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

Percentage Accurate: 69.8% → 95.3%

Time: 15.0s

Alternatives: 10

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: 69.8% 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: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+48}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-283}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y + x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6.6e+48)
   (* 2.0 (pow (exp (* 0.25 (- (log (- y)) (log (/ -1.0 x))))) 2.0))
   (if (<= y 3.65e-283)
     (* 2.0 (sqrt (fma x y (* z (+ y x)))))
     (* 2.0 (* (sqrt z) (sqrt (+ y x)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.6e+48) {
		tmp = 2.0 * pow(exp((0.25 * (log(-y) - log((-1.0 / x))))), 2.0);
	} else if (y <= 3.65e-283) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt((y + x)));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -6.6e+48)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(-y)) - log(Float64(-1.0 / x))))) ^ 2.0));
	elseif (y <= 3.65e-283)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(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, -6.6e+48], 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, 3.65e-283], N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.60000000000000045e48

   1. Initial program 54.3%

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

      1. associate-+l+ 54.3%

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

      2. +-commutative 54.3%

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

      3. distribute-rgt-in 54.3%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in z around 0 30.9%

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

      1. add-sqr-sqrt 30.7%

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

      2. pow2 30.7%

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

      3. pow1/2 30.8%

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

      4. metadata-eval 30.8%

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

      5. sqrt-pow1 30.8%

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

      6. *-commutative 30.8%

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

      7. metadata-eval 30.8%

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

      8. metadata-eval 30.8%

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

   7. Applied egg-rr 30.8%

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

   8. Taylor expanded in x around -inf 41.9%

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

   if -6.60000000000000045e48 < y < 3.65e-283

   1. Initial program 80.6%

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

      1. +-commutative 80.6%

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

      2. *-commutative 80.6%

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

      3. +-commutative 80.6%

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

      4. *-commutative 80.6%

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

      5. *-commutative 80.6%

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

      6. associate-+l+ 80.6%

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

      7. +-commutative 80.6%

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

      8. *-commutative 80.6%

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

      9. fma-define 80.6%

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

      10. +-commutative 80.6%

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

      11. distribute-lft-out 80.6%

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

   3. Simplified 80.6%

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

   if 3.65e-283 < y

   1. Initial program 71.2%

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

      1. associate-+l+ 71.2%

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

      2. +-commutative 71.2%

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

      3. distribute-rgt-in 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in z around inf 46.8%

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

      1. *-commutative 46.8%

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

      2. sqrt-prod 46.2%

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

      3. +-commutative 46.2%

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

   7. Applied egg-rr 46.2%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+48}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-283}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y + x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.15e-276)
   (* 2.0 (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0))
   (* 2.0 (* (sqrt (fma x (/ y z) (+ y x))) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.15e-276) {
		tmp = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	} else {
		tmp = 2.0 * (sqrt(fma(x, (y / z), (y + x))) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.15e-276)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(sqrt(fma(x, Float64(y / z), Float64(y + x))) * sqrt(z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.15e-276], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(x * N[(y / z), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.14999999999999991e-276

   1. Initial program 67.4%

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

      1. +-commutative 67.4%

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

      2. *-commutative 67.4%

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

      3. +-commutative 67.4%

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

      4. *-commutative 67.4%

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

      5. *-commutative 67.4%

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

      6. associate-+l+ 67.4%

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

      7. +-commutative 67.4%

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

      8. *-commutative 67.4%

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

      9. fma-define 67.4%

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

      10. +-commutative 67.4%

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

      11. distribute-lft-out 67.5%

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

   3. Simplified 67.5%

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

      1. fma-undefine 67.4%

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

      2. distribute-rgt-in 67.4%

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

      3. associate-+l+ 67.4%

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

      4. add-sqr-sqrt 66.9%

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

      5. pow2 66.9%

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

      6. pow1/2 66.9%

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

      7. sqrt-pow1 67.0%

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

      8. associate-+l+ 67.0%

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

      9. distribute-rgt-in 67.0%

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

      10. fma-undefine 67.0%

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

      11. metadata-eval 67.0%

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

   6. Applied egg-rr 67.0%

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

   7. Taylor expanded in x around -inf 44.1%

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

   if -1.14999999999999991e-276 < y

   1. Initial program 72.4%

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

      1. +-commutative 72.4%

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

      2. *-commutative 72.4%

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

      3. +-commutative 72.4%

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

      4. *-commutative 72.4%

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

      5. *-commutative 72.4%

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

      6. associate-+l+ 72.4%

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

      7. +-commutative 72.4%

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

      8. *-commutative 72.4%

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

      9. fma-define 72.4%

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

      10. +-commutative 72.4%

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

      11. distribute-lft-out 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in z around inf 66.3%

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

      1. associate-+r+ 66.3%

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

      2. associate-/l* 64.3%

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

   7. Simplified 64.3%

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

      1. sqrt-prod 51.3%

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

      2. *-commutative 51.3%

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

      3. +-commutative 51.3%

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

      4. fma-define 51.3%

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

      5. +-commutative 51.3%

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

   9. Applied egg-rr 51.3%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-290}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y + x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.7e-290)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt z) (sqrt (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.7e-290) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt((y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.7d-290) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt((y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.7e-290) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt((y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.7e-290:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt((y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.7e-290)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.7e-290)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt((y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2.7e-290], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.69999999999999999e-290

