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

Percentage Accurate: 71.2% → 94.6%

Time: 16.7s

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: 71.2% 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: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -62000000:\\ \;\;\;\;2 \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{\left(x + z\right) \cdot {y}^{3}}}\right) - \sqrt{\frac{x + z}{y}}\right)\right)\\ \mathbf{elif}\;y \leq -4.25 \cdot 10^{-187}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -1.38 \cdot 10^{-258}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -62000000.0)
   (*
    2.0
    (*
     y
     (-
      (* 0.5 (* (* x z) (sqrt (/ 1.0 (* (+ x z) (pow y 3.0))))))
      (sqrt (/ (+ x z) y)))))
   (if (<= y -4.25e-187)
     (* 2.0 (sqrt (* x (+ y z))))
     (if (<= y -1.38e-258)
       (* 2.0 (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0))
       (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -62000000.0) {
		tmp = 2.0 * (y * ((0.5 * ((x * z) * sqrt((1.0 / ((x + z) * pow(y, 3.0)))))) - sqrt(((x + z) / y))));
	} else if (y <= -4.25e-187) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -1.38e-258) {
		tmp = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	} else {
		tmp = 2.0 * (sqrt((y + x)) * 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 <= (-62000000.0d0)) then
        tmp = 2.0d0 * (y * ((0.5d0 * ((x * z) * sqrt((1.0d0 / ((x + z) * (y ** 3.0d0)))))) - sqrt(((x + z) / y))))
    else if (y <= (-4.25d-187)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-1.38d-258)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log((-y - z)) - log(((-1.0d0) / x))))) ** 2.0d0)
    else
        tmp = 2.0d0 * (sqrt((y + x)) * 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 <= -62000000.0) {
		tmp = 2.0 * (y * ((0.5 * ((x * z) * Math.sqrt((1.0 / ((x + z) * Math.pow(y, 3.0)))))) - Math.sqrt(((x + z) / y))));
	} else if (y <= -4.25e-187) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -1.38e-258) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((-y - z)) - Math.log((-1.0 / x))))), 2.0);
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -62000000.0:
		tmp = 2.0 * (y * ((0.5 * ((x * z) * math.sqrt((1.0 / ((x + z) * math.pow(y, 3.0)))))) - math.sqrt(((x + z) / y))))
	elif y <= -4.25e-187:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -1.38e-258:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log((-y - z)) - math.log((-1.0 / x))))), 2.0)
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -62000000.0)
		tmp = Float64(2.0 * Float64(y * Float64(Float64(0.5 * Float64(Float64(x * z) * sqrt(Float64(1.0 / Float64(Float64(x + z) * (y ^ 3.0)))))) - sqrt(Float64(Float64(x + z) / y)))));
	elseif (y <= -4.25e-187)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -1.38e-258)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * 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 <= -62000000.0)
		tmp = 2.0 * (y * ((0.5 * ((x * z) * sqrt((1.0 / ((x + z) * (y ^ 3.0)))))) - sqrt(((x + z) / y))));
	elseif (y <= -4.25e-187)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -1.38e-258)
		tmp = 2.0 * (exp((0.25 * (log((-y - z)) - log((-1.0 / x))))) ^ 2.0);
	else
		tmp = 2.0 * (sqrt((y + x)) * 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, -62000000.0], N[(2.0 * N[(y * N[(N[(0.5 * N[(N[(x * z), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(x + z), $MachinePrecision] * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4.25e-187], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.38e-258], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.2e7

   1. Initial program 63.2%

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

      1. +-commutative 63.2%

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

      2. associate-+r+ 63.2%

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

      3. *-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. +-commutative 63.2%

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

      6. *-commutative 63.2%

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

      7. *-commutative 63.2%

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

      8. associate-+l+ 63.2%

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

      9. +-commutative 63.2%

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

      10. *-commutative 63.2%

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

      11. associate-+l+ 63.2%

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

      12. *-commutative 63.2%

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

      13. *-commutative 63.2%

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

      14. +-commutative 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in y around inf 0.9%

      \[\leadsto 2 \cdot \color{blue}{\left(y \cdot \left(\sqrt{\frac{x + z}{y}} + 0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(x + z\right)}}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 0.9%

         \[\leadsto 2 \cdot \left(y \cdot \left(\sqrt{\frac{\color{blue}{z + x}}{y}} + 0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(x + z\right)}}\right)\right)\right) \]

      2. *-commutative 0.9%

         \[\leadsto 2 \cdot \left(y \cdot \left(\sqrt{\frac{z + x}{y}} + 0.5 \cdot \left(\color{blue}{\left(z \cdot x\right)} \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(x + z\right)}}\right)\right)\right) \]

