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

Percentage Accurate: 69.5% → 94.1%

Time: 15.4s

Alternatives: 13

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: 69.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
end

Wolfram

code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 12:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot \left(\sqrt{\frac{y + x}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.5e+60)
   (* 2.0 (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0))
   (if (<= y 12.0)
     (* 2.0 (sqrt (+ (+ (* y x) (* z x)) (* y z))))
     (*
      2.0
      (*
       z
       (+
        (sqrt (/ (+ y x) z))
        (* 0.5 (* x (* y (sqrt (/ (/ 1.0 (pow z 3.0)) (+ y x))))))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.5e+60) {
		tmp = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	} else if (y <= 12.0) {
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = 2.0 * (z * (sqrt(((y + x) / z)) + (0.5 * (x * (y * sqrt(((1.0 / pow(z, 3.0)) / (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 <= (-3.5d+60)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log((-y - z)) - log(((-1.0d0) / x))))) ** 2.0d0)
    else if (y <= 12.0d0) then
        tmp = 2.0d0 * sqrt((((y * x) + (z * x)) + (y * z)))
    else
        tmp = 2.0d0 * (z * (sqrt(((y + x) / z)) + (0.5d0 * (x * (y * sqrt(((1.0d0 / (z ** 3.0d0)) / (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 <= -3.5e+60) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((-y - z)) - Math.log((-1.0 / x))))), 2.0);
	} else if (y <= 12.0) {
		tmp = 2.0 * Math.sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = 2.0 * (z * (Math.sqrt(((y + x) / z)) + (0.5 * (x * (y * Math.sqrt(((1.0 / Math.pow(z, 3.0)) / (y + x))))))));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -3.5e+60:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log((-y - z)) - math.log((-1.0 / x))))), 2.0)
	elif y <= 12.0:
		tmp = 2.0 * math.sqrt((((y * x) + (z * x)) + (y * z)))
	else:
		tmp = 2.0 * (z * (math.sqrt(((y + x) / z)) + (0.5 * (x * (y * math.sqrt(((1.0 / math.pow(z, 3.0)) / (y + x))))))))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3.5e+60)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0));
	elseif (y <= 12.0)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(y * x) + Float64(z * x)) + Float64(y * z))));
	else
		tmp = Float64(2.0 * Float64(z * Float64(sqrt(Float64(Float64(y + x) / z)) + Float64(0.5 * Float64(x * Float64(y * sqrt(Float64(Float64(1.0 / (z ^ 3.0)) / 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 <= -3.5e+60)
		tmp = 2.0 * (exp((0.25 * (log((-y - z)) - log((-1.0 / x))))) ^ 2.0);
	elseif (y <= 12.0)
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	else
		tmp = 2.0 * (z * (sqrt(((y + x) / z)) + (0.5 * (x * (y * sqrt(((1.0 / (z ^ 3.0)) / (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, -3.5e+60], 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], If[LessEqual[y, 12.0], N[(2.0 * N[Sqrt[N[(N[(N[(y * x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z * N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(x * N[(y * N[Sqrt[N[(N[(1.0 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5000000000000002e60

   1. Initial program 50.0%

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

      1. distribute-lft-out 50.0%

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

      2. *-commutative 50.0%

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

   4. Applied egg-rr 50.0%

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

      1. add-sqr-sqrt 49.7%

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

      2. pow2 49.7%

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

      3. pow1/2 49.7%

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

      4. sqrt-pow1 49.8%

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

      5. +-commutative 49.8%

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

      6. *-commutative 49.8%

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

      7. fma-define 50.6%

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

      8. *-commutative 50.6%

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

      9. metadata-eval 50.6%

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

   6. Applied egg-rr 50.6%

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

   7. Taylor expanded in x around -inf 48.6%

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

   if -3.5000000000000002e60 < y < 12

   1. Initial program 85.8%

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

   if 12 < y

   1. Initial program 60.3%

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

      1. associate-+l+ 60.3%

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

      2. *-commutative 60.3%

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

      3. *-commutative 60.3%

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

      4. *-commutative 60.3%

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

      5. +-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate-+l+ 60.3%

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

      8. *-commutative 60.3%

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

      9. *-commutative 60.3%

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

      10. +-commutative 60.3%

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

      11. +-commutative 60.3%

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

      12. *-commutative 60.3%

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

      13. associate-+l+ 60.3%

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

      14. +-commutative 60.3%

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

      15. distribute-rgt-in 60.3%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in z around inf 38.0%

