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

Percentage Accurate: 70.2% → 97.5%

Time: 14.5s

Alternatives: 11

Speedup: 1.0×

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: 70.2% 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: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \frac{\sqrt{z - x}}{{\left(\frac{-1}{y}\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(y + x\right)}^{0.5} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e-278)
   (* 2.0 (/ (sqrt (- z x)) (pow (/ -1.0 y) 0.5)))
   (* 2.0 (* (pow (+ y x) 0.5) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e-278) {
		tmp = 2.0 * (sqrt((z - x)) / pow((-1.0 / y), 0.5));
	} else {
		tmp = 2.0 * (pow((y + x), 0.5) * sqrt(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 <= (-3.2d-278)) then
        tmp = 2.0d0 * (sqrt((z - x)) / (((-1.0d0) / y) ** 0.5d0))
    else
        tmp = 2.0d0 * (((y + x) ** 0.5d0) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e-278) {
		tmp = 2.0 * (Math.sqrt((z - x)) / Math.pow((-1.0 / y), 0.5));
	} else {
		tmp = 2.0 * (Math.pow((y + x), 0.5) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -3.2e-278:
		tmp = 2.0 * (math.sqrt((z - x)) / math.pow((-1.0 / y), 0.5))
	else:
		tmp = 2.0 * (math.pow((y + x), 0.5) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e-278)
		tmp = Float64(2.0 * Float64(sqrt(Float64(z - x)) / (Float64(-1.0 / y) ^ 0.5)));
	else
		tmp = Float64(2.0 * Float64((Float64(y + x) ^ 0.5) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.2e-278)
		tmp = 2.0 * (sqrt((z - x)) / ((-1.0 / y) ^ 0.5));
	else
		tmp = 2.0 * (((y + x) ^ 0.5) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -3.2e-278], N[(2.0 * N[(N[Sqrt[N[(z - x), $MachinePrecision]], $MachinePrecision] / N[Power[N[(-1.0 / y), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(y + x), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.20000000000000018e-278

   1. Initial program 65.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 65.7%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 65.7%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{x \cdot y + \left(x \cdot z + y \cdot z\right)}\right)\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}\right)}\right)\right) \]

   6. Applied egg-rr 65.5%

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

   7. Taylor expanded in y around -inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{y \cdot \left(-1 \cdot x + -1 \cdot z\right)}\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{y}}{-1 \cdot x + -1 \cdot z}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{y}\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + -1 \cdot x\right)\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z - x\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(-1 \cdot z\right), x\right)\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), x\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(0 - z\right), x\right)\right)\right)\right)\right) \]

      10. --lowering--.f64 50.4%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), x\right)\right)\right)\right)\right) \]

   9. Simplified 50.4%

      \[\leadsto 2 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\frac{-1}{y}}{\left(0 - z\right) - x}}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\frac{\sqrt{\frac{-1}{y}}}{\color{blue}{\sqrt{\left(0 - z\right) - x}}}}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{\left(0 - z\right) - x}}{\color{blue}{\sqrt{\frac{-1}{y}}}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\sqrt{\left(0 - z\right) - x}\right), \color{blue}{\left(\sqrt{\frac{-1}{y}}\right)}\right)\right) \]

   11. Applied egg-rr 38.7%

       \[\leadsto 2 \cdot \color{blue}{\frac{\sqrt{z - x}}{{\left(\frac{-1}{y}\right)}^{0.5}}} \]

   if -3.20000000000000018e-278 < y

   1. Initial program 72.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(x \cdot z\right)}, \mathsf{*.f64}\left(y, z\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(z \cdot x\right), \mathsf{*.f64}\left(y, z\right)\right)\right)\right) \]

      2. *-lowering-*.f64 49.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, x\right), \mathsf{*.f64}\left(y, z\right)\right)\right)\right) \]