      3. +-commutative 0.9%

         \[\leadsto 2 \cdot \left(y \cdot \left(\sqrt{\frac{z + x}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \color{blue}{\left(z + x\right)}}}\right)\right)\right) \]

   7. Simplified 0.9%

      \[\leadsto 2 \cdot \color{blue}{\left(y \cdot \left(\sqrt{\frac{z + x}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right)} \]

   8. Taylor expanded in y around -inf 0.0%

      \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{\sqrt{\frac{x + z}{y}} \cdot {\left(\sqrt{-1}\right)}^{2}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{x + z}{y}}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

      2. unpow2 0.0%

         \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{x + z}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

      3. rem-square-sqrt 80.6%

         \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{x + z}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

   10. Simplified 80.6%

       \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{-1 \cdot \sqrt{\frac{x + z}{y}}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

   if -6.2e7 < y < -4.2499999999999999e-187

   1. Initial program 89.2%

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

      1. +-commutative 89.2%

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

      2. associate-+r+ 89.2%

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

      3. *-commutative 89.2%

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

      4. +-commutative 89.2%

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

      5. +-commutative 89.2%

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

      6. *-commutative 89.2%

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

      7. *-commutative 89.2%

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

      8. associate-+l+ 89.2%

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

      9. +-commutative 89.2%

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

      10. *-commutative 89.2%

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

      11. associate-+l+ 89.2%

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

      12. *-commutative 89.2%

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

      13. *-commutative 89.2%

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

      14. +-commutative 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in x around inf 63.3%

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

   if -4.2499999999999999e-187 < y < -1.38e-258

   1. Initial program 54.3%

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

      1. +-commutative 54.3%

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

      2. associate-+r+ 54.3%

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

      3. *-commutative 54.3%

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

      4. +-commutative 54.3%

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

      5. +-commutative 54.3%

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

      6. *-commutative 54.3%

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

      7. *-commutative 54.3%

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

      8. associate-+l+ 54.3%

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

      9. +-commutative 54.3%

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

      10. *-commutative 54.3%

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

      11. associate-+l+ 54.3%

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

      12. *-commutative 54.3%

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

      13. *-commutative 54.3%

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

      14. +-commutative 54.3%

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

   3. Simplified 54.3%

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

      1. add-sqr-sqrt 54.3%

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

      2. pow2 54.3%

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

      3. pow1/2 54.3%

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

      4. sqrt-pow1 54.3%

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

      5. distribute-rgt-in 54.3%

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

      6. associate-+r+ 54.3%

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

      7. *-commutative 54.3%

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

      8. distribute-lft-in 54.3%

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

      9. fma-define 54.3%

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

      10. metadata-eval 54.3%

          \[\leadsto 2 \cdot {\left({\left(\mathsf{fma}\left(y, x + z, x \cdot z\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   6. Applied egg-rr 54.3%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y, x + z, x \cdot z\right)\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in x around -inf 45.6%

      \[\leadsto 2 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}}^{2} \]

   if -1.38e-258 < y

   1. Initial program 76.2%

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

      1. +-commutative 76.2%

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

      2. associate-+r+ 76.2%

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

      3. *-commutative 76.2%

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

      4. +-commutative 76.2%

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

      5. +-commutative 76.2%

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

      6. *-commutative 76.2%

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

      7. *-commutative 76.2%

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

      8. associate-+l+ 76.2%

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

      9. +-commutative 76.2%

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

      10. *-commutative 76.2%

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

      11. associate-+l+ 76.2%

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

      12. *-commutative 76.2%

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

      13. *-commutative 76.2%

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

      14. +-commutative 76.2%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in z around inf 54.8%