   1. Initial program 68.8%

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

      1. associate-+l+ 68.8%

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

      2. +-commutative 68.8%

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

      3. distribute-rgt-in 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in x around inf 45.9%

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

      1. +-commutative 45.9%

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

   7. Simplified 45.9%

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

   if 2.69999999999999999e-290 < y

   1. Initial program 71.5%

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

      1. associate-+l+ 71.5%

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

      2. +-commutative 71.5%

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

      3. distribute-rgt-in 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in z around inf 47.2%

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

      1. *-commutative 47.2%

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

      2. sqrt-prod 45.9%

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

      3. +-commutative 45.9%

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

   7. Applied egg-rr 45.9%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-290}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y + x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 49000000000:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 49000000000.0)
   (* 2.0 (sqrt (+ (* x (+ y z)) (* y z))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 49000000000.0) {
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 49000000000.0d0) then
        tmp = 2.0d0 * sqrt(((x * (y + z)) + (y * z)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 49000000000.0) {
		tmp = 2.0 * Math.sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 49000000000.0:
		tmp = 2.0 * math.sqrt(((x * (y + z)) + (y * z)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 49000000000.0)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * Float64(y + z)) + Float64(y * z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 49000000000.0)
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 49000000000.0], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9e10

   1. Initial program 74.7%

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

      1. distribute-lft-out 74.7%

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

      2. *-commutative 74.7%

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

   4. Applied egg-rr 74.7%

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

   if 4.9e10 < y

   1. Initial program 56.3%

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

      1. +-commutative 56.3%

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

      2. *-commutative 56.3%

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

      3. +-commutative 56.3%

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

      4. *-commutative 56.3%

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

      5. *-commutative 56.3%

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

      6. associate-+l+ 56.3%

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

      7. +-commutative 56.3%

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

      8. *-commutative 56.3%

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

      9. fma-define 56.4%

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

      10. +-commutative 56.4%

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

      11. distribute-lft-out 56.5%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in z around inf 45.7%

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

      1. associate-+r+ 45.7%

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

      2. associate-/l* 41.2%

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

   7. Simplified 41.2%

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

      1. sqrt-prod 51.5%

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

      2. *-commutative 51.5%

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

      3. +-commutative 51.5%

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

      4. fma-define 51.5%

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

      5. +-commutative 51.5%

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

   9. Applied egg-rr 51.5%

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

   10. Taylor expanded in x around 0 37.5%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 49000000000:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-282}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1e-282) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-282) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1d-282) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-282) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1e-282:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1e-282)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e-282)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1e-282], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e-282

   1. Initial program 69.0%

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

      1. associate-+l+ 69.0%

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

      2. +-commutative 69.0%

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

      3. distribute-rgt-in 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in x around inf 46.3%

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

      1. +-commutative 46.3%

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

   7. Simplified 46.3%

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

   if 1e-282 < y

   1. Initial program 71.2%

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

      1. associate-+l+ 71.2%

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

      2. +-commutative 71.2%

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

      3. distribute-rgt-in 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in z around inf 46.8%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-282}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-283}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 7.5e-283) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e-283) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 7.5d-283) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e-283) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 7.5e-283:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 7.5e-283)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.5e-283)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 7.5e-283], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000001e-283

   1. Initial program 69.0%

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

      1. associate-+l+ 69.0%

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

      2. +-commutative 69.0%

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

      3. distribute-rgt-in 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in x around inf 46.3%

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

      1. +-commutative 46.3%

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

   7. Simplified 46.3%

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

   if 7.5000000000000001e-283 < y

   1. Initial program 71.2%

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

      1. associate-+l+ 71.2%

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

      2. +-commutative 71.2%

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

      3. distribute-rgt-in 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in x around 0 21.3%

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

      1. *-commutative 21.3%

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

   7. Simplified 21.3%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-283}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% 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 70.1%

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

   1. associate-+l+ 70.1%

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

   2. +-commutative 70.1%

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

   3. distribute-rgt-in 70.1%

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

3. Simplified 70.1%

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

5. Final simplification 70.1%

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

Alternative 8: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{y \cdot \left(x \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310) (sqrt (* y (* x 4.0))) (* 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 = sqrt((y * (x * 4.0)));
	} 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 = sqrt((y * (x * 4.0d0)))
    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 = Math.sqrt((y * (x * 4.0)));
	} 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 = math.sqrt((y * (x * 4.0)))
	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 = sqrt(Float64(y * Float64(x * 4.0)));
	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 = sqrt((y * (x * 4.0)));
	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[Sqrt[N[(y * N[(x * 4.0), $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 69.0%

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

      1. associate-+l+ 69.0%

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

      2. +-commutative 69.0%

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

      3. distribute-rgt-in 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in z around 0 27.1%