   5. Simplified 49.9%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(z \cdot x + y \cdot z\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(x \cdot z + y \cdot z\right)}^{\frac{1}{2}}\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(z \cdot \left(x + y\right)\right)}^{\frac{1}{2}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\left(x + y\right) \cdot z\right)}^{\frac{1}{2}}\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(x + y\right)}^{\frac{1}{2}} \cdot \color{blue}{{z}^{\frac{1}{2}}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({\left(x + y\right)}^{\frac{1}{2}}\right), \color{blue}{\left({z}^{\frac{1}{2}}\right)}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x + y\right), \frac{1}{2}\right), \left({\color{blue}{z}}^{\frac{1}{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{1}{2}\right), \left({z}^{\frac{1}{2}}\right)\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{1}{2}\right), \left(\sqrt{z}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 55.2%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(z\right)\right)\right) \]

   7. Applied egg-rr 55.2%

      \[\leadsto 2 \cdot \color{blue}{\left({\left(x + y\right)}^{0.5} \cdot \sqrt{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \frac{\sqrt{z - x}}{{\left(\frac{-1}{y}\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(y + x\right)}^{0.5} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z + x}}{y}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(y + x\right)}^{0.5} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e-278)
   (* 2.0 (pow (/ (/ 1.0 (+ z x)) y) -0.5))
   (* 2.0 (* (pow (+ y x) 0.5) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e-278) {
		tmp = 2.0 * pow(((1.0 / (z + x)) / y), -0.5);
	} else {
		tmp = 2.0 * (pow((y + x), 0.5) * sqrt(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 <= (-3.2d-278)) then
        tmp = 2.0d0 * (((1.0d0 / (z + x)) / y) ** (-0.5d0))
    else
        tmp = 2.0d0 * (((y + x) ** 0.5d0) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e-278) {
		tmp = 2.0 * Math.pow(((1.0 / (z + x)) / y), -0.5);
	} else {
		tmp = 2.0 * (Math.pow((y + x), 0.5) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -3.2e-278:
		tmp = 2.0 * math.pow(((1.0 / (z + x)) / y), -0.5)
	else:
		tmp = 2.0 * (math.pow((y + x), 0.5) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e-278)
		tmp = Float64(2.0 * (Float64(Float64(1.0 / Float64(z + x)) / y) ^ -0.5));
	else
		tmp = Float64(2.0 * Float64((Float64(y + x) ^ 0.5) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.2e-278)
		tmp = 2.0 * (((1.0 / (z + x)) / y) ^ -0.5);
	else
		tmp = 2.0 * (((y + x) ^ 0.5) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -3.2e-278], N[(2.0 * N[Power[N[(N[(1.0 / N[(z + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(y + x), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.20000000000000018e-278

   1. Initial program 65.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 65.7%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 65.7%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{x \cdot y + \left(x \cdot z + y \cdot z\right)}\right)\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}\right)}\right)\right) \]

   6. Applied egg-rr 65.5%

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

   7. Taylor expanded in y around -inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{y \cdot \left(-1 \cdot x + -1 \cdot z\right)}\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{y}}{-1 \cdot x + -1 \cdot z}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{y}\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + -1 \cdot x\right)\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z - x\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(-1 \cdot z\right), x\right)\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), x\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(0 - z\right), x\right)\right)\right)\right)\right) \]

      10. --lowering--.f64 50.4%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), x\right)\right)\right)\right)\right) \]

   9. Simplified 50.4%

      \[\leadsto 2 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\frac{-1}{y}}{\left(0 - z\right) - x}}}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{{\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\color{blue}{\frac{1}{2}}}}\right)\right) \]

       2. pow-flip N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\frac{-1}{2}}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{-1}{\left(\left(0 - z\right) - x\right) \cdot y}\right), \frac{-1}{2}\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{-1}{\left(0 - z\right) - x}}{y}\right), \frac{-1}{2}\right)\right) \]

       7. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}}{y}\right), \frac{-1}{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{1}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}}{y}\right), \frac{-1}{2}\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