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

      1. +-commutative 54.8%

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

   7. Simplified 54.8%

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

      1. *-commutative 54.8%

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

      2. sqrt-prod 57.9%

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

   9. Applied egg-rr 57.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -62000000:\\ \;\;\;\;2 \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{\left(x + z\right) \cdot {y}^{3}}}\right) - \sqrt{\frac{x + z}{y}}\right)\right)\\ \mathbf{elif}\;y \leq -4.25 \cdot 10^{-187}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -1.38 \cdot 10^{-258}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -90000000:\\ \;\;\;\;2 \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{\left(x + z\right) \cdot {y}^{3}}}\right) - \sqrt{\frac{x + z}{y}}\right)\right)\\ \mathbf{elif}\;y \leq -4.25 \cdot 10^{-187}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-262}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -90000000.0)
   (*
    2.0
    (*
     y
     (-
      (* 0.5 (* (* x z) (sqrt (/ 1.0 (* (+ x z) (pow y 3.0))))))
      (sqrt (/ (+ x z) y)))))
   (if (<= y -4.25e-187)
     (* 2.0 (sqrt (* x (+ y z))))
     (if (<= y -8.2e-262)
       (* 2.0 (pow (exp (* 0.25 (- (log (- (- z) x)) (log (/ -1.0 y))))) 2.0))
       (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -90000000.0) {
		tmp = 2.0 * (y * ((0.5 * ((x * z) * sqrt((1.0 / ((x + z) * pow(y, 3.0)))))) - sqrt(((x + z) / y))));
	} else if (y <= -4.25e-187) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -8.2e-262) {
		tmp = 2.0 * pow(exp((0.25 * (log((-z - x)) - log((-1.0 / y))))), 2.0);
	} else {
		tmp = 2.0 * (sqrt((y + x)) * 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 <= (-90000000.0d0)) then
        tmp = 2.0d0 * (y * ((0.5d0 * ((x * z) * sqrt((1.0d0 / ((x + z) * (y ** 3.0d0)))))) - sqrt(((x + z) / y))))
    else if (y <= (-4.25d-187)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-8.2d-262)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log((-z - x)) - log(((-1.0d0) / y))))) ** 2.0d0)
    else
        tmp = 2.0d0 * (sqrt((y + x)) * 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 <= -90000000.0) {
		tmp = 2.0 * (y * ((0.5 * ((x * z) * Math.sqrt((1.0 / ((x + z) * Math.pow(y, 3.0)))))) - Math.sqrt(((x + z) / y))));
	} else if (y <= -4.25e-187) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -8.2e-262) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((-z - x)) - Math.log((-1.0 / y))))), 2.0);
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -90000000.0:
		tmp = 2.0 * (y * ((0.5 * ((x * z) * math.sqrt((1.0 / ((x + z) * math.pow(y, 3.0)))))) - math.sqrt(((x + z) / y))))
	elif y <= -4.25e-187:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -8.2e-262:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log((-z - x)) - math.log((-1.0 / y))))), 2.0)
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -90000000.0)
		tmp = Float64(2.0 * Float64(y * Float64(Float64(0.5 * Float64(Float64(x * z) * sqrt(Float64(1.0 / Float64(Float64(x + z) * (y ^ 3.0)))))) - sqrt(Float64(Float64(x + z) / y)))));
	elseif (y <= -4.25e-187)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -8.2e-262)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-z) - x)) - log(Float64(-1.0 / y))))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * 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 <= -90000000.0)
		tmp = 2.0 * (y * ((0.5 * ((x * z) * sqrt((1.0 / ((x + z) * (y ^ 3.0)))))) - sqrt(((x + z) / y))));
	elseif (y <= -4.25e-187)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -8.2e-262)
		tmp = 2.0 * (exp((0.25 * (log((-z - x)) - log((-1.0 / y))))) ^ 2.0);
	else
		tmp = 2.0 * (sqrt((y + x)) * 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, -90000000.0], N[(2.0 * N[(y * N[(N[(0.5 * N[(N[(x * z), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(x + z), $MachinePrecision] * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4.25e-187], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -8.2e-262], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[((-z) - x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9e7

   1. Initial program 63.2%

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

      1. +-commutative 63.2%

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

      2. associate-+r+ 63.2%

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

      3. *-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. +-commutative 63.2%

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

      6. *-commutative 63.2%

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

      7. *-commutative 63.2%

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

      8. associate-+l+ 63.2%

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

      9. +-commutative 63.2%

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

      10. *-commutative 63.2%

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

      11. associate-+l+ 63.2%

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

      12. *-commutative 63.2%

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

      13. *-commutative 63.2%

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

      14. +-commutative 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in y around inf 0.9%

      \[\leadsto 2 \cdot \color{blue}{\left(y \cdot \left(\sqrt{\frac{x + z}{y}} + 0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(x + z\right)}}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 0.9%

         \[\leadsto 2 \cdot \left(y \cdot \left(\sqrt{\frac{\color{blue}{z + x}}{y}} + 0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(x + z\right)}}\right)\right)\right) \]

      2. *-commutative 0.9%

         \[\leadsto 2 \cdot \left(y \cdot \left(\sqrt{\frac{z + x}{y}} + 0.5 \cdot \left(\color{blue}{\left(z \cdot x\right)} \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(x + z\right)}}\right)\right)\right) \]

      3. +-commutative 0.9%

         \[\leadsto 2 \cdot \left(y \cdot \left(\sqrt{\frac{z + x}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \color{blue}{\left(z + x\right)}}}\right)\right)\right) \]

