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

Average Accuracy: 68.7% → 94.8%

Time: 17.7s

Precision: binary64

Cost: 20364

\[ \begin{array}{c}[x, y, z] = \mathsf{sort}([x, y, z])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-43}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(y - z\right) \cdot \left(y + z\right)}}{\sqrt{\frac{y - z}{x}}}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+154)
   (* 2.0 (exp (* (- (log (- x)) (log (/ -1.0 y))) 0.5)))
   (if (<= y -4.8e-43)
     (* 2.0 (/ (sqrt (* (- y z) (+ y z))) (sqrt (/ (- y z) x))))
     (if (<= y 5.6e-271)
       (* 2.0 (exp (* 0.5 (- (log (- (- y) z)) (log (/ -1.0 x))))))
       (* 2.0 (* (sqrt z) (sqrt y)))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+154) {
		tmp = 2.0 * exp(((log(-x) - log((-1.0 / y))) * 0.5));
	} else if (y <= -4.8e-43) {
		tmp = 2.0 * (sqrt(((y - z) * (y + z))) / sqrt(((y - z) / x)));
	} else if (y <= 5.6e-271) {
		tmp = 2.0 * exp((0.5 * (log((-y - z)) - log((-1.0 / x)))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	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
    code = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end 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 <= (-1.4d+154)) then
        tmp = 2.0d0 * exp(((log(-x) - log(((-1.0d0) / y))) * 0.5d0))
    else if (y <= (-4.8d-43)) then
        tmp = 2.0d0 * (sqrt(((y - z) * (y + z))) / sqrt(((y - z) / x)))
    else if (y <= 5.6d-271) then
        tmp = 2.0d0 * exp((0.5d0 * (log((-y - z)) - log(((-1.0d0) / x)))))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+154) {
		tmp = 2.0 * Math.exp(((Math.log(-x) - Math.log((-1.0 / y))) * 0.5));
	} else if (y <= -4.8e-43) {
		tmp = 2.0 * (Math.sqrt(((y - z) * (y + z))) / Math.sqrt(((y - z) / x)));
	} else if (y <= 5.6e-271) {
		tmp = 2.0 * Math.exp((0.5 * (Math.log((-y - z)) - Math.log((-1.0 / x)))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= -1.4e+154:
		tmp = 2.0 * math.exp(((math.log(-x) - math.log((-1.0 / y))) * 0.5))
	elif y <= -4.8e-43:
		tmp = 2.0 * (math.sqrt(((y - z) * (y + z))) / math.sqrt(((y - z) / x)))
	elif y <= 5.6e-271:
		tmp = 2.0 * math.exp((0.5 * (math.log((-y - z)) - math.log((-1.0 / x)))))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+154)
		tmp = Float64(2.0 * exp(Float64(Float64(log(Float64(-x)) - log(Float64(-1.0 / y))) * 0.5)));
	elseif (y <= -4.8e-43)
		tmp = Float64(2.0 * Float64(sqrt(Float64(Float64(y - z) * Float64(y + z))) / sqrt(Float64(Float64(y - z) / x))));
	elseif (y <= 5.6e-271)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.4e+154)
		tmp = 2.0 * exp(((log(-x) - log((-1.0 / y))) * 0.5));
	elseif (y <= -4.8e-43)
		tmp = 2.0 * (sqrt(((y - z) * (y + z))) / sqrt(((y - z) / x)));
	elseif (y <= 5.6e-271)
		tmp = 2.0 * exp((0.5 * (log((-y - z)) - log((-1.0 / x)))));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
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]

↓

code[x_, y_, z_] := If[LessEqual[y, -1.4e+154], N[(2.0 * N[Exp[N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4.8e-43], N[(2.0 * N[(N[Sqrt[N[(N[(y - z), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(y - z), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e-271], N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	68.7%
Target	82.1%
Herbie	94.8%

\[\begin{array}{l} \mathbf{if}\;z < 7.636950090573675 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if y < -1.4e154