       10. associate--l- N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(0 - \left(z + x\right)\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

       11. sub0-neg N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(z + x\right)\right)\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

       12. remove-double-neg N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{z + x}\right), y\right), \frac{-1}{2}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(z + x\right)\right), y\right), \frac{-1}{2}\right)\right) \]

       14. +-lowering-+.f64 50.4%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(z, x\right)\right), y\right), \frac{-1}{2}\right)\right) \]

   11. Applied egg-rr 50.4%

       \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\frac{1}{z + x}}{y}\right)}^{-0.5}} \]

   if -3.20000000000000018e-278 < y

   1. Initial program 72.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(x \cdot z\right)}, \mathsf{*.f64}\left(y, z\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(z \cdot x\right), \mathsf{*.f64}\left(y, z\right)\right)\right)\right) \]

      2. *-lowering-*.f64 49.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, x\right), \mathsf{*.f64}\left(y, z\right)\right)\right)\right) \]

   5. Simplified 49.9%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(z \cdot x + y \cdot z\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(x \cdot z + y \cdot z\right)}^{\frac{1}{2}}\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(z \cdot \left(x + y\right)\right)}^{\frac{1}{2}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\left(x + y\right) \cdot z\right)}^{\frac{1}{2}}\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(x + y\right)}^{\frac{1}{2}} \cdot \color{blue}{{z}^{\frac{1}{2}}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({\left(x + y\right)}^{\frac{1}{2}}\right), \color{blue}{\left({z}^{\frac{1}{2}}\right)}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x + y\right), \frac{1}{2}\right), \left({\color{blue}{z}}^{\frac{1}{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{1}{2}\right), \left({z}^{\frac{1}{2}}\right)\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{1}{2}\right), \left(\sqrt{z}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 55.2%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(z\right)\right)\right) \]

   7. Applied egg-rr 55.2%

      \[\leadsto 2 \cdot \color{blue}{\left({\left(x + y\right)}^{0.5} \cdot \sqrt{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z + x}}{y}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(y + x\right)}^{0.5} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z + x}}{y}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5.8e-271)
   (* 2.0 (pow (/ (/ 1.0 (+ z x)) y) -0.5))
   (* (* 2.0 (sqrt z)) (sqrt y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.8e-271) {
		tmp = 2.0 * pow(((1.0 / (z + x)) / y), -0.5);
	} 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 <= 5.8d-271) then
        tmp = 2.0d0 * (((1.0d0 / (z + x)) / y) ** (-0.5d0))
    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 <= 5.8e-271) {
		tmp = 2.0 * Math.pow(((1.0 / (z + x)) / y), -0.5);
	} 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 <= 5.8e-271:
		tmp = 2.0 * math.pow(((1.0 / (z + x)) / y), -0.5)
	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 <= 5.8e-271)
		tmp = Float64(2.0 * (Float64(Float64(1.0 / Float64(z + x)) / y) ^ -0.5));
	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 <= 5.8e-271)
		tmp = 2.0 * (((1.0 / (z + x)) / y) ^ -0.5);
	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, 5.8e-271], N[(2.0 * N[Power[N[(N[(1.0 / N[(z + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], -0.5], $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 < 5.80000000000000028e-271

   1. Initial program 68.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 68.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 68.3%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{x \cdot y + \left(x \cdot z + y \cdot z\right)}\right)\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}\right)}\right)\right) \]

   6. Applied egg-rr 68.1%

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

   7. Taylor expanded in y around -inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{y \cdot \left(-1 \cdot x + -1 \cdot z\right)}\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{y}}{-1 \cdot x + -1 \cdot z}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{y}\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + -1 \cdot x\right)\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z - x\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(-1 \cdot z\right), x\right)\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), x\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(0 - z\right), x\right)\right)\right)\right)\right) \]

      10. --lowering--.f64 44.7%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), x\right)\right)\right)\right)\right) \]