   7. Simplified 0.9%

      \[\leadsto 2 \cdot \color{blue}{\left(y \cdot \left(\sqrt{\frac{z + x}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right)} \]

   8. Taylor expanded in y around -inf 0.0%

      \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{\sqrt{\frac{x + z}{y}} \cdot {\left(\sqrt{-1}\right)}^{2}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{x + z}{y}}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

      2. unpow2 0.0%

         \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{x + z}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

      3. rem-square-sqrt 80.6%

         \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{x + z}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

   10. Simplified 80.6%

       \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{-1 \cdot \sqrt{\frac{x + z}{y}}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

   if -9e7 < y < -4.2499999999999999e-187

   1. Initial program 89.2%

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

      1. +-commutative 89.2%

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

      2. associate-+r+ 89.2%

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

      3. *-commutative 89.2%

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

      4. +-commutative 89.2%

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

      5. +-commutative 89.2%

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

      6. *-commutative 89.2%

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

      7. *-commutative 89.2%

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

      8. associate-+l+ 89.2%

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

      9. +-commutative 89.2%

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

      10. *-commutative 89.2%

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

      11. associate-+l+ 89.2%

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

      12. *-commutative 89.2%

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

      13. *-commutative 89.2%

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

      14. +-commutative 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in x around inf 63.3%

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

   if -4.2499999999999999e-187 < y < -8.20000000000000052e-262

   1. Initial program 54.3%

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

      1. +-commutative 54.3%

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

      2. associate-+r+ 54.3%

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

      3. *-commutative 54.3%

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

      4. +-commutative 54.3%

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

      5. +-commutative 54.3%

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

      6. *-commutative 54.3%

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

      7. *-commutative 54.3%

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

      8. associate-+l+ 54.3%

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

      9. +-commutative 54.3%

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

      10. *-commutative 54.3%

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

      11. associate-+l+ 54.3%

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

      12. *-commutative 54.3%

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

      13. *-commutative 54.3%

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

      14. +-commutative 54.3%

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

   3. Simplified 54.3%

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

      1. add-sqr-sqrt 54.3%

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

      2. pow2 54.3%

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

      3. pow1/2 54.3%

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

      4. sqrt-pow1 54.3%

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

      5. distribute-rgt-in 54.3%

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

      6. associate-+r+ 54.3%

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

      7. *-commutative 54.3%

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

      8. distribute-lft-in 54.3%

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

      9. fma-define 54.3%

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

      10. metadata-eval 54.3%

          \[\leadsto 2 \cdot {\left({\left(\mathsf{fma}\left(y, x + z, x \cdot z\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   6. Applied egg-rr 54.3%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y, x + z, x \cdot z\right)\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in y around -inf 32.2%

      \[\leadsto 2 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-1 \cdot \left(x + z\right)\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right)}\right)}}^{2} \]

   if -8.20000000000000052e-262 < y

   1. Initial program 76.2%

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

      1. +-commutative 76.2%

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

      2. associate-+r+ 76.2%

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

      3. *-commutative 76.2%

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

      4. +-commutative 76.2%

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

      5. +-commutative 76.2%

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

      6. *-commutative 76.2%

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

      7. *-commutative 76.2%

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

      8. associate-+l+ 76.2%

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

      9. +-commutative 76.2%

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

      10. *-commutative 76.2%

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

      11. associate-+l+ 76.2%

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

      12. *-commutative 76.2%

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

      13. *-commutative 76.2%

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

      14. +-commutative 76.2%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in z around inf 54.8%