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

      1. +-commutative 38.0%

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

      2. associate-*l* 48.7%

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

      3. associate-/r* 48.7%

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

      4. +-commutative 48.7%

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

   7. Simplified 48.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 12:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot \left(\sqrt{\frac{y + x}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 3000000:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot \left(\sqrt{\frac{y + x}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4e+60)
   (* 2.0 (pow (exp (* 0.25 (- (log (- y)) (log (/ -1.0 x))))) 2.0))
   (if (<= y 3000000.0)
     (* 2.0 (sqrt (+ (+ (* y x) (* z x)) (* y z))))
     (*
      2.0
      (*
       z
       (+
        (sqrt (/ (+ y x) z))
        (* 0.5 (* x (* y (sqrt (/ (/ 1.0 (pow z 3.0)) (+ y x))))))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+60) {
		tmp = 2.0 * pow(exp((0.25 * (log(-y) - log((-1.0 / x))))), 2.0);
	} else if (y <= 3000000.0) {
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = 2.0 * (z * (sqrt(((y + x) / z)) + (0.5 * (x * (y * sqrt(((1.0 / pow(z, 3.0)) / (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 <= (-4d+60)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log(-y) - log(((-1.0d0) / x))))) ** 2.0d0)
    else if (y <= 3000000.0d0) then
        tmp = 2.0d0 * sqrt((((y * x) + (z * x)) + (y * z)))
    else
        tmp = 2.0d0 * (z * (sqrt(((y + x) / z)) + (0.5d0 * (x * (y * sqrt(((1.0d0 / (z ** 3.0d0)) / (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 <= -4e+60) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log(-y) - Math.log((-1.0 / x))))), 2.0);
	} else if (y <= 3000000.0) {
		tmp = 2.0 * Math.sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = 2.0 * (z * (Math.sqrt(((y + x) / z)) + (0.5 * (x * (y * Math.sqrt(((1.0 / Math.pow(z, 3.0)) / (y + x))))))));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -4e+60:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log(-y) - math.log((-1.0 / x))))), 2.0)
	elif y <= 3000000.0:
		tmp = 2.0 * math.sqrt((((y * x) + (z * x)) + (y * z)))
	else:
		tmp = 2.0 * (z * (math.sqrt(((y + x) / z)) + (0.5 * (x * (y * math.sqrt(((1.0 / math.pow(z, 3.0)) / (y + x))))))))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -4e+60)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(-y)) - log(Float64(-1.0 / x))))) ^ 2.0));
	elseif (y <= 3000000.0)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(y * x) + Float64(z * x)) + Float64(y * z))));
	else
		tmp = Float64(2.0 * Float64(z * Float64(sqrt(Float64(Float64(y + x) / z)) + Float64(0.5 * Float64(x * Float64(y * sqrt(Float64(Float64(1.0 / (z ^ 3.0)) / 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 <= -4e+60)
		tmp = 2.0 * (exp((0.25 * (log(-y) - log((-1.0 / x))))) ^ 2.0);
	elseif (y <= 3000000.0)
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	else
		tmp = 2.0 * (z * (sqrt(((y + x) / z)) + (0.5 * (x * (y * sqrt(((1.0 / (z ^ 3.0)) / (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, -4e+60], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[(-y)], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3000000.0], N[(2.0 * N[Sqrt[N[(N[(N[(y * x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z * N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(x * N[(y * N[Sqrt[N[(N[(1.0 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999998e60

   1. Initial program 50.0%

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

      1. associate-+l+ 50.0%

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

      2. *-commutative 50.0%

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

      3. *-commutative 50.0%

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

      4. *-commutative 50.0%

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

      5. +-commutative 50.0%

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

      6. +-commutative 50.0%

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

      7. associate-+l+ 50.0%

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

      8. *-commutative 50.0%

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

      9. *-commutative 50.0%

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

      10. +-commutative 50.0%

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

      11. +-commutative 50.0%

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

      12. *-commutative 50.0%

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

      13. associate-+l+ 50.0%

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

      14. +-commutative 50.0%

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

      15. distribute-rgt-in 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in z around 0 31.6%