   9. Simplified 44.7%

      \[\leadsto 2 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\frac{-1}{y}}{\left(0 - z\right) - x}}}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{{\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\color{blue}{\frac{1}{2}}}}\right)\right) \]

       2. pow-flip N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\frac{-1}{2}}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{-1}{\left(\left(0 - z\right) - x\right) \cdot y}\right), \frac{-1}{2}\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{-1}{\left(0 - z\right) - x}}{y}\right), \frac{-1}{2}\right)\right) \]

       7. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}}{y}\right), \frac{-1}{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{1}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}}{y}\right), \frac{-1}{2}\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

       10. associate--l- N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(0 - \left(z + x\right)\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

       11. sub0-neg N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(z + x\right)\right)\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

       12. remove-double-neg N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{z + x}\right), y\right), \frac{-1}{2}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(z + x\right)\right), y\right), \frac{-1}{2}\right)\right) \]

       14. +-lowering-+.f64 44.9%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(z, x\right)\right), y\right), \frac{-1}{2}\right)\right) \]

   11. Applied egg-rr 44.9%

       \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\frac{1}{z + x}}{y}\right)}^{-0.5}} \]

   if 5.80000000000000028e-271 < y

   1. Initial program 70.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 70.1%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 70.1%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \frac{x \cdot x - y \cdot y}{x - y}\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \frac{1}{\frac{x - y}{x \cdot x - y \cdot y}}\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\frac{z}{\frac{x - y}{x \cdot x - y \cdot y}}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{/.f64}\left(z, \left(\frac{x - y}{x \cdot x - y \cdot y}\right)\right)\right)\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{/.f64}\left(z, \left(\frac{1}{\frac{x \cdot x - y \cdot y}{x - y}}\right)\right)\right)\right)\right) \]

      6. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{/.f64}\left(z, \left(\frac{1}{x + y}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(1, \left(x + y\right)\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 70.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot z\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot y\right)\right)\right) \]

      4. *-lowering-*.f64 24.4%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, y\right)\right)\right) \]

   9. Simplified 24.4%

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

       1. pow1/2 N/A

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

       2. unpow-prod-down N/A

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

       3. metadata-eval N/A

          \[\leadsto 2 \cdot \left({z}^{\left(\frac{1}{4} + \frac{1}{4}\right)} \cdot {y}^{\frac{1}{2}}\right) \]

       4. pow-prod-up N/A

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

       5. pow-prod-down N/A

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

       6. sqr-neg N/A

          \[\leadsto 2 \cdot \left({\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\frac{1}{4}} \cdot {y}^{\frac{1}{2}}\right) \]

       7. sub0-neg N/A

          \[\leadsto 2 \cdot \left({\left(\left(0 - z\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\frac{1}{4}} \cdot {y}^{\frac{1}{2}}\right) \]

       8. sub0-neg N/A

          \[\leadsto 2 \cdot \left({\left(\left(0 - z\right) \cdot \left(0 - z\right)\right)}^{\frac{1}{4}} \cdot {y}^{\frac{1}{2}}\right) \]

       9. pow-prod-down N/A

          \[\leadsto 2 \cdot \left(\left({\left(0 - z\right)}^{\frac{1}{4}} \cdot {\left(0 - z\right)}^{\frac{1}{4}}\right) \cdot {\color{blue}{y}}^{\frac{1}{2}}\right) \]

       10. pow-prod-up N/A

           \[\leadsto 2 \cdot \left({\left(0 - z\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)} \cdot {\color{blue}{y}}^{\frac{1}{2}}\right) \]

       11. metadata-eval N/A

           \[\leadsto 2 \cdot \left({\left(0 - z\right)}^{\frac{1}{2}} \cdot {y}^{\frac{1}{2}}\right) \]

       12. pow1/2 N/A

           \[\leadsto 2 \cdot \left(\sqrt{0 - z} \cdot {\color{blue}{y}}^{\frac{1}{2}}\right) \]