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

      1. +-commutative 54.8%

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

   7. Simplified 54.8%

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

      1. *-commutative 54.8%

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

      2. sqrt-prod 57.9%

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

   9. Applied egg-rr 57.9%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -90000000:\\ \;\;\;\;2 \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{\left(x + z\right) \cdot {y}^{3}}}\right) - \sqrt{\frac{x + z}{y}}\right)\right)\\ \mathbf{elif}\;y \leq -4.25 \cdot 10^{-187}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-262}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -46000000:\\ \;\;\;\;2 \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{\left(x + z\right) \cdot {y}^{3}}}\right) - \sqrt{\frac{x + z}{y}}\right)\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -46000000.0)
   (*
    2.0
    (*
     y
     (-
      (* 0.5 (* (* x z) (sqrt (/ 1.0 (* (+ x z) (pow y 3.0))))))
      (sqrt (/ (+ x z) y)))))
   (if (<= y 6.6e-281)
     (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
     (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -46000000.0) {
		tmp = 2.0 * (y * ((0.5 * ((x * z) * sqrt((1.0 / ((x + z) * pow(y, 3.0)))))) - sqrt(((x + z) / y))));
	} else if (y <= 6.6e-281) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * 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 <= (-46000000.0d0)) then
        tmp = 2.0d0 * (y * ((0.5d0 * ((x * z) * sqrt((1.0d0 / ((x + z) * (y ** 3.0d0)))))) - sqrt(((x + z) / y))))
    else if (y <= 6.6d-281) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * 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 <= -46000000.0) {
		tmp = 2.0 * (y * ((0.5 * ((x * z) * Math.sqrt((1.0 / ((x + z) * Math.pow(y, 3.0)))))) - Math.sqrt(((x + z) / y))));
	} else if (y <= 6.6e-281) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -46000000.0:
		tmp = 2.0 * (y * ((0.5 * ((x * z) * math.sqrt((1.0 / ((x + z) * math.pow(y, 3.0)))))) - math.sqrt(((x + z) / y))))
	elif y <= 6.6e-281:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -46000000.0)
		tmp = Float64(2.0 * Float64(y * Float64(Float64(0.5 * Float64(Float64(x * z) * sqrt(Float64(1.0 / Float64(Float64(x + z) * (y ^ 3.0)))))) - sqrt(Float64(Float64(x + z) / y)))));
	elseif (y <= 6.6e-281)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * 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 <= -46000000.0)
		tmp = 2.0 * (y * ((0.5 * ((x * z) * sqrt((1.0 / ((x + z) * (y ^ 3.0)))))) - sqrt(((x + z) / y))));
	elseif (y <= 6.6e-281)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	else
		tmp = 2.0 * (sqrt((y + x)) * 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, -46000000.0], N[(2.0 * N[(y * N[(N[(0.5 * N[(N[(x * z), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(x + z), $MachinePrecision] * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.6e-281], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6e7

   1. Initial program 63.2%

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

      1. +-commutative 63.2%

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

      2. associate-+r+ 63.2%

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

      3. *-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. +-commutative 63.2%

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

      6. *-commutative 63.2%

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

      7. *-commutative 63.2%

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

      8. associate-+l+ 63.2%

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

      9. +-commutative 63.2%

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

      10. *-commutative 63.2%

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

      11. associate-+l+ 63.2%

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

      12. *-commutative 63.2%

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

      13. *-commutative 63.2%

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

      14. +-commutative 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in y around inf 0.9%

      \[\leadsto 2 \cdot \color{blue}{\left(y \cdot \left(\sqrt{\frac{x + z}{y}} + 0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(x + z\right)}}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 0.9%

         \[\leadsto 2 \cdot \left(y \cdot \left(\sqrt{\frac{\color{blue}{z + x}}{y}} + 0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(x + z\right)}}\right)\right)\right) \]

      2. *-commutative 0.9%

         \[\leadsto 2 \cdot \left(y \cdot \left(\sqrt{\frac{z + x}{y}} + 0.5 \cdot \left(\color{blue}{\left(z \cdot x\right)} \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(x + z\right)}}\right)\right)\right) \]

      3. +-commutative 0.9%

         \[\leadsto 2 \cdot \left(y \cdot \left(\sqrt{\frac{z + x}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \color{blue}{\left(z + x\right)}}}\right)\right)\right) \]

   7. Simplified 0.9%

      \[\leadsto 2 \cdot \color{blue}{\left(y \cdot \left(\sqrt{\frac{z + x}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right)} \]

   8. Taylor expanded in y around -inf 0.0%

      \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{\sqrt{\frac{x + z}{y}} \cdot {\left(\sqrt{-1}\right)}^{2}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{x + z}{y}}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

      2. unpow2 0.0%

         \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{x + z}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

      3. rem-square-sqrt 80.6%

         \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{x + z}{y}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

   10. Simplified 80.6%

       \[\leadsto 2 \cdot \left(y \cdot \left(\color{blue}{-1 \cdot \sqrt{\frac{x + z}{y}}} + 0.5 \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{{y}^{3} \cdot \left(z + x\right)}}\right)\right)\right) \]