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

      1. *-commutative 31.6%

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

   7. Simplified 31.6%

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

      1. add-sqr-sqrt 31.3%

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

      2. pow2 31.3%

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

      3. pow1/2 31.9%

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

      4. sqrt-pow1 32.0%

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

      5. metadata-eval 32.0%

         \[\leadsto 2 \cdot {\left({\left(y \cdot x\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   9. Applied egg-rr 32.0%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(y \cdot x\right)}^{0.25}\right)}^{2}} \]

   10. Taylor expanded in x around -inf 46.7%

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

   if -3.9999999999999998e60 < y < 3e6

   1. Initial program 85.8%

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

   if 3e6 < y

   1. Initial program 60.3%

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

      1. associate-+l+ 60.3%

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

      2. *-commutative 60.3%

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

      3. *-commutative 60.3%

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

      4. *-commutative 60.3%

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

      5. +-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate-+l+ 60.3%

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

      8. *-commutative 60.3%

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

      9. *-commutative 60.3%

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

      10. +-commutative 60.3%

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

      11. +-commutative 60.3%

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

      12. *-commutative 60.3%

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

      13. associate-+l+ 60.3%

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

      14. +-commutative 60.3%

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

      15. distribute-rgt-in 60.3%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in z around inf 38.0%

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

      1. +-commutative 38.0%

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

      2. associate-*l* 48.7%

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

      3. associate-/r* 48.7%

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

      4. +-commutative 48.7%

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

   7. Simplified 48.7%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 3000000:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot \left(\sqrt{\frac{y + x}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot \left(\sqrt{\frac{y + x}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5)
   (* 2.0 (sqrt (fma x z (* y (+ z x)))))
   (*
    2.0
    (*
     z
     (+
      (sqrt (/ (+ y x) z))
      (* 0.5 (* x (* y (sqrt (/ (/ 1.0 (pow z 3.0)) (+ y x)))))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = 2.0 * (z * (sqrt(((y + x) / z)) + (0.5 * (x * (y * sqrt(((1.0 / pow(z, 3.0)) / (y + x))))))));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(2.0 * Float64(z * Float64(sqrt(Float64(Float64(y + x) / z)) + Float64(0.5 * Float64(x * Float64(y * sqrt(Float64(Float64(1.0 / (z ^ 3.0)) / Float64(y + x)))))))));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.5], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z * N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(x * N[(y * N[Sqrt[N[(N[(1.0 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5

   1. Initial program 76.8%

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

      1. associate-+l+ 76.8%

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

      2. *-commutative 76.8%

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

      3. *-commutative 76.8%

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

      4. *-commutative 76.8%

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

      5. +-commutative 76.8%

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

      6. +-commutative 76.8%

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

      7. +-commutative 76.8%

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

      8. *-commutative 76.8%

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

      9. *-commutative 76.8%

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

      10. associate-+l+ 76.8%

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

      11. +-commutative 76.8%

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

      12. fma-define 76.8%

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

      13. distribute-lft-out 76.9%

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

   3. Simplified 76.9%

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

   if 1.5 < y

   1. Initial program 60.3%

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

      1. associate-+l+ 60.3%

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

      2. *-commutative 60.3%

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

      3. *-commutative 60.3%

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

      4. *-commutative 60.3%

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

      5. +-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate-+l+ 60.3%

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

      8. *-commutative 60.3%

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

      9. *-commutative 60.3%

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

      10. +-commutative 60.3%

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

      11. +-commutative 60.3%

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

      12. *-commutative 60.3%

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

      13. associate-+l+ 60.3%

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

      14. +-commutative 60.3%

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

      15. distribute-rgt-in 60.3%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in z around inf 38.0%

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

      1. +-commutative 38.0%

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

      2. associate-*l* 48.7%

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

      3. associate-/r* 48.7%

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

      4. +-commutative 48.7%

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

   7. Simplified 48.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot \left(\sqrt{\frac{y + x}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{\frac{1}{{z}^{3}}}{y + x}}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, z + x\right)} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5.5e+31)
   (* 2.0 (sqrt (fma x z (* y (+ z x)))))
   (* 2.0 (* (sqrt (fma x (/ z y) (+ z x))) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.5e+31) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = 2.0 * (sqrt(fma(x, (z / y), (z + x))) * sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 5.5e+31)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(fma(x, Float64(z / y), Float64(z + x))) * sqrt(y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 5.5e+31], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(x * N[(z / y), $MachinePrecision] + N[(z + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.50000000000000002e31

