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

Percentage Accurate: 70.7% → 94.1%

Time: 14.7s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.7% 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, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.7 \cdot 10^{+47}:\\ \;\;\;\;{\left(\sqrt[3]{2} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{3}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6.7e+47)
   (pow
    (*
     (cbrt 2.0)
     (exp (* 0.16666666666666666 (- (log (- (- x) z)) (log (/ -1.0 y))))))
    3.0)
   (if (<= y 2.25e-256)
     (* 2.0 (sqrt (fma x y (* z (+ y x)))))
     (* 2.0 (* (sqrt z) (sqrt y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.7e+47) {
		tmp = pow((cbrt(2.0) * exp((0.16666666666666666 * (log((-x - z)) - log((-1.0 / y)))))), 3.0);
	} else if (y <= 2.25e-256) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -6.7e+47)
		tmp = Float64(cbrt(2.0) * exp(Float64(0.16666666666666666 * Float64(log(Float64(Float64(-x) - z)) - log(Float64(-1.0 / y)))))) ^ 3.0;
	elseif (y <= 2.25e-256)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * 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, -6.7e+47], N[Power[N[(N[Power[2.0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[Log[N[((-x) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[y, 2.25e-256], 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[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.69999999999999973e47

   1. Initial program 46.9%

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

      1. +-commutative 46.9%

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

      2. associate-+r+ 46.9%

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

      3. *-commutative 46.9%

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

      4. +-commutative 46.9%

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

      5. associate-+l+ 46.9%

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

      6. *-commutative 46.9%

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

      7. distribute-rgt-in 46.9%

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

   3. Simplified 46.9%

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

      1. add-sqr-sqrt 46.6%

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

      2. sqrt-unprod 46.9%

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

      3. swap-sqr 46.9%

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

      4. add-sqr-sqrt 46.9%

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

      5. distribute-rgt-in 46.9%

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

      6. associate-+r+ 46.9%

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

      7. *-commutative 46.9%

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

      8. distribute-lft-in 47.0%

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

      9. +-commutative 47.0%

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

      10. fma-undefine 47.1%

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

      11. add-sqr-sqrt 47.1%

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

      12. swap-sqr 47.1%

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

      13. sqrt-unprod 46.8%

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

      14. add-sqr-sqrt 47.1%

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

   6. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}}\right)}^{3}} \]

   7. Taylor expanded in y around -inf 83.8%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{2} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-1 \cdot \left(x + z\right)\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right)}\right)}}^{3} \]

   if -6.69999999999999973e47 < y < 2.2500000000000001e-256

   1. Initial program 84.5%

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

      1. associate-+l+ 84.5%

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

      2. *-commutative 84.5%

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

      3. *-commutative 84.5%

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

      4. *-commutative 84.5%

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

      5. +-commutative 84.5%

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

      6. +-commutative 84.5%

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

      7. associate-+l+ 84.5%

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

      8. *-commutative 84.5%

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

      9. *-commutative 84.5%

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

      10. +-commutative 84.5%

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

      11. +-commutative 84.5%

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

      12. *-commutative 84.5%

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

      13. *-commutative 84.5%

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

      14. associate-+l+ 84.5%

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

      15. +-commutative 84.5%

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

      16. *-commutative 84.5%

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

      17. fma-define 84.5%

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

   3. Simplified 84.5%

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

   if 2.2500000000000001e-256 < y

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

      2. associate-+r+ 71.3%

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

      3. *-commutative 71.3%

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

      4. +-commutative 71.3%

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

      5. associate-+l+ 71.3%

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

      6. *-commutative 71.3%

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

      7. 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. Step-by-step derivation

      1. add-sqr-sqrt 70.9%

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

      2. sqrt-unprod 70.5%

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

      3. swap-sqr 70.5%

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

      4. add-sqr-sqrt 70.5%

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

      5. distribute-rgt-in 70.4%

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

      6. associate-+r+ 70.4%

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

      7. *-commutative 70.4%

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

      8. distribute-lft-in 70.5%

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

      9. +-commutative 70.5%

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

      10. fma-undefine 70.7%

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

      11. add-sqr-sqrt 70.7%

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

      12. swap-sqr 70.7%

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

      13. sqrt-unprod 71.1%

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

      14. add-sqr-sqrt 71.5%

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

   6. Applied egg-rr 70.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}}\right)}^{3}} \]

   7. Taylor expanded in x around 0 27.8%

      \[\leadsto \color{blue}{\sqrt{y \cdot z} \cdot {\left(\sqrt[3]{2}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 28.6%