       13. associate-*r* N/A

           \[\leadsto \left(2 \cdot \sqrt{0 - z}\right) \cdot \color{blue}{{y}^{\frac{1}{2}}} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \sqrt{0 - z}\right), \color{blue}{\left({y}^{\frac{1}{2}}\right)}\right) \]

   11. Applied egg-rr 38.5%

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

Alternative 4: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x + z \cdot x\right) + y \cdot z \leq 2 \cdot 10^{+298}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\sqrt{\frac{\frac{-1}{y}}{0 - \left(z + x\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ (+ (* y x) (* z x)) (* y z)) 2e+298)
   (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
   (* 2.0 (/ 1.0 (sqrt (/ (/ -1.0 y) (- 0.0 (+ z x))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((((y * x) + (z * x)) + (y * z)) <= 2e+298) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (1.0 / sqrt(((-1.0 / y) / (0.0 - (z + 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 * x) + (z * x)) + (y * z)) <= 2d+298) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    else
        tmp = 2.0d0 * (1.0d0 / sqrt((((-1.0d0) / y) / (0.0d0 - (z + 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 * x) + (z * x)) + (y * z)) <= 2e+298) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (1.0 / Math.sqrt(((-1.0 / y) / (0.0 - (z + x)))));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (((y * x) + (z * x)) + (y * z)) <= 2e+298:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	else:
		tmp = 2.0 * (1.0 / math.sqrt(((-1.0 / y) / (0.0 - (z + x)))))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(y * x) + Float64(z * x)) + Float64(y * z)) <= 2e+298)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(1.0 / sqrt(Float64(Float64(-1.0 / y) / Float64(0.0 - Float64(z + 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 * x) + (z * x)) + (y * z)) <= 2e+298)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	else
		tmp = 2.0 * (1.0 / sqrt(((-1.0 / y) / (0.0 - (z + 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[N[(N[(N[(y * x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision], 2e+298], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[Sqrt[N[(N[(-1.0 / y), $MachinePrecision] / N[(0.0 - N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)) < 1.9999999999999999e298

   1. Initial program 96.5%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 96.5%

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

   if 1.9999999999999999e298 < (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))

   1. Initial program 10.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 11.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 11.0%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{x \cdot y + \left(x \cdot z + y \cdot z\right)}\right)\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}\right)}\right)\right) \]

   6. Applied egg-rr 11.0%

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

   7. Taylor expanded in y around -inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{y \cdot \left(-1 \cdot x + -1 \cdot z\right)}\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{y}}{-1 \cdot x + -1 \cdot z}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{y}\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + -1 \cdot x\right)\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z - x\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(-1 \cdot z\right), x\right)\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), x\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(0 - z\right), x\right)\right)\right)\right)\right) \]

      10. --lowering--.f64 16.0%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), x\right)\right)\right)\right)\right) \]

   9. Simplified 16.0%

      \[\leadsto 2 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\frac{-1}{y}}{\left(0 - z\right) - x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x + z \cdot x\right) + y \cdot z \leq 2 \cdot 10^{+298}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\sqrt{\frac{\frac{-1}{y}}{0 - \left(z + x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x + z \cdot x\right) + y \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z + x}}{y}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ (+ (* y x) (* z x)) (* y z)) 5e+305)
   (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
   (* 2.0 (pow (/ (/ 1.0 (+ z x)) y) -0.5))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((((y * x) + (z * x)) + (y * z)) <= 5e+305) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * pow(((1.0 / (z + x)) / y), -0.5);
	}
	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 * x) + (z * x)) + (y * z)) <= 5d+305) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    else
        tmp = 2.0d0 * (((1.0d0 / (z + x)) / y) ** (-0.5d0))
    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 * x) + (z * x)) + (y * z)) <= 5e+305) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * Math.pow(((1.0 / (z + x)) / y), -0.5);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (((y * x) + (z * x)) + (y * z)) <= 5e+305:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	else:
		tmp = 2.0 * math.pow(((1.0 / (z + x)) / y), -0.5)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(y * x) + Float64(z * x)) + Float64(y * z)) <= 5e+305)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / Float64(z + x)) / y) ^ -0.5));
	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 * x) + (z * x)) + (y * z)) <= 5e+305)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	else
		tmp = 2.0 * (((1.0 / (z + x)) / y) ^ -0.5);
	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[N[(N[(N[(y * x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision], 5e+305], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / N[(z + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)) < 5.00000000000000009e305