   if -4.6e7 < y < 6.5999999999999998e-281

   1. Initial program 81.1%

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

      1. +-commutative 81.1%

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

      2. associate-+r+ 81.1%

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

      3. *-commutative 81.1%

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

      4. +-commutative 81.1%

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

      5. +-commutative 81.1%

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

      6. *-commutative 81.1%

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

      7. *-commutative 81.1%

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

      8. associate-+l+ 81.1%

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

      9. +-commutative 81.1%

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

      10. *-commutative 81.1%

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

      11. associate-+l+ 81.1%

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

      12. *-commutative 81.1%

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

      13. *-commutative 81.1%

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

      14. +-commutative 81.1%

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

   3. Simplified 81.1%

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

   if 6.5999999999999998e-281 < y

   1. Initial program 75.9%

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

      1. +-commutative 75.9%

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

      2. associate-+r+ 75.9%

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

      3. *-commutative 75.9%

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

      4. +-commutative 75.9%

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

      5. +-commutative 75.9%

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

      6. *-commutative 75.9%

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

      7. *-commutative 75.9%

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

      8. associate-+l+ 75.9%

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

      9. +-commutative 75.9%

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

      10. *-commutative 75.9%

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

      11. associate-+l+ 75.9%

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

      12. *-commutative 75.9%

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

      13. *-commutative 75.9%

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

      14. +-commutative 75.9%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in z around inf 52.3%

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

      1. +-commutative 52.3%

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

   7. Simplified 52.3%

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

      1. *-commutative 52.3%

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

      2. sqrt-prod 56.5%

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

   9. Applied egg-rr 56.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -46000000:\\ \;\;\;\;2 \cdot \left(y \cdot \left(0.5 \cdot \left(\left(x \cdot z\right) \cdot \sqrt{\frac{1}{\left(x + z\right) \cdot {y}^{3}}}\right) - \sqrt{\frac{x + z}{y}}\right)\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.6e-281)
   (* 2.0 (sqrt (fma x z (* y (+ x z)))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.6e-281) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (x + z))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6.6e-281)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(x + z)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 6.6e-281], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5999999999999998e-281

   1. Initial program 73.0%

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

      1. associate-+l+ 73.0%

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

      2. *-commutative 73.0%

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

      3. *-commutative 73.0%

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

      4. *-commutative 73.0%

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

      5. +-commutative 73.0%

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

      6. +-commutative 73.0%

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

      7. +-commutative 73.0%

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

      8. *-commutative 73.0%

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

      9. *-commutative 73.0%

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

      10. associate-+l+ 73.0%

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

      11. +-commutative 73.0%

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

      12. fma-define 73.0%

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

      13. distribute-lft-out 73.0%

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

   3. Simplified 73.0%

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

   if 6.5999999999999998e-281 < y

   1. Initial program 75.9%

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

      1. +-commutative 75.9%

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

      2. associate-+r+ 75.9%

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

      3. *-commutative 75.9%

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

      4. +-commutative 75.9%

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

      5. +-commutative 75.9%

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

      6. *-commutative 75.9%

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

      7. *-commutative 75.9%

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

      8. associate-+l+ 75.9%

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

      9. +-commutative 75.9%

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

      10. *-commutative 75.9%

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

      11. associate-+l+ 75.9%

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

      12. *-commutative 75.9%

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

      13. *-commutative 75.9%

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

      14. +-commutative 75.9%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in z around inf 52.3%

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

      1. +-commutative 52.3%

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

   7. Simplified 52.3%

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

      1. *-commutative 52.3%

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

      2. sqrt-prod 56.5%

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

   9. Applied egg-rr 56.5%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.6e-281)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.6e-281) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * 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 <= 6.6d-281) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * 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 <= 6.6e-281) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 6.6e-281:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6.6e-281)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * 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 <= 6.6e-281)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt((y + x)) * 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, 6.6e-281], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5999999999999998e-281

   1. Initial program 73.0%

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

      1. +-commutative 73.0%

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

      2. associate-+r+ 73.0%

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

      3. *-commutative 73.0%

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

      4. +-commutative 73.0%

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

      5. +-commutative 73.0%

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

      6. *-commutative 73.0%

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

      7. *-commutative 73.0%

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

      8. associate-+l+ 73.0%

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

      9. +-commutative 73.0%

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

      10. *-commutative 73.0%

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

      11. associate-+l+ 73.0%

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

      12. *-commutative 73.0%

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

      13. *-commutative 73.0%

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

      14. +-commutative 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in x around inf 50.8%

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

   if 6.5999999999999998e-281 < y

   1. Initial program 75.9%

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

      1. +-commutative 75.9%

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

      2. associate-+r+ 75.9%

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

      3. *-commutative 75.9%

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

      4. +-commutative 75.9%

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

      5. +-commutative 75.9%

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

      6. *-commutative 75.9%

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

      7. *-commutative 75.9%

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

      8. associate-+l+ 75.9%

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

      9. +-commutative 75.9%

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

      10. *-commutative 75.9%

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

      11. associate-+l+ 75.9%

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

      12. *-commutative 75.9%

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

      13. *-commutative 75.9%

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

      14. +-commutative 75.9%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in z around inf 52.3%