   1. Initial program 76.7%

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

      1. associate-+l+ 76.7%

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

      2. *-commutative 76.7%

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

      3. *-commutative 76.7%

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

      4. *-commutative 76.7%

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

      5. +-commutative 76.7%

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

      6. +-commutative 76.7%

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

      7. +-commutative 76.7%

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

      8. *-commutative 76.7%

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

      9. *-commutative 76.7%

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

      10. associate-+l+ 76.7%

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

      11. +-commutative 76.7%

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

      12. fma-define 76.7%

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

      13. distribute-lft-out 76.9%

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

   3. Simplified 76.9%

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

   if 5.50000000000000002e31 < y

   1. Initial program 58.2%

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

      1. associate-+l+ 58.2%

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

      2. *-commutative 58.2%

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

      3. *-commutative 58.2%

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

      4. *-commutative 58.2%

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

      5. +-commutative 58.2%

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

      6. +-commutative 58.2%

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

      7. associate-+l+ 58.2%

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

      8. *-commutative 58.2%

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

      9. *-commutative 58.2%

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

      10. +-commutative 58.2%

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

      11. +-commutative 58.2%

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

      12. *-commutative 58.2%

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

      13. associate-+l+ 58.2%

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

      14. +-commutative 58.2%

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

      15. distribute-rgt-in 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in y around inf 58.4%

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

      1. associate-+r+ 58.4%

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

      2. +-commutative 58.4%

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

      3. associate-/l* 58.5%

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

   7. Simplified 58.5%

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

      1. *-commutative 58.5%

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

      2. sqrt-prod 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-define 99.4%

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

   9. Applied egg-rr 99.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, z + x\right)} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.2e+29)
   (* 2.0 (sqrt (fma x z (* y (+ z x)))))
   (* 2.0 (* (sqrt y) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e+29) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = 2.0 * (sqrt(y) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.2e+29)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(y) * 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, 4.2e+29], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[y], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2000000000000003e29

   1. Initial program 76.6%

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

      1. associate-+l+ 76.6%

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

      2. *-commutative 76.6%

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

      3. *-commutative 76.6%

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

      4. *-commutative 76.6%

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

      5. +-commutative 76.6%

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

      6. +-commutative 76.6%

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

      7. +-commutative 76.6%

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

      8. *-commutative 76.6%

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

      9. *-commutative 76.6%

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

      10. associate-+l+ 76.6%

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

      11. +-commutative 76.6%

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

      12. fma-define 76.6%

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

      13. distribute-lft-out 76.8%

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

   3. Simplified 76.8%

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

   if 4.2000000000000003e29 < y

   1. Initial program 58.9%

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

      1. associate-+l+ 58.9%

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

      2. *-commutative 58.9%

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

      3. *-commutative 58.9%

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

      4. *-commutative 58.9%

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

      5. +-commutative 58.9%

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

      6. +-commutative 58.9%

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

      7. associate-+l+ 58.9%

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

      8. *-commutative 58.9%

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

      9. *-commutative 58.9%

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

      10. +-commutative 58.9%

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

      11. +-commutative 58.9%

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

      12. *-commutative 58.9%

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

      13. associate-+l+ 58.9%

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

      14. +-commutative 58.9%

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

      15. distribute-rgt-in 58.9%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in x around 0 22.2%