   1. Initial program 96.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 96.6%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 96.6%

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

   if 5.00000000000000009e305 < (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))

   1. Initial program 5.1%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 5.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 5.3%

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

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{x \cdot y + \left(x \cdot z + y \cdot z\right)}\right)\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}\right)\right) \]

      5. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}\right)}\right)\right) \]

   6. Applied egg-rr 5.3%

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

   7. Taylor expanded in y around -inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{y \cdot \left(-1 \cdot x + -1 \cdot z\right)}\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{y}}{-1 \cdot x + -1 \cdot z}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{y}\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + -1 \cdot x\right)\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z - x\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(-1 \cdot z\right), x\right)\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), x\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(0 - z\right), x\right)\right)\right)\right)\right) \]

      10. --lowering--.f64 11.9%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), x\right)\right)\right)\right)\right) \]

   9. Simplified 11.9%

      \[\leadsto 2 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\frac{-1}{y}}{\left(0 - z\right) - x}}}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{{\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\color{blue}{\frac{1}{2}}}}\right)\right) \]

       2. pow-flip N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\frac{-1}{2}}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{-1}{\left(\left(0 - z\right) - x\right) \cdot y}\right), \frac{-1}{2}\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{-1}{\left(0 - z\right) - x}}{y}\right), \frac{-1}{2}\right)\right) \]

       7. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}}{y}\right), \frac{-1}{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{1}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}}{y}\right), \frac{-1}{2}\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

       10. associate--l- N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(0 - \left(z + x\right)\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

       11. sub0-neg N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(z + x\right)\right)\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

       12. remove-double-neg N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{z + x}\right), y\right), \frac{-1}{2}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(z + x\right)\right), y\right), \frac{-1}{2}\right)\right) \]

       14. +-lowering-+.f64 11.9%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(z, x\right)\right), y\right), \frac{-1}{2}\right)\right) \]

   11. Applied egg-rr 11.9%

       \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\frac{1}{z + x}}{y}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x + z \cdot x\right) + y \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z + x}}{y}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{y + x}}{z}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.65e-298)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (pow (/ (/ 1.0 (+ y x)) z) -0.5))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.65e-298) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * pow(((1.0 / (y + x)) / z), -0.5);
	}
	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 <= 2.65d-298) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (((1.0d0 / (y + x)) / z) ** (-0.5d0))
    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 <= 2.65e-298) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.pow(((1.0 / (y + x)) / z), -0.5);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.65e-298:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.pow(((1.0 / (y + x)) / z), -0.5)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.65e-298)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / Float64(y + x)) / z) ^ -0.5));
	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 <= 2.65e-298)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (((1.0 / (y + x)) / z) ^ -0.5);
	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, 2.65e-298], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.65000000000000001e-298

   1. Initial program 66.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 66.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot \left(y + z\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot \left(y + z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(y + z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 47.7%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, z\right)\right)\right)\right) \]

   7. Simplified 47.7%

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

   if 2.65000000000000001e-298 < y

   1. Initial program 71.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(x \cdot z\right)}, \mathsf{*.f64}\left(y, z\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(z \cdot x\right), \mathsf{*.f64}\left(y, z\right)\right)\right)\right) \]

      2. *-lowering-*.f64 47.4%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, x\right), \mathsf{*.f64}\left(y, z\right)\right)\right)\right) \]