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

      1. +-commutative 52.3%

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

   7. Simplified 52.3%

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

      1. *-commutative 52.3%

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

      2. sqrt-prod 56.5%

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

   9. Applied egg-rr 56.5%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{+35}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 7.6e+35)
   (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.6e+35) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} 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 <= 7.6d+35) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    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 <= 7.6e+35) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} 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 <= 7.6e+35:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	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 <= 7.6e+35)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(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 <= 7.6e+35)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	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, 7.6e+35], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5999999999999999e35

   1. Initial program 78.8%

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

      1. +-commutative 78.8%

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

      2. associate-+r+ 78.8%

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

      3. *-commutative 78.8%

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

      4. +-commutative 78.8%

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

      5. +-commutative 78.8%

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

      6. *-commutative 78.8%

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

      7. *-commutative 78.8%

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

      8. associate-+l+ 78.8%

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

      9. +-commutative 78.8%

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

      10. *-commutative 78.8%

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

      11. associate-+l+ 78.8%

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

      12. *-commutative 78.8%

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

      13. *-commutative 78.8%

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

      14. +-commutative 78.8%

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

   3. Simplified 78.8%

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

   if 7.5999999999999999e35 < y

   1. Initial program 60.2%

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

      1. +-commutative 60.2%

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

      2. associate-+r+ 60.2%

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

      3. *-commutative 60.2%

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

      4. +-commutative 60.2%

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

      5. +-commutative 60.2%

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

      6. *-commutative 60.2%

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

      7. *-commutative 60.2%

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

      8. associate-+l+ 60.2%

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

      9. +-commutative 60.2%

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

      10. *-commutative 60.2%

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

      11. associate-+l+ 60.2%

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

      12. *-commutative 60.2%

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

      13. *-commutative 60.2%

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

      14. +-commutative 60.2%

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

   3. Simplified 60.4%

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

      1. add-sqr-sqrt 60.1%

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

      2. pow2 60.1%

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

      3. pow1/2 60.1%

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

      4. sqrt-pow1 60.1%

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

      5. distribute-rgt-in 59.9%

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

      6. associate-+r+ 59.9%

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

      7. *-commutative 59.9%

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

      8. distribute-lft-in 59.9%

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

      9. fma-define 60.3%

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

      10. metadata-eval 60.3%

          \[\leadsto 2 \cdot {\left({\left(\mathsf{fma}\left(y, x + z, x \cdot z\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   6. Applied egg-rr 60.3%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y, x + z, x \cdot z\right)\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in x around 0 31.0%

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

      1. *-commutative 31.0%

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

   9. Simplified 31.0%

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

       1. sqrt-prod 53.1%

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

   11. Applied egg-rr 53.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{+35}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.6e-281) (* 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 <= 6.6e-281) {
		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 <= 6.6d-281) 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 <= 6.6e-281) {
		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 <= 6.6e-281:
		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 <= 6.6e-281)
		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 <= 6.6e-281)
		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, 6.6e-281], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5999999999999998e-281

   1. Initial program 73.0%

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

      1. +-commutative 73.0%

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

      2. associate-+r+ 73.0%

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

      3. *-commutative 73.0%

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

      4. +-commutative 73.0%

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

      5. +-commutative 73.0%

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

      6. *-commutative 73.0%

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

      7. *-commutative 73.0%

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

      8. associate-+l+ 73.0%

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

      9. +-commutative 73.0%

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

      10. *-commutative 73.0%

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

      11. associate-+l+ 73.0%

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

      12. *-commutative 73.0%

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

      13. *-commutative 73.0%

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

      14. +-commutative 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in x around inf 50.8%

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

   if 6.5999999999999998e-281 < y

   1. Initial program 75.9%

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

      1. +-commutative 75.9%

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

      2. associate-+r+ 75.9%

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

      3. *-commutative 75.9%

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

      4. +-commutative 75.9%

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

      5. +-commutative 75.9%

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

      6. *-commutative 75.9%

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

      7. *-commutative 75.9%

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

      8. associate-+l+ 75.9%

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

      9. +-commutative 75.9%

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

      10. *-commutative 75.9%

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

      11. associate-+l+ 75.9%

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

      12. *-commutative 75.9%

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

      13. *-commutative 75.9%

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

      14. +-commutative 75.9%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in x around 0 29.5%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.4% accurate, 1.0× speedup

Math

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

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-296) (* 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-296) {
		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-296)) 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-296) {
		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-296:
		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-296)
		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-296)
		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-296], 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 < -2e-296

