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

Percentage Accurate: 71.1% → 86.2%

Time: 10.7s

Alternatives: 8

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: 71.1% 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: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \frac{1}{y + z}\\ t_1 := \frac{1}{y \cdot z}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\mathsf{fma}\left(x, t\_1, t\_0\right)}}{\sqrt{t\_1 \cdot t\_0}}\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-272}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + \frac{y + z}{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + \mathsf{fma}\left(x, \frac{y}{z}, x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ y z))) (t_1 (/ 1.0 (* y z))))
   (if (<= y -6.5e+101)
     (* 2.0 (/ (sqrt (fma x t_1 t_0)) (sqrt (* t_1 t_0))))
     (if (<= y -2.05e-272)
       (* 2.0 (sqrt (+ (* y z) (/ (+ y z) (/ 1.0 x)))))
       (* (* 2.0 (sqrt z)) (sqrt (+ y (fma x (/ y z) x))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 1.0 / (y + z);
	double t_1 = 1.0 / (y * z);
	double tmp;
	if (y <= -6.5e+101) {
		tmp = 2.0 * (sqrt(fma(x, t_1, t_0)) / sqrt((t_1 * t_0)));
	} else if (y <= -2.05e-272) {
		tmp = 2.0 * sqrt(((y * z) + ((y + z) / (1.0 / x))));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((y + fma(x, (y / z), x)));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(1.0 / Float64(y + z))
	t_1 = Float64(1.0 / Float64(y * z))
	tmp = 0.0
	if (y <= -6.5e+101)
		tmp = Float64(2.0 * Float64(sqrt(fma(x, t_1, t_0)) / sqrt(Float64(t_1 * t_0))));
	elseif (y <= -2.05e-272)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * z) + Float64(Float64(y + z) / Float64(1.0 / x)))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(Float64(y + fma(x, Float64(y / 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_] := Block[{t$95$0 = N[(1.0 / N[(y + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+101], N[(2.0 * N[(N[Sqrt[N[(x * t$95$1 + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.05e-272], N[(2.0 * N[Sqrt[N[(N[(y * z), $MachinePrecision] + N[(N[(y + z), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(y + N[(x * N[(y / z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.50000000000000016e101

   1. Initial program 36.7%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{y \cdot y - z \cdot z}{y - z}} \cdot x + y \cdot z} \]

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 10.0

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

   4. Applied rewrites 10.0%

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

   6. Applied rewrites 36.8%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{1}{\frac{\frac{1}{y + z}}{x \cdot 1}}} + y \cdot z} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. lift-*.f64 N/A

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

      8. remove-double-div N/A

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

      9. lift-/.f64 N/A

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

      10. frac-add N/A

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

      11. sqrt-div N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 17.5%

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

   if -6.50000000000000016e101 < y < -2.0499999999999999e-272

   1. Initial program 82.2%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{y \cdot y - z \cdot z}{y - z}} \cdot x + y \cdot z} \]

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 56.3

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

   4. Applied rewrites 56.3%

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

   6. Applied rewrites 82.3%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{1}{\frac{\frac{1}{y + z}}{x \cdot 1}}} + y \cdot z} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. remove-double-div N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-rgt-identity N/A

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

      11. lower-/.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -2.0499999999999999e-272 < y

   1. Initial program 69.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 58.8

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

   5. Applied rewrites 58.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. pow1/2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 55.2

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

      14. lift-+.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. associate-+r+ N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-fma.f64 55.2

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

   7. Applied rewrites 55.2%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\mathsf{fma}\left(x, \frac{1}{y \cdot z}, \frac{1}{y + z}\right)}}{\sqrt{\frac{1}{y \cdot z} \cdot \frac{1}{y + z}}}\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-272}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + \frac{y + z}{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + \mathsf{fma}\left(x, \frac{y}{z}, x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-201}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + \frac{y + z}{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + \mathsf{fma}\left(x, \frac{y}{z}, x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.55e-201)
   (* 2.0 (sqrt (+ (* y z) (/ (+ y z) (/ 1.0 x)))))
   (* (* 2.0 (sqrt z)) (sqrt (+ y (fma x (/ y z) x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.55e-201) {
		tmp = 2.0 * sqrt(((y * z) + ((y + z) / (1.0 / x))));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((y + fma(x, (y / z), x)));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 1.55e-201)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * z) + Float64(Float64(y + z) / Float64(1.0 / x)))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(Float64(y + fma(x, Float64(y / 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[z, 1.55e-201], N[(2.0 * N[Sqrt[N[(N[(y * z), $MachinePrecision] + N[(N[(y + z), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(y + N[(x * N[(y / z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.5499999999999999e-201