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

      1. *-commutative 22.2%

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

   7. Simplified 22.2%

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

      1. sqrt-prod 43.3%

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

   9. Applied egg-rr 43.3%

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

       1. *-commutative 43.3%

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

   11. Simplified 43.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + \left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.45e+30)
   (* 2.0 (sqrt (+ (* y z) (* (+ y z) x))))
   (* 2.0 (* (sqrt y) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.45e+30) {
		tmp = 2.0 * sqrt(((y * z) + ((y + z) * x)));
	} else {
		tmp = 2.0 * (sqrt(y) * 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 <= 1.45d+30) then
        tmp = 2.0d0 * sqrt(((y * z) + ((y + z) * x)))
    else
        tmp = 2.0d0 * (sqrt(y) * 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 <= 1.45e+30) {
		tmp = 2.0 * Math.sqrt(((y * z) + ((y + z) * x)));
	} else {
		tmp = 2.0 * (Math.sqrt(y) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.45e+30:
		tmp = 2.0 * math.sqrt(((y * z) + ((y + z) * x)))
	else:
		tmp = 2.0 * (math.sqrt(y) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.45e+30)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * z) + Float64(Float64(y + z) * x))));
	else
		tmp = Float64(2.0 * Float64(sqrt(y) * 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 <= 1.45e+30)
		tmp = 2.0 * sqrt(((y * z) + ((y + z) * x)));
	else
		tmp = 2.0 * (sqrt(y) * 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, 1.45e+30], N[(2.0 * N[Sqrt[N[(N[(y * z), $MachinePrecision] + N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[y], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4499999999999999e30

   1. Initial program 76.6%

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

      1. distribute-lft-out 76.6%

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

      2. *-commutative 76.6%

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

   4. Applied egg-rr 76.6%

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

   if 1.4499999999999999e30 < y

   1. Initial program 58.9%

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

      1. associate-+l+ 58.9%

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

      2. *-commutative 58.9%

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

      3. *-commutative 58.9%

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

      4. *-commutative 58.9%

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

      5. +-commutative 58.9%

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

      6. +-commutative 58.9%

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

      7. associate-+l+ 58.9%

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

      8. *-commutative 58.9%

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

      9. *-commutative 58.9%

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

      10. +-commutative 58.9%

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

      11. +-commutative 58.9%

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

      12. *-commutative 58.9%

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

      13. associate-+l+ 58.9%

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

      14. +-commutative 58.9%

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

      15. distribute-rgt-in 58.9%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in x around 0 22.2%

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

      1. *-commutative 22.2%

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

   7. Simplified 22.2%

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

      1. sqrt-prod 43.3%

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

   9. Applied egg-rr 43.3%

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

       1. *-commutative 43.3%

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

   11. Simplified 43.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + \left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot \left(x + x \cdot \frac{z}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -8.8e-276)
   (* 2.0 (sqrt (* y (+ x (* x (/ z y))))))
   (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.8e-276) {
		tmp = 2.0 * sqrt((y * (x + (x * (z / y)))));
	} 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 <= (-8.8d-276)) then
        tmp = 2.0d0 * sqrt((y * (x + (x * (z / y)))))
    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 <= -8.8e-276) {
		tmp = 2.0 * Math.sqrt((y * (x + (x * (z / y)))));
	} 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 <= -8.8e-276:
		tmp = 2.0 * math.sqrt((y * (x + (x * (z / y)))))
	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 <= -8.8e-276)
		tmp = Float64(2.0 * sqrt(Float64(y * Float64(x + Float64(x * Float64(z / y))))));
	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 <= -8.8e-276)
		tmp = 2.0 * sqrt((y * (x + (x * (z / y)))));
	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, -8.8e-276], N[(2.0 * N[Sqrt[N[(y * N[(x + N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $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 < -8.79999999999999923e-276

   1. Initial program 72.1%

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

      1. associate-+l+ 72.1%

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

      2. *-commutative 72.1%

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

      3. *-commutative 72.1%

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

      4. *-commutative 72.1%

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

      5. +-commutative 72.1%

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

      6. +-commutative 72.1%

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

      7. associate-+l+ 72.1%

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

      8. *-commutative 72.1%

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

      9. *-commutative 72.1%

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

      10. +-commutative 72.1%

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

      11. +-commutative 72.1%

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

      12. *-commutative 72.1%

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

      13. associate-+l+ 72.1%

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

      14. +-commutative 72.1%

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

      15. distribute-rgt-in 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in y around inf 69.9%

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

      1. associate-+r+ 69.9%

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

      2. +-commutative 69.9%

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

      3. associate-/l* 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in z around 0 44.6%

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

   if -8.79999999999999923e-276 < y

   1. Initial program 73.1%

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

      1. associate-+l+ 73.1%

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

      2. *-commutative 73.1%

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

      3. *-commutative 73.1%

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

      4. *-commutative 73.1%

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

      5. +-commutative 73.1%

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

      6. +-commutative 73.1%

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

      7. associate-+l+ 73.1%

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

      8. *-commutative 73.1%

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

      9. *-commutative 73.1%

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

      10. +-commutative 73.1%

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

      11. +-commutative 73.1%

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

      12. *-commutative 73.1%

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

      13. associate-+l+ 73.1%

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

      14. +-commutative 73.1%

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

      15. distribute-rgt-in 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in z around inf 47.8%

