z / x), $MachinePrecision] + 1.0), $MachinePrecision] * y + z), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e9

   1. Initial program 51.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 51.2

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

      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 51.4

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

      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 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in y around -inf

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

   10. Applied rewrites 84.3%

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

   if -1.7e9 < y < 2.9000000000000001e-294

   1. Initial program 79.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if 2.9000000000000001e-294 < y

   1. Initial program 66.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 32.7%

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

   10. Applied rewrites 35.4%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1700000000:\\ \;\;\;\;\left(\left(\sqrt{\frac{z + x}{y}} \cdot -1\right) \cdot 2\right) \cdot y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(\frac{z}{x} + 1, y, z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\frac{\sqrt{-\left(x + z\right)}}{\sqrt{-y}} \cdot -1\right) \cdot 2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (* (* (* (/ (sqrt (- (+ x z))) (sqrt (- y))) -1.0) 2.0) y)
   (* (/ (* (sqrt y) 2.0) (sqrt z)) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = (((sqrt(-(x + z)) / sqrt(-y)) * -1.0) * 2.0) * y;
	} else {
		tmp = ((sqrt(y) * 2.0) / sqrt(z)) * z;
	}
	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 <= (-2d-310)) then
        tmp = (((sqrt(-(x + z)) / sqrt(-y)) * (-1.0d0)) * 2.0d0) * y
    else
        tmp = ((sqrt(y) * 2.0d0) / 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 <= -2e-310) {
		tmp = (((Math.sqrt(-(x + z)) / Math.sqrt(-y)) * -1.0) * 2.0) * y;
	} else {
		tmp = ((Math.sqrt(y) * 2.0) / Math.sqrt(z)) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2e-310:
		tmp = (((math.sqrt(-(x + z)) / math.sqrt(-y)) * -1.0) * 2.0) * y
	else:
		tmp = ((math.sqrt(y) * 2.0) / math.sqrt(z)) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(-Float64(x + z))) / sqrt(Float64(-y))) * -1.0) * 2.0) * y);
	else
		tmp = Float64(Float64(Float64(sqrt(y) * 2.0) / 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 <= -2e-310)
		tmp = (((sqrt(-(x + z)) / sqrt(-y)) * -1.0) * 2.0) * y;
	else
		tmp = ((sqrt(y) * 2.0) / 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, -2e-310], N[(N[(N[(N[(N[Sqrt[(-N[(x + z), $MachinePrecision])], $MachinePrecision] / N[Sqrt[(-y)], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] * 2.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[Sqrt[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 63.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 63.2

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

      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 63.3

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

      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 63.3

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in y around -inf

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

   10. Applied rewrites 64.9%

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

   12. Applied rewrites 69.7%

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

   if -1.999999999999994e-310 < y

   1. Initial program 66.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 36.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 32.0%

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

   10. Applied rewrites 34.7%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\frac{\sqrt{-\left(x + z\right)}}{\sqrt{-y}} \cdot -1\right) \cdot 2\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1700000000:\\ \;\;\;\;\left(\left(\sqrt{\frac{z + x}{y}} \cdot -1\right) \cdot 2\right) \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x + y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1700000000.0)
   (* (* (* (sqrt (/ (+ z x) y)) -1.0) 2.0) y)
   (if (<= y 1.9e+17)
     (* (sqrt (fma (+ y x) z (* y x))) 2.0)
     (* (* (sqrt (/ (+ x y) z)) 2.0) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1700000000.0) {
		tmp = ((sqrt(((z + x) / y)) * -1.0) * 2.0) * y;
	} else if (y <= 1.9e+17) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 2.0;
	} else {
		tmp = (sqrt(((x + y) / z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1700000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(z + x) / y)) * -1.0) * 2.0) * y);
	elseif (y <= 1.9e+17)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(x + y) / z)) * 2.0) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1700000000.0], N[(N[(N[(N[Sqrt[N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision] * 2.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.9e+17], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e9