   1. Initial program 70.6%

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

         \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{y \cdot y - z \cdot z}{y - z}} \cdot x + y \cdot z} \]

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 42.9

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

   4. Applied rewrites 42.9%

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

   6. Applied rewrites 70.6%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{1}{\frac{\frac{1}{y + z}}{x \cdot 1}}} + y \cdot z} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. remove-double-div N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-rgt-identity N/A

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

      11. lower-/.f64 70.6

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

   8. Applied rewrites 70.6%

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

   if 1.5499999999999999e-201 < z

   1. Initial program 61.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 57.7

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

   5. Applied rewrites 57.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. pow1/2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 87.4

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

      14. lift-+.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. associate-+r+ N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-fma.f64 87.4

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

   7. Applied rewrites 87.4%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-201}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + \frac{y + z}{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + \mathsf{fma}\left(x, \frac{y}{z}, x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-286}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.1e-286)
   (* 2.0 (sqrt (* x (+ y z))))
   (* (* 2.0 (sqrt z)) (sqrt (+ y x)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e-286) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((y + x));
	}
	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 <= 1.1d-286) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = (2.0d0 * sqrt(z)) * sqrt((y + x))
    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 <= 1.1e-286) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.sqrt((y + x));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.1e-286:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.sqrt((y + x))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.1e-286)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(Float64(y + x)));
	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 <= 1.1e-286)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = (2.0 * sqrt(z)) * sqrt((y + x));
	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, 1.1e-286], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e-286

   1. Initial program 67.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 47.7

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

   5. Applied rewrites 47.7%

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

   if 1.1e-286 < y

   1. Initial program 67.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 37.8

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

   5. Applied rewrites 37.8%

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

      1. lift-+.f64 N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 44.9

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

   7. Applied rewrites 44.9%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-286}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e-286) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt(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 <= 6d-286) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = (2.0d0 * sqrt(z)) * sqrt(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 <= 6e-286) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.sqrt(y);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 6e-286:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.sqrt(y)
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if y < 6.0000000000000001e-286

   1. Initial program 67.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 47.7

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

   5. Applied rewrites 47.7%

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

   if 6.0000000000000001e-286 < y

   1. Initial program 67.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 37.8

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

   5. Applied rewrites 37.8%

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

      1. lift-+.f64 N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 44.9

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

   7. Applied rewrites 44.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 32.3

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

   10. Applied rewrites 32.3%

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

Alternative 5: 71.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-302}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-302) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-302) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	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 <= (-5d-302)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-302) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -5e-302:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-302)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-302)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -5e-302], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000033e-302

   1. Initial program 65.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 44.1

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

   5. Applied rewrites 44.1%

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

   if -5.00000000000000033e-302 < y

   1. Initial program 68.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 42.0

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

   5. Applied rewrites 42.0%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-302}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e-286) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((y * 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 <= 6d-286) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((y * 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 <= 6e-286) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 6e-286:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if y < 6.0000000000000001e-286

   1. Initial program 67.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 47.7

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

   5. Applied rewrites 47.7%

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

   if 6.0000000000000001e-286 < y

   1. Initial program 67.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 22.0

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

   5. Applied rewrites 22.0%

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

Alternative 7: 69.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-272}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.05e-272) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.05e-272) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((y * 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 <= (-2.05d-272)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.05e-272) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2.05e-272:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.05e-272)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.05e-272)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -2.05e-272], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0499999999999999e-272

   1. Initial program 65.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 26.6

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

   5. Applied rewrites 26.6%

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

   if -2.0499999999999999e-272 < y

   1. Initial program 69.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 20.1

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

   5. Applied rewrites 20.1%

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

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-272}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 67.3%

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

3. Taylor expanded in z around 0

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

   1. lower-*.f64 26.5

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

5. Applied rewrites 26.5%

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

6. Final simplification 26.5%

   \[\leadsto 2 \cdot \sqrt{y \cdot x} \]
7. 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 2024214 
(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)))))