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

   9. Simplified 28.6%

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

       1. *-commutative 28.6%

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

       2. sqrt-prod 35.5%

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

   11. Applied egg-rr 35.5%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.7 \cdot 10^{+47}:\\ \;\;\;\;{\left(\sqrt[3]{2} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{3}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-268}:\\ \;\;\;\;{\left(\sqrt[3]{2} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e-268)
   (pow
    (*
     (cbrt 2.0)
     (exp (* 0.16666666666666666 (- (log (- (- y) z)) (log (/ -1.0 x))))))
    3.0)
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e-268) {
		tmp = pow((cbrt(2.0) * exp((0.16666666666666666 * (log((-y - z)) - log((-1.0 / x)))))), 3.0);
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e-268) {
		tmp = Math.pow((Math.cbrt(2.0) * Math.exp((0.16666666666666666 * (Math.log((-y - z)) - Math.log((-1.0 / x)))))), 3.0);
	} 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 <= 2.1e-268)
		tmp = Float64(cbrt(2.0) * exp(Float64(0.16666666666666666 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x)))))) ^ 3.0;
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * 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, 2.1e-268], N[Power[N[(N[Power[2.0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $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 < 2.09999999999999998e-268

   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. +-commutative 69.0%

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

      2. associate-+r+ 69.0%

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

      3. *-commutative 69.0%

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

      4. +-commutative 69.0%

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

      5. associate-+l+ 69.0%

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

      6. *-commutative 69.0%

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

      7. 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. Step-by-step derivation

      1. add-sqr-sqrt 68.6%

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

      2. sqrt-unprod 69.0%

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

      3. swap-sqr 69.0%

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

      4. add-sqr-sqrt 69.0%

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

      5. distribute-rgt-in 69.0%

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

      6. associate-+r+ 69.0%

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

      7. *-commutative 69.0%

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

      8. distribute-lft-in 69.0%

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

      9. +-commutative 69.0%

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

      10. fma-undefine 69.1%

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

      11. add-sqr-sqrt 69.1%

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

      12. swap-sqr 69.1%

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

      13. sqrt-unprod 68.7%

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

      14. add-sqr-sqrt 69.1%

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

   6. Applied egg-rr 67.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}}\right)}^{3}} \]

   7. Taylor expanded in x around -inf 40.7%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{2} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}}^{3} \]

   if 2.09999999999999998e-268 < 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. +-commutative 71.5%

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

      2. associate-+r+ 71.5%

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

      3. *-commutative 71.5%

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

      4. +-commutative 71.5%

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

      5. associate-+l+ 71.5%

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

      6. *-commutative 71.5%

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

      7. distribute-rgt-in 71.6%

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

   3. Simplified 71.6%

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

      1. add-sqr-sqrt 71.1%

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

      2. sqrt-unprod 70.8%

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

      3. swap-sqr 70.8%

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

      4. add-sqr-sqrt 70.8%

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

      5. distribute-rgt-in 70.7%

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

      6. associate-+r+ 70.7%

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

      7. *-commutative 70.7%

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

      8. distribute-lft-in 70.8%

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

      9. +-commutative 70.8%

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

      10. fma-undefine 71.0%

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

      11. add-sqr-sqrt 71.0%

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

      12. swap-sqr 71.0%

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

      13. sqrt-unprod 71.3%

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

      14. add-sqr-sqrt 71.8%

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

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}}\right)}^{3}} \]

   7. Taylor expanded in x around 0 28.4%

      \[\leadsto \color{blue}{\sqrt{y \cdot z} \cdot {\left(\sqrt[3]{2}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 29.2%

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

   9. Simplified 29.2%

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

       1. *-commutative 29.2%

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

       2. sqrt-prod 36.1%

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

   11. Applied egg-rr 36.1%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-268}:\\ \;\;\;\;{\left(\sqrt[3]{2} \cdot e^{0.16666666666666666 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{\left(z + x\right) \cdot {y}^{3}}} - 2 \cdot \sqrt{\frac{z + x}{y}}\right)\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

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+31)
   (*
    y
    (-
     (* (* z x) (sqrt (/ 1.0 (* (+ z x) (pow y 3.0)))))
     (* 2.0 (sqrt (/ (+ z x) y)))))
   (if (<= y 2.25e-256)
     (* 2.0 (sqrt (fma x y (* z (+ y x)))))
     (* 2.0 (* (sqrt z) (sqrt y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.45e+31) {
		tmp = y * (((z * x) * sqrt((1.0 / ((z + x) * pow(y, 3.0))))) - (2.0 * sqrt(((z + x) / y))));
	} else if (y <= 2.25e-256) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.45e+31)
		tmp = Float64(y * Float64(Float64(Float64(z * x) * sqrt(Float64(1.0 / Float64(Float64(z + x) * (y ^ 3.0))))) - Float64(2.0 * sqrt(Float64(Float64(z + x) / y)))));
	elseif (y <= 2.25e-256)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * 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, -1.45e+31], N[(y * N[(N[(N[(z * x), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(z + x), $MachinePrecision] * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[Sqrt[N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e-256], 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[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45e31