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

      1. +-commutative 47.8%

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

   7. Simplified 47.8%

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

Alternative 8: 69.7% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-294) (* 2.0 (sqrt (* (+ y z) x))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-294) {
		tmp = 2.0 * sqrt(((y + z) * x));
	} 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 <= (-1d-294)) then
        tmp = 2.0d0 * sqrt(((y + z) * x))
    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 <= -1e-294) {
		tmp = 2.0 * Math.sqrt(((y + z) * x));
	} 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 <= -1e-294:
		tmp = 2.0 * math.sqrt(((y + z) * x))
	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 <= -1e-294)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + z) * x)));
	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 <= -1e-294)
		tmp = 2.0 * sqrt(((y + z) * x));
	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, -1e-294], N[(2.0 * N[Sqrt[N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000002e-294

   1. Initial program 72.3%

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

      1. associate-+l+ 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

      4. *-commutative 72.3%

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

      5. +-commutative 72.3%

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

      6. +-commutative 72.3%

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

      7. associate-+l+ 72.3%

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

      8. *-commutative 72.3%

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

      9. *-commutative 72.3%

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

      10. +-commutative 72.3%

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

      11. +-commutative 72.3%

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

      12. *-commutative 72.3%

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

      13. associate-+l+ 72.3%

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

      14. +-commutative 72.3%

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

      15. distribute-rgt-in 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x around inf 48.1%

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

   if -1.00000000000000002e-294 < y

   1. Initial program 72.9%

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

      1. associate-+l+ 72.9%

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

      2. *-commutative 72.9%

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

      3. *-commutative 72.9%

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

      4. *-commutative 72.9%

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

      5. +-commutative 72.9%

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

      6. +-commutative 72.9%

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

      7. associate-+l+ 72.9%

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

      8. *-commutative 72.9%

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

      9. *-commutative 72.9%

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

      10. +-commutative 72.9%

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

      11. +-commutative 72.9%

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

      12. *-commutative 72.9%

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

      13. associate-+l+ 72.9%

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

      14. +-commutative 72.9%

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

      15. distribute-rgt-in 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in z around inf 47.4%

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

      1. +-commutative 47.4%

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

   7. Simplified 47.4%

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

4. Final simplification 47.7%

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

Alternative 9: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-299}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.25e-299) (* 2.0 (sqrt (* (+ y z) x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e-299) {
		tmp = 2.0 * sqrt(((y + z) * 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 <= 1.25d-299) then
        tmp = 2.0d0 * sqrt(((y + z) * 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 <= 1.25e-299) {
		tmp = 2.0 * Math.sqrt(((y + z) * 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 <= 1.25e-299:
		tmp = 2.0 * math.sqrt(((y + z) * 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 <= 1.25e-299)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + z) * 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 <= 1.25e-299)
		tmp = 2.0 * sqrt(((y + z) * 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, 1.25e-299], N[(2.0 * N[Sqrt[N[(N[(y + z), $MachinePrecision] * 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 < 1.24999999999999989e-299

   1. Initial program 73.1%

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

      1. associate-+l+ 73.1%

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

      2. *-commutative 73.1%

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

      3. *-commutative 73.1%

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

      4. *-commutative 73.1%

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

      5. +-commutative 73.1%

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

      6. +-commutative 73.1%

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

      7. associate-+l+ 73.1%

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

      8. *-commutative 73.1%

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

      9. *-commutative 73.1%

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

      10. +-commutative 73.1%

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

      11. +-commutative 73.1%

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

      12. *-commutative 73.1%

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

      13. associate-+l+ 73.1%

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

      14. +-commutative 73.1%

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

      15. distribute-rgt-in 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around inf 49.7%

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

   if 1.24999999999999989e-299 < y

   1. Initial program 72.0%

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

      1. associate-+l+ 72.0%

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

      2. *-commutative 72.0%

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

      3. *-commutative 72.0%

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

      4. *-commutative 72.0%

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

      5. +-commutative 72.0%

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

      6. +-commutative 72.0%

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

      7. associate-+l+ 72.0%

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

      8. *-commutative 72.0%

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

      9. *-commutative 72.0%

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

      10. +-commutative 72.0%

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

      11. +-commutative 72.0%

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

      12. *-commutative 72.0%

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

      13. associate-+l+ 72.0%

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

      14. +-commutative 72.0%

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

      15. distribute-rgt-in 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in x around 0 24.1%