   1. Initial program 51.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 51.2

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

      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 51.4

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

      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 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in y around -inf

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

   10. Applied rewrites 84.3%

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

   if -1.7e9 < y < 1.9e17

   1. Initial program 81.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 81.2

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

      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 81.2

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

      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 81.2

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

   4. Applied rewrites 81.2%

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

   if 1.9e17 < y

   1. Initial program 54.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1700000000:\\ \;\;\;\;\left(\left(\sqrt{\frac{z + x}{y}} \cdot -1\right) \cdot 2\right) \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x + y}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1700000000:\\ \;\;\;\;\left(\sqrt{\frac{x}{y}} \cdot -2\right) \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x + y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1700000000.0)
   (* (* (sqrt (/ x y)) -2.0) y)
   (if (<= y 1.9e+17)
     (* (sqrt (fma (+ y x) z (* y x))) 2.0)
     (* (* (sqrt (/ (+ x y) z)) 2.0) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1700000000.0) {
		tmp = (sqrt((x / y)) * -2.0) * y;
	} else if (y <= 1.9e+17) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 2.0;
	} else {
		tmp = (sqrt(((x + y) / z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1700000000.0)
		tmp = Float64(Float64(sqrt(Float64(x / y)) * -2.0) * y);
	elseif (y <= 1.9e+17)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(x + y) / z)) * 2.0) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1700000000.0], N[(N[(N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.9e+17], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e9

   1. Initial program 51.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 51.2

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

      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 51.4

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

      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 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in y around -inf

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

   10. Applied rewrites 84.3%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 40.4%

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

   if -1.7e9 < y < 1.9e17

   1. Initial program 81.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 81.2

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

      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 81.2

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

      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 81.2

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

   4. Applied rewrites 81.2%

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

   if 1.9e17 < y

   1. Initial program 54.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.8%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1700000000:\\ \;\;\;\;\left(\sqrt{\frac{x}{y}} \cdot -2\right) \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x + y}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1700000000:\\ \;\;\;\;\left(\sqrt{\frac{x}{y}} \cdot -2\right) \cdot y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{z}{y}} \cdot 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1700000000.0)
   (* (* (sqrt (/ x y)) -2.0) y)
   (if (<= y 2.4e+36)
     (* (sqrt (fma (+ y x) z (* y x))) 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 <= -1700000000.0) {
		tmp = (sqrt((x / y)) * -2.0) * y;
	} else if (y <= 2.4e+36) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 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 <= -1700000000.0)
		tmp = Float64(Float64(sqrt(Float64(x / y)) * -2.0) * y);
	elseif (y <= 2.4e+36)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 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, -1700000000.0], N[(N[(N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.4e+36], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $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 3 regimes

2. if y < -1.7e9

   1. Initial program 51.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 51.2

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

      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 51.4

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

      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 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in y around -inf

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

   10. Applied rewrites 84.3%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 40.4%

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

   if -1.7e9 < y < 2.39999999999999992e36

   1. Initial program 81.8%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 81.8

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

      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 81.9

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

      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 81.9

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

   4. Applied rewrites 81.9%

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

   if 2.39999999999999992e36 < y

   1. Initial program 51.1%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 51.1

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

      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 51.4

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

      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 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 82.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 43.9%

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

Alternative 6: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1700000000:\\ \;\;\;\;\left(\sqrt{\frac{x}{y}} \cdot -2\right) \cdot y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1700000000.0)
   (* (* (sqrt (/ x y)) -2.0) y)
   (if (<= y 2.4e+36)
     (* (sqrt (fma (+ y x) z (* y x))) 2.0)
     (* (* (sqrt (/ y z)) 2.0) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1700000000.0) {
		tmp = (sqrt((x / y)) * -2.0) * y;
	} else if (y <= 2.4e+36) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 2.0;
	} else {
		tmp = (sqrt((y / z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1700000000.0)
		tmp = Float64(Float64(sqrt(Float64(x / y)) * -2.0) * y);
	elseif (y <= 2.4e+36)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(y / z)) * 2.0) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1700000000.0], N[(N[(N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.4e+36], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e9