   5. Simplified 47.4%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot x + z \cdot y}\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \left(x + y\right)}\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\left(z \cdot \left(x + y\right)\right) \cdot 1}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\left(z \cdot \left(x + y\right)\right) \cdot \frac{x - y}{x - y}}\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{\left(z \cdot \left(x + y\right)\right) \cdot \left(x - y\right)}{x - y}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{x - y}{\left(z \cdot \left(x + y\right)\right) \cdot \left(x - y\right)}}}\right)\right) \]

      7. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{{\left(\frac{x - y}{\left(z \cdot \left(x + y\right)\right) \cdot \left(x - y\right)}\right)}^{-1}}\right)\right) \]

      8. sqrt-pow1 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\frac{x - y}{\left(z \cdot \left(x + y\right)\right) \cdot \left(x - y\right)}\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{x - y}{\left(z \cdot \left(x + y\right)\right) \cdot \left(x - y\right)}\right), \color{blue}{\left(\frac{-1}{2}\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{x - y}{\left(x - y\right) \cdot \left(z \cdot \left(x + y\right)\right)}\right), \left(\frac{-1}{2}\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{x - y}{x - y}}{z \cdot \left(x + y\right)}\right), \left(\frac{\color{blue}{-1}}{2}\right)\right)\right) \]

      12. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{1}{z \cdot \left(x + y\right)}\right), \left(\frac{-1}{2}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(z \cdot \left(x + y\right)\right)\right), \left(\frac{\color{blue}{-1}}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right), \left(\frac{-1}{2}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\frac{-1}{2}\right)\right)\right) \]

      16. metadata-eval 47.6%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \frac{-1}{2}\right)\right) \]

   7. Applied egg-rr 47.6%

      \[\leadsto 2 \cdot \color{blue}{{\left(\frac{1}{z \cdot \left(x + y\right)}\right)}^{-0.5}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{1}{\left(x + y\right) \cdot z}\right), \frac{-1}{2}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{1}{x + y}}{z}\right), \frac{-1}{2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{x + y}\right), z\right), \frac{-1}{2}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x + y\right)\right), z\right), \frac{-1}{2}\right)\right) \]

      5. +-lowering-+.f64 49.4%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), z\right), \frac{-1}{2}\right)\right) \]

   9. Applied egg-rr 49.4%

      \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\frac{1}{x + y}}{z}\right)}}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{y + x}}{z}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-278) (* 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 <= -2e-278) {
		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 <= (-2d-278)) 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 <= -2e-278) {
		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 <= -2e-278:
		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 <= -2e-278)
		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 <= -2e-278)
		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, -2e-278], 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 < -1.99999999999999988e-278

   1. Initial program 65.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 65.7%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 65.7%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot \left(y + z\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot \left(y + z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(y + z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 45.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, z\right)\right)\right)\right) \]

   7. Simplified 45.0%

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

   if -1.99999999999999988e-278 < y

   1. Initial program 72.2%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 72.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 72.3%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 50.1%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 50.1%

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

Alternative 8: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e-296) (* 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 <= 2.8e-296) {
		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 <= 2.8d-296) 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 <= 2.8e-296) {
		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 <= 2.8e-296:
		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 <= 2.8e-296)
		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 <= 2.8e-296)
		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, 2.8e-296], 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 < 2.7999999999999999e-296

   1. Initial program 67.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 67.6%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 67.6%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot \left(y + z\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot \left(y + z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(y + z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 48.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, z\right)\right)\right)\right) \]

   7. Simplified 48.9%

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

   if 2.7999999999999999e-296 < y

   1. Initial program 70.8%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 70.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 70.9%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot z\right)\right)\right) \]

      3. *-lowering-*.f64 23.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, z\right)\right)\right) \]

   7. Simplified 23.9%

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

Alternative 9: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310) (* 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 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 66.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 66.6%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 66.6%

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

   5. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot y}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 26.4%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, x\right)\right)\right) \]