   1. Initial program 0.0%

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

   2. Simplified 0.0%

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

      [Start] 0.0	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 0.0	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 0.0%

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

   4. Applied egg-rr 0.0%

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

      [Start] 0.0	\[ 2 \cdot \sqrt{\left(y + z\right) \cdot x} \]
      pow1/2 [=>] 0.0	\[ 2 \cdot \color{blue}{{\left(\left(y + z\right) \cdot x\right)}^{0.5}} \]
      pow-to-exp [=>] 0.0	\[ 2 \cdot \color{blue}{e^{\log \left(\left(y + z\right) \cdot x\right) \cdot 0.5}} \]

   5. Taylor expanded in y around -inf 87.8%

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

   6. Simplified 87.8%

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

      [Start] 87.8	\[ 2 \cdot e^{\left(-1 \cdot \log \left(\frac{-1}{y}\right) + \log \left(-1 \cdot x\right)\right) \cdot 0.5} \]
      +-commutative [=>] 87.8	\[ 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot x\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right)} \cdot 0.5} \]
      mul-1-neg [=>] 87.8	\[ 2 \cdot e^{\left(\log \left(-1 \cdot x\right) + \color{blue}{\left(-\log \left(\frac{-1}{y}\right)\right)}\right) \cdot 0.5} \]
      unsub-neg [=>] 87.8	\[ 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot x\right) - \log \left(\frac{-1}{y}\right)\right)} \cdot 0.5} \]
      mul-1-neg [=>] 87.8	\[ 2 \cdot e^{\left(\log \color{blue}{\left(-x\right)} - \log \left(\frac{-1}{y}\right)\right) \cdot 0.5} \]

   if -1.4e154 < y < -4.8000000000000004e-43

   1. Initial program 69.6%

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

   2. Simplified 69.6%

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

      [Start] 69.6	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 69.6	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 69.5%

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

   4. Applied egg-rr 51.3%

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

      [Start] 69.5	\[ 2 \cdot \sqrt{\left(y + z\right) \cdot x} \]
      flip-+ [=>] 69.5	\[ 2 \cdot \sqrt{\color{blue}{\frac{y \cdot y - z \cdot z}{y - z}} \cdot x} \]
      associate-*l/ [=>] 51.3	\[ 2 \cdot \sqrt{\color{blue}{\frac{\left(y \cdot y - z \cdot z\right) \cdot x}{y - z}}} \]

   5. Applied egg-rr 98.9%

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

      [Start] 51.3	\[ 2 \cdot \sqrt{\frac{\left(y \cdot y - z \cdot z\right) \cdot x}{y - z}} \]
      associate-/l* [=>] 68.8	\[ 2 \cdot \sqrt{\color{blue}{\frac{y \cdot y - z \cdot z}{\frac{y - z}{x}}}} \]
      sqrt-div [=>] 98.9	\[ 2 \cdot \color{blue}{\frac{\sqrt{y \cdot y - z \cdot z}}{\sqrt{\frac{y - z}{x}}}} \]
      difference-of-squares [=>] 98.9	\[ 2 \cdot \frac{\sqrt{\color{blue}{\left(y + z\right) \cdot \left(y - z\right)}}}{\sqrt{\frac{y - z}{x}}} \]
      *-commutative [=>] 98.9	\[ 2 \cdot \frac{\sqrt{\color{blue}{\left(y - z\right) \cdot \left(y + z\right)}}}{\sqrt{\frac{y - z}{x}}} \]

   6. Simplified 98.9%

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

      [Start] 98.9	\[ 2 \cdot \frac{\sqrt{\left(y - z\right) \cdot \left(y + z\right)}}{\sqrt{\frac{y - z}{x}}} \]
      +-commutative [=>] 98.9	\[ 2 \cdot \frac{\sqrt{\left(y - z\right) \cdot \color{blue}{\left(z + y\right)}}}{\sqrt{\frac{y - z}{x}}} \]

   if -4.8000000000000004e-43 < y < 5.5999999999999995e-271

   1. Initial program 93.1%

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

   2. Simplified 93.1%

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

      [Start] 93.1	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 93.1	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 91.5%