   1. Initial program 51.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 51.2

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

      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 51.4

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

      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 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in y around -inf

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

   10. Applied rewrites 84.3%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 40.4%

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

   if -1.7e9 < y < 2.39999999999999992e36

   1. Initial program 81.8%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 81.8

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

      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 81.9

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

      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 81.9

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

   4. Applied rewrites 81.9%

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

   if 2.39999999999999992e36 < y

   1. Initial program 51.1%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 43.9%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1700000000:\\ \;\;\;\;\left(\sqrt{\frac{x}{y}} \cdot -2\right) \cdot y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e+36)
   (* (sqrt (fma (+ y x) z (* y x))) 2.0)
   (* (* (sqrt (/ y z)) 2.0) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e+36) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 2.0;
	} else {
		tmp = (sqrt((y / z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e+36)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(y / z)) * 2.0) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2.4e+36], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999992e36

   1. Initial program 69.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 69.4

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

      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 69.5

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

      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 69.5

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

   4. Applied rewrites 69.5%

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

   if 2.39999999999999992e36 < y

   1. Initial program 51.1%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 43.9%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5.2e-271)
   (* 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 <= -5.2e-271) {
		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.
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 <= (-5.2d-271)) 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 <= -5.2e-271) {
		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 <= -5.2e-271:
		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 <= -5.2e-271)
		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 <= -5.2e-271)
		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, -5.2e-271], 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 < -5.2e-271

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \color{blue}{\sqrt{x \cdot \left(y + z\right)}} \]
   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 38.3

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

   5. Applied rewrites 38.3%

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

   if -5.2e-271 < y

   1. Initial program 67.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 47.2

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

   5. Applied rewrites 47.2%

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

Alternative 9: 69.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5.2e-271) (* 2.0 (sqrt (* (+ z 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 <= -5.2e-271) {
		tmp = 2.0 * sqrt(((z + 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.
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 <= (-5.2d-271)) then
        tmp = 2.0d0 * sqrt(((z + 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 <= -5.2e-271) {
		tmp = 2.0 * Math.sqrt(((z + 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 <= -5.2e-271:
		tmp = 2.0 * math.sqrt(((z + 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 <= -5.2e-271)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + 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 <= -5.2e-271)
		tmp = 2.0 * sqrt(((z + 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, -5.2e-271], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * 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 < -5.2e-271

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \color{blue}{\sqrt{x \cdot \left(y + z\right)}} \]
   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 38.3

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

   5. Applied rewrites 38.3%

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

   if -5.2e-271 < y

   1. Initial program 67.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 21.8

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

   5. Applied rewrites 21.8%

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

Alternative 10: 70.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 64.8%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
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 64.8

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

   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 65.0

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

   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 65.0

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

4. Applied rewrites 65.0%

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

Alternative 11: 68.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5.2e-271) (* 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 <= -5.2e-271) {
		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.
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 <= (-5.2d-271)) 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 <= -5.2e-271) {
		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 <= -5.2e-271:
		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 <= -5.2e-271)
		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 <= -5.2e-271)
		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, -5.2e-271], 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 < -5.2e-271

   1. Initial program 62.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \color{blue}{\sqrt{x \cdot y}} \]
   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 24.9

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

   5. Applied rewrites 24.9%

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

   if -5.2e-271 < y

   1. Initial program 67.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 21.8

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

   5. Applied rewrites 21.8%

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

Alternative 12: 35.9% 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.
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((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 64.8%

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

3. Taylor expanded in z around 0

   \[\leadsto 2 \cdot \color{blue}{\sqrt{x \cdot y}} \]
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 23.6

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

5. Applied rewrites 23.6%

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

Developer Target 1: 83.3% 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 2024299 
(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)))))