   7. Simplified 26.4%

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

   if -4.999999999999985e-310 < y

   1. Initial program 71.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 71.8%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 71.8%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot z\right)\right)\right) \]

      3. *-lowering-*.f64 23.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, z\right)\right)\right) \]

   7. Simplified 23.2%

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

Alternative 10: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ 2 \cdot {\left(\frac{\frac{1}{z + x}}{y}\right)}^{-0.5} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* 2.0 (pow (/ (/ 1.0 (+ z x)) y) -0.5)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 2.0 * pow(((1.0 / (z + x)) / y), -0.5);
}

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 * (((1.0d0 / (z + x)) / y) ** (-0.5d0))
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return 2.0 * Math.pow(((1.0 / (z + x)) / y), -0.5);
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 2.0 * math.pow(((1.0 / (z + x)) / y), -0.5)

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(2.0 * (Float64(Float64(1.0 / Float64(z + x)) / y) ^ -0.5))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 2.0 * (((1.0 / (z + x)) / y) ^ -0.5);
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[Power[N[(N[(1.0 / N[(z + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.1%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f64 69.2%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

3. Simplified 69.2%

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

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{x \cdot y + \left(x \cdot z + y \cdot z\right)}\right)\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}\right)\right) \]

   5. sqrt-div N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}\right)}\right)\right) \]

6. Applied egg-rr 68.9%

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

7. Taylor expanded in y around -inf

   \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{y \cdot \left(-1 \cdot x + -1 \cdot z\right)}\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{y}}{-1 \cdot x + -1 \cdot z}\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{y}\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot x + -1 \cdot z\right)\right)\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + -1 \cdot x\right)\right)\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \left(-1 \cdot z - x\right)\right)\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(-1 \cdot z\right), x\right)\right)\right)\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), x\right)\right)\right)\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\left(0 - z\right), x\right)\right)\right)\right)\right) \]

   10. --lowering--.f64 47.5%

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, y\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), x\right)\right)\right)\right)\right) \]

9. Simplified 47.5%

   \[\leadsto 2 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\frac{-1}{y}}{\left(0 - z\right) - x}}}} \]
10. Step-by-step derivation

    1. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{{\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\color{blue}{\frac{1}{2}}}}\right)\right) \]

    2. pow-flip N/A

       \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right)}^{\frac{-1}{2}}\right)\right) \]

    4. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{-1}{y}}{\left(0 - z\right) - x}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

    5. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{-1}{\left(\left(0 - z\right) - x\right) \cdot y}\right), \frac{-1}{2}\right)\right) \]

    6. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{-1}{\left(0 - z\right) - x}}{y}\right), \frac{-1}{2}\right)\right) \]

    7. frac-2neg N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}}{y}\right), \frac{-1}{2}\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\frac{\frac{1}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}}{y}\right), \frac{-1}{2}\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(\left(0 - z\right) - x\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

    10. associate--l- N/A

        \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(0 - \left(z + x\right)\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

    11. sub0-neg N/A

        \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(z + x\right)\right)\right)\right)}\right), y\right), \frac{-1}{2}\right)\right) \]

    12. remove-double-neg N/A

        \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{z + x}\right), y\right), \frac{-1}{2}\right)\right) \]

    13. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(z + x\right)\right), y\right), \frac{-1}{2}\right)\right) \]

    14. +-lowering-+.f64 47.6%

        \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(z, x\right)\right), y\right), \frac{-1}{2}\right)\right) \]

11. Applied egg-rr 47.6%

    \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\frac{1}{z + x}}{y}\right)}^{-0.5}} \]
12. Add Preprocessing

Alternative 11: 35.4% accurate, 1.1× 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 69.1%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f64 69.2%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

3. Simplified 69.2%

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

5. Taylor expanded in z around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot y}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 25.5%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, x\right)\right)\right) \]

7. Simplified 25.5%

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

Developer Target 1: 82.2% accurate, 0.1× 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 2024191 
(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)))))