   1. Initial program 50.2%

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

      1. +-commutative 50.2%

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

      2. associate-+r+ 50.2%

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

      3. *-commutative 50.2%

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

      4. +-commutative 50.2%

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

      5. associate-+l+ 50.2%

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

      6. *-commutative 50.2%

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

      7. distribute-rgt-in 50.2%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in y around inf 0.8%

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

   6. Taylor expanded in y around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 81.8%

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

      4. +-commutative 81.8%

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

   8. Simplified 81.8%

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

   if -1.45e31 < y < 2.2500000000000001e-256

   1. Initial program 83.8%

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

      1. associate-+l+ 83.8%

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

      2. *-commutative 83.8%

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

      3. *-commutative 83.8%

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

      4. *-commutative 83.8%

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

      5. +-commutative 83.8%

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

      6. +-commutative 83.8%

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

      7. associate-+l+ 83.8%

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

      8. *-commutative 83.8%

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

      9. *-commutative 83.8%

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

      10. +-commutative 83.8%

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

      11. +-commutative 83.8%

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

      12. *-commutative 83.8%

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

      13. *-commutative 83.8%

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

      14. associate-+l+ 83.8%

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

      15. +-commutative 83.8%

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

      16. *-commutative 83.8%

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

      17. fma-define 83.8%

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

   3. Simplified 83.8%

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

   if 2.2500000000000001e-256 < y

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

      2. associate-+r+ 71.3%

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

      3. *-commutative 71.3%

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

      4. +-commutative 71.3%

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

      5. associate-+l+ 71.3%

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

      6. *-commutative 71.3%

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

      7. 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. Step-by-step derivation

      1. add-sqr-sqrt 70.9%

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

      2. sqrt-unprod 70.5%

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

      3. swap-sqr 70.5%

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

      4. add-sqr-sqrt 70.5%

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

      5. distribute-rgt-in 70.4%

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

      6. associate-+r+ 70.4%

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

      7. *-commutative 70.4%

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

      8. distribute-lft-in 70.5%

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

      9. +-commutative 70.5%

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

      10. fma-undefine 70.7%

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

      11. add-sqr-sqrt 70.7%

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

      12. swap-sqr 70.7%

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

      13. sqrt-unprod 71.1%

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

      14. add-sqr-sqrt 71.5%

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

   6. Applied egg-rr 70.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}}\right)}^{3}} \]

   7. Taylor expanded in x around 0 27.8%

      \[\leadsto \color{blue}{\sqrt{y \cdot z} \cdot {\left(\sqrt[3]{2}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 28.6%

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

   9. Simplified 28.6%

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

       1. *-commutative 28.6%

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

       2. sqrt-prod 35.5%

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

   11. Applied egg-rr 35.5%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{\left(z + x\right) \cdot {y}^{3}}} - 2 \cdot \sqrt{\frac{z + x}{y}}\right)\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.25e-256)
   (* 2.0 (sqrt (fma x z (* y (+ z x)))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.25e-256) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.25e-256)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * 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, 2.25e-256], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.2500000000000001e-256

   1. Initial program 69.2%

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

      1. associate-+l+ 69.2%

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

      2. *-commutative 69.2%

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

      3. *-commutative 69.2%

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

      4. *-commutative 69.2%

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

      5. +-commutative 69.2%

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

      6. +-commutative 69.2%

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

      7. +-commutative 69.2%

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

      8. *-commutative 69.2%

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

      9. *-commutative 69.2%

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

      10. associate-+l+ 69.2%

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

      11. +-commutative 69.2%

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

      12. fma-define 69.2%

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

      13. distribute-lft-out 69.3%

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

   3. Simplified 69.3%

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

   if 2.2500000000000001e-256 < y

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

      2. associate-+r+ 71.3%

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

      3. *-commutative 71.3%

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

      4. +-commutative 71.3%

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

      5. associate-+l+ 71.3%

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

      6. *-commutative 71.3%

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

      7. 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. Step-by-step derivation

      1. add-sqr-sqrt 70.9%

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

      2. sqrt-unprod 70.5%

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

      3. swap-sqr 70.5%

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

      4. add-sqr-sqrt 70.5%

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

      5. distribute-rgt-in 70.4%

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

      6. associate-+r+ 70.4%

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

      7. *-commutative 70.4%

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

      8. distribute-lft-in 70.5%

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

      9. +-commutative 70.5%

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

      10. fma-undefine 70.7%

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

      11. add-sqr-sqrt 70.7%

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

      12. swap-sqr 70.7%

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

      13. sqrt-unprod 71.1%

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

      14. add-sqr-sqrt 71.5%

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

   6. Applied egg-rr 70.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}}\right)}^{3}} \]

   7. Taylor expanded in x around 0 27.8%

      \[\leadsto \color{blue}{\sqrt{y \cdot z} \cdot {\left(\sqrt[3]{2}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 28.6%