   1. Initial program 72.4%

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

      1. +-commutative 72.4%

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

      2. associate-+r+ 72.4%

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

      3. *-commutative 72.4%

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

      4. +-commutative 72.4%

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

      5. +-commutative 72.4%

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

      6. *-commutative 72.4%

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

      7. *-commutative 72.4%

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

      8. associate-+l+ 72.4%

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

      9. +-commutative 72.4%

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

      10. *-commutative 72.4%

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

      11. associate-+l+ 72.4%

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

      12. *-commutative 72.4%

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

      13. *-commutative 72.4%

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

      14. +-commutative 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in x around inf 49.1%

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

   if -2e-296 < y

   1. Initial program 76.2%

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

      1. +-commutative 76.2%

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

      2. associate-+r+ 76.2%

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

      3. *-commutative 76.2%

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

      4. +-commutative 76.2%

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

      5. +-commutative 76.2%

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

      6. *-commutative 76.2%

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

      7. *-commutative 76.2%

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

      8. associate-+l+ 76.2%

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

      9. +-commutative 76.2%

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

      10. *-commutative 76.2%

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

      11. associate-+l+ 76.2%

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

      12. *-commutative 76.2%

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

      13. *-commutative 76.2%

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

      14. +-commutative 76.2%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in z around inf 53.7%

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

      1. +-commutative 53.7%

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

   7. Simplified 53.7%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Initial program 74.4%

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

   1. +-commutative 74.4%

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

   2. associate-+r+ 74.4%

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

   3. *-commutative 74.4%

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

   4. +-commutative 74.4%

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

   5. +-commutative 74.4%

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

   6. *-commutative 74.4%

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

   7. *-commutative 74.4%

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

   8. associate-+l+ 74.4%

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

   9. +-commutative 74.4%

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

   10. *-commutative 74.4%

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

   11. associate-+l+ 74.4%

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

   12. *-commutative 74.4%

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

   13. *-commutative 74.4%

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

   14. +-commutative 74.4%

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

3. Simplified 74.5%

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

5. Final simplification 74.5%

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

Alternative 10: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-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 <= -4e-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 <= (-4d-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 <= -4e-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 <= -4e-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 <= -4e-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 <= -4e-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, -4e-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 < -3.999999999999988e-310

   1. Initial program 72.4%

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

      1. +-commutative 72.4%

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

      2. associate-+r+ 72.4%

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

      3. *-commutative 72.4%

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

      4. +-commutative 72.4%

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

      5. +-commutative 72.4%

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

      6. *-commutative 72.4%

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

      7. *-commutative 72.4%

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

      8. associate-+l+ 72.4%

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

      9. +-commutative 72.4%

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

      10. *-commutative 72.4%

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

      11. associate-+l+ 72.4%

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

      12. *-commutative 72.4%

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

      13. *-commutative 72.4%

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

      14. +-commutative 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in z around 0 30.5%

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

      1. *-commutative 30.5%

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

   7. Simplified 30.5%

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

   if -3.999999999999988e-310 < y

   1. Initial program 76.2%

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

      1. +-commutative 76.2%

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

      2. associate-+r+ 76.2%

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

      3. *-commutative 76.2%

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

      4. +-commutative 76.2%

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

      5. +-commutative 76.2%

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

      6. *-commutative 76.2%

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

      7. *-commutative 76.2%

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

      8. associate-+l+ 76.2%

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

      9. +-commutative 76.2%

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

      10. *-commutative 76.2%

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

      11. associate-+l+ 76.2%

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

      12. *-commutative 76.2%

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

      13. *-commutative 76.2%

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

      14. +-commutative 76.2%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 28.4%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.5% 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 74.4%

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

   1. +-commutative 74.4%

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

   2. associate-+r+ 74.4%

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

   3. *-commutative 74.4%

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

   4. +-commutative 74.4%

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

   5. +-commutative 74.4%

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

   6. *-commutative 74.4%

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

   7. *-commutative 74.4%

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

   8. associate-+l+ 74.4%

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

   9. +-commutative 74.4%

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

   10. *-commutative 74.4%

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

   11. associate-+l+ 74.4%

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

   12. *-commutative 74.4%

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

   13. *-commutative 74.4%

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

   14. +-commutative 74.4%

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

3. Simplified 74.5%

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

5. Taylor expanded in z around 0 27.8%

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

   1. *-commutative 27.8%

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

7. Simplified 27.8%

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

8. Final simplification 27.8%

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

Developer target: 83.4% 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 2024115 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (if (< z 7.636950090573675e+176) (* 2.0 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25))) (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25)))) 2.0))

  (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))