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

   4. Applied egg-rr 85.6%

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

      [Start] 91.5	\[ 2 \cdot \sqrt{\left(y + z\right) \cdot x} \]
      pow1/2 [=>] 91.5	\[ 2 \cdot \color{blue}{{\left(\left(y + z\right) \cdot x\right)}^{0.5}} \]
      pow-to-exp [=>] 85.6	\[ 2 \cdot \color{blue}{e^{\log \left(\left(y + z\right) \cdot x\right) \cdot 0.5}} \]

   5. Taylor expanded in x around -inf 88.4%

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

   6. Simplified 88.4%

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

      [Start] 88.4	\[ 2 \cdot e^{\left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot \left(y + z\right)\right)\right) \cdot 0.5} \]
      +-commutative [=>] 88.4	\[ 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot \left(y + z\right)\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]
      mul-1-neg [=>] 88.4	\[ 2 \cdot e^{\left(\log \left(-1 \cdot \left(y + z\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{x}\right)\right)}\right) \cdot 0.5} \]
      unsub-neg [=>] 88.4	\[ 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot \left(y + z\right)\right) - \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]
      mul-1-neg [=>] 88.4	\[ 2 \cdot e^{\left(\log \color{blue}{\left(-\left(y + z\right)\right)} - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]
      neg-sub0 [=>] 88.4	\[ 2 \cdot e^{\left(\log \color{blue}{\left(0 - \left(y + z\right)\right)} - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]
      associate--r+ [=>] 88.4	\[ 2 \cdot e^{\left(\log \color{blue}{\left(\left(0 - y\right) - z\right)} - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]
      neg-sub0 [<=] 88.4	\[ 2 \cdot e^{\left(\log \left(\color{blue}{\left(-y\right)} - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]

   if 5.5999999999999995e-271 < y

   1. Initial program 69.1%

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

   2. Simplified 69.1%

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

      [Start] 69.1	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 69.1	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around 0 66.8%

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

   4. Applied egg-rr 96.7%

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

      [Start] 66.8	\[ 2 \cdot \sqrt{y \cdot z} \]
      sqrt-prod [=>] 96.7	\[ 2 \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{z}\right)} \]
      *-commutative [=>] 96.7	\[ 2 \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{y}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-43}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(y - z\right) \cdot \left(y + z\right)}}{\sqrt{\frac{y - z}{x}}}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.6%
Cost	19972

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot \left(\sqrt{y \cdot y - z \cdot z} \cdot \sqrt{\frac{x}{y - z}}\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-262}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + \frac{y}{\frac{1}{x + z}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Accuracy	83.5%
Cost	14665

\[\begin{array}{l} t_0 := \left(x \cdot z + y \cdot x\right) + y \cdot z\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \end{array} \]

Alternative 3
Accuracy	88.7%
Cost	13892

\[\begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+39}:\\ \;\;\;\;2 \cdot \left(\sqrt{y \cdot y - z \cdot z} \cdot \sqrt{\frac{x}{y - z}}\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-262}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + \frac{y}{\frac{1}{x + z}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4
Accuracy	88.6%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+37}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(y - z\right) \cdot \left(y + z\right)}}{\sqrt{\frac{y}{x}}}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-262}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + \frac{y}{\frac{1}{x + z}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 5
Accuracy	68.6%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 6
Accuracy	68.7%
Cost	7104

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

Alternative 7
Accuracy	68.7%
Cost	7104

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

Alternative 8
Accuracy	67.6%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-288}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 9
Accuracy	68.6%
Cost	6980

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

Alternative 10
Accuracy	66.5%
Cost	6852

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

Alternative 11
Accuracy	34.1%
Cost	6720

\[2 \cdot \sqrt{y \cdot x} \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

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

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