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

      1. add-sqr-sqrt 26.9%

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

      2. pow2 26.9%

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

      3. pow1/2 27.0%

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

      4. metadata-eval 27.0%

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

      5. sqrt-pow1 27.0%

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

      6. *-commutative 27.0%

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

      7. metadata-eval 27.0%

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

      8. metadata-eval 27.0%

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

   7. Applied egg-rr 27.0%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(y \cdot x\right)}^{0.25}\right)}^{2}} \]
   8. Step-by-step derivation

      1. pow-pow 27.1%

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

      2. metadata-eval 27.1%

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

      3. pow1/2 27.1%

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

      4. pow1 27.1%

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

      5. metadata-eval 27.1%

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

      6. metadata-eval 27.1%

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

      7. pow-sqr 27.0%

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

      8. pow-prod-down 27.1%

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

      9. *-commutative 27.1%

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

      10. *-commutative 27.1%

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

      11. swap-sqr 27.1%

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

      12. add-sqr-sqrt 27.1%

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

      13. metadata-eval 27.1%

          \[\leadsto {\left(\left(y \cdot x\right) \cdot \color{blue}{4}\right)}^{\left(0.25 \cdot 2\right)} \]

      14. metadata-eval 27.1%

          \[\leadsto {\left(\left(y \cdot x\right) \cdot 4\right)}^{\color{blue}{0.5}} \]

   9. Applied egg-rr 27.1%

      \[\leadsto \color{blue}{{\left(\left(y \cdot x\right) \cdot 4\right)}^{0.5}} \]
   10. Step-by-step derivation

       1. unpow1/2 27.1%

          \[\leadsto \color{blue}{\sqrt{\left(y \cdot x\right) \cdot 4}} \]

       2. associate-*l* 27.1%

          \[\leadsto \sqrt{\color{blue}{y \cdot \left(x \cdot 4\right)}} \]

   11. Simplified 27.1%

       \[\leadsto \color{blue}{\sqrt{y \cdot \left(x \cdot 4\right)}} \]

   if -4.999999999999985e-310 < y

   1. Initial program 71.2%

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

      1. associate-+l+ 71.2%

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

      2. +-commutative 71.2%

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

      3. distribute-rgt-in 71.2%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in x around 0 20.9%

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

      1. *-commutative 20.9%

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

   7. Simplified 20.9%

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

4. Final simplification 23.9%

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

Alternative 9: 35.8% 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 70.1%

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

   1. associate-+l+ 70.1%

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

   2. +-commutative 70.1%

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

   3. distribute-rgt-in 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in z around 0 25.8%

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

6. Final simplification 25.8%

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

Alternative 10: 35.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \sqrt{y \cdot \left(x \cdot 4\right)} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (sqrt (* y (* x 4.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return sqrt((y * (x * 4.0)));
}

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 = sqrt((y * (x * 4.0d0)))
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return Math.sqrt((y * (x * 4.0)));
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return math.sqrt((y * (x * 4.0)))

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return sqrt(Float64(y * Float64(x * 4.0)))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = sqrt((y * (x * 4.0)));
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[Sqrt[N[(y * N[(x * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 70.1%

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

   1. associate-+l+ 70.1%

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

   2. +-commutative 70.1%

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

   3. distribute-rgt-in 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in z around 0 25.8%

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

   1. add-sqr-sqrt 25.6%

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

   2. pow2 25.6%

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

   3. pow1/2 25.7%

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

   4. metadata-eval 25.7%

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

   5. sqrt-pow1 25.7%

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

   6. *-commutative 25.7%

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

   7. metadata-eval 25.7%

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

   8. metadata-eval 25.7%

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

7. Applied egg-rr 25.7%

   \[\leadsto 2 \cdot \color{blue}{{\left({\left(y \cdot x\right)}^{0.25}\right)}^{2}} \]
8. Step-by-step derivation

   1. pow-pow 25.9%

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

   2. metadata-eval 25.9%

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

   3. pow1/2 25.8%

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

   4. pow1 25.8%

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

   5. metadata-eval 25.8%

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

   6. metadata-eval 25.8%

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

   7. pow-sqr 25.7%

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

   8. pow-prod-down 25.8%

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

   9. *-commutative 25.8%

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

   10. *-commutative 25.8%

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

   11. swap-sqr 25.8%

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

   12. add-sqr-sqrt 25.9%

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

   13. metadata-eval 25.9%

       \[\leadsto {\left(\left(y \cdot x\right) \cdot \color{blue}{4}\right)}^{\left(0.25 \cdot 2\right)} \]

   14. metadata-eval 25.9%

       \[\leadsto {\left(\left(y \cdot x\right) \cdot 4\right)}^{\color{blue}{0.5}} \]

9. Applied egg-rr 25.9%

   \[\leadsto \color{blue}{{\left(\left(y \cdot x\right) \cdot 4\right)}^{0.5}} \]
10. Step-by-step derivation

    1. unpow1/2 25.8%

       \[\leadsto \color{blue}{\sqrt{\left(y \cdot x\right) \cdot 4}} \]

    2. associate-*l* 25.8%

       \[\leadsto \sqrt{\color{blue}{y \cdot \left(x \cdot 4\right)}} \]

11. Simplified 25.8%

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

Developer Target 1: 82.3% 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 2024136 
(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)))))