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

      1. *-commutative 24.1%

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

   7. Simplified 24.1%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-299}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 1.0× speedup

Math

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

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 z) (* (+ y z) x)))))

C

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

Python

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

Julia

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

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 2.0 * sqrt(((y * z) + ((y + z) * 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 * z), $MachinePrecision] + N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

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

   1. distribute-lft-out 72.6%

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

   2. *-commutative 72.6%

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

4. Applied egg-rr 72.6%

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

5. Final simplification 72.6%

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

Alternative 11: 69.6% 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 72.6%

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

   1. associate-+l+ 72.6%

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

   2. *-commutative 72.6%

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

   3. *-commutative 72.6%

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

   4. *-commutative 72.6%

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

   5. +-commutative 72.6%

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

   6. +-commutative 72.6%

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

   7. associate-+l+ 72.6%

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

   8. *-commutative 72.6%

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

   9. *-commutative 72.6%

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

   10. +-commutative 72.6%

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

   11. +-commutative 72.6%

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

   12. *-commutative 72.6%

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

   13. associate-+l+ 72.6%

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

   14. +-commutative 72.6%

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

   15. distribute-rgt-in 72.6%

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

3. Simplified 72.6%

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

5. Final simplification 72.6%

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

Alternative 12: 67.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-310)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 72.5%

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

      1. associate-+l+ 72.5%

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

      2. *-commutative 72.5%

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

      3. *-commutative 72.5%

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

      4. *-commutative 72.5%

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

      5. +-commutative 72.5%

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

      6. +-commutative 72.5%

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

      7. associate-+l+ 72.5%

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

      8. *-commutative 72.5%

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

      9. *-commutative 72.5%

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

      10. +-commutative 72.5%

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

      11. +-commutative 72.5%

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

      12. *-commutative 72.5%

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

      13. associate-+l+ 72.5%

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

      14. +-commutative 72.5%

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

      15. distribute-rgt-in 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in z around 0 29.1%

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

      1. *-commutative 29.1%

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

   7. Simplified 29.1%

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

   if -4.999999999999985e-310 < y

   1. Initial program 72.7%

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

      1. associate-+l+ 72.7%

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

      2. *-commutative 72.7%

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

      3. *-commutative 72.7%

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

      4. *-commutative 72.7%

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

      5. +-commutative 72.7%

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

      6. +-commutative 72.7%

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

      7. associate-+l+ 72.7%

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

      8. *-commutative 72.7%

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

      9. *-commutative 72.7%

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

      10. +-commutative 72.7%

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

      11. +-commutative 72.7%

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

      12. *-commutative 72.7%

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

      13. associate-+l+ 72.7%

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

      14. +-commutative 72.7%

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

      15. distribute-rgt-in 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in x around 0 23.6%

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

      1. *-commutative 23.6%

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

   7. Simplified 23.6%

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

4. Final simplification 26.4%

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

Alternative 13: 35.3% 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 72.6%

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

   1. associate-+l+ 72.6%

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

   2. *-commutative 72.6%

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

   3. *-commutative 72.6%

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

   4. *-commutative 72.6%

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

   5. +-commutative 72.6%

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

   6. +-commutative 72.6%

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

   7. associate-+l+ 72.6%

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

   8. *-commutative 72.6%

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

   9. *-commutative 72.6%

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

   10. +-commutative 72.6%

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

   11. +-commutative 72.6%

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

   12. *-commutative 72.6%

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

   13. associate-+l+ 72.6%

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

   14. +-commutative 72.6%

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

   15. distribute-rgt-in 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in z around 0 28.6%

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

   1. *-commutative 28.6%

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

7. Simplified 28.6%

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

Developer Target 1: 82.1% 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 2024137 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 763695009057367500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 2 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 1/4 (* (* (pow y -3/4) (* (pow z -3/4) x)) (+ y z))) (* (pow z 1/4) (pow y 1/4))) (+ (* 1/4 (* (* (pow y -3/4) (* (pow z -3/4) x)) (+ y z))) (* (pow z 1/4) (pow y 1/4)))) 2)))

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