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

   9. Simplified 28.6%

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

       1. *-commutative 28.6%

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

       2. sqrt-prod 35.5%

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

   11. Applied egg-rr 35.5%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-269}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y \cdot \left(1 + \frac{z}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e-269)
   (* 2.0 (sqrt (* x (* y (+ 1.0 (/ z y))))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e-269) {
		tmp = 2.0 * sqrt((x * (y * (1.0 + (z / y)))));
	} 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 <= 9.2d-269) then
        tmp = 2.0d0 * sqrt((x * (y * (1.0d0 + (z / y)))))
    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 <= 9.2e-269) {
		tmp = 2.0 * Math.sqrt((x * (y * (1.0 + (z / y)))));
	} 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 <= 9.2e-269:
		tmp = 2.0 * math.sqrt((x * (y * (1.0 + (z / y)))))
	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 <= 9.2e-269)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y * Float64(1.0 + Float64(z / y))))));
	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 <= 9.2e-269)
		tmp = 2.0 * sqrt((x * (y * (1.0 + (z / y)))));
	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, 9.2e-269], N[(2.0 * N[Sqrt[N[(x * N[(y * N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.1999999999999999e-269

   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%

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

      3. *-commutative 69.0%

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

      4. *-commutative 69.0%

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

      5. +-commutative 69.0%

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

      6. +-commutative 69.0%

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

      7. associate-+l+ 69.0%

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

      8. *-commutative 69.0%

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

      9. *-commutative 69.0%

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

      10. +-commutative 69.0%

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

      11. +-commutative 69.0%

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

      12. *-commutative 69.0%

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

      13. *-commutative 69.0%

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

      14. associate-+l+ 69.0%

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

      15. +-commutative 69.0%

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

      16. *-commutative 69.0%

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

      17. fma-define 69.0%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in y around inf 59.8%

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

      1. associate-+r+ 59.8%

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

      2. +-commutative 59.8%

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

      3. associate-/l* 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in x around inf 45.6%

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

   if 9.1999999999999999e-269 < 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. +-commutative 71.5%

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

      2. associate-+r+ 71.5%

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

      3. *-commutative 71.5%

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

      4. +-commutative 71.5%

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

      5. associate-+l+ 71.5%

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

      6. *-commutative 71.5%

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

      7. distribute-rgt-in 71.6%

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

   3. Simplified 71.6%

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

      1. add-sqr-sqrt 71.1%

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

      2. sqrt-unprod 70.8%

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

      3. swap-sqr 70.8%

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

      4. add-sqr-sqrt 70.8%

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

      5. distribute-rgt-in 70.7%

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

      6. associate-+r+ 70.7%

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

      7. *-commutative 70.7%

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

      8. distribute-lft-in 70.8%

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

      9. +-commutative 70.8%

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

      10. fma-undefine 71.0%

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

      11. add-sqr-sqrt 71.0%

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

      12. swap-sqr 71.0%

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

      13. sqrt-unprod 71.3%

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

      14. add-sqr-sqrt 71.8%

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

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}}\right)}^{3}} \]

   7. Taylor expanded in x around 0 28.4%

      \[\leadsto \color{blue}{\sqrt{y \cdot z} \cdot {\left(\sqrt[3]{2}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 29.2%

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

   9. Simplified 29.2%

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

       1. *-commutative 29.2%

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

       2. sqrt-prod 36.1%

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

   11. Applied egg-rr 36.1%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-269}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y \cdot \left(1 + \frac{z}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y \cdot \left(1 + \frac{z}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e-281)
   (* 2.0 (sqrt (* x (* y (+ 1.0 (/ 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 <= -2.5e-281) {
		tmp = 2.0 * sqrt((x * (y * (1.0 + (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 <= (-2.5d-281)) then
        tmp = 2.0d0 * sqrt((x * (y * (1.0d0 + (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 <= -2.5e-281) {
		tmp = 2.0 * Math.sqrt((x * (y * (1.0 + (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 <= -2.5e-281:
		tmp = 2.0 * math.sqrt((x * (y * (1.0 + (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 <= -2.5e-281)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y * Float64(1.0 + 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 <= -2.5e-281)
		tmp = 2.0 * sqrt((x * (y * (1.0 + (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, -2.5e-281], N[(2.0 * N[Sqrt[N[(x * N[(y * N[(1.0 + 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 < -2.4999999999999999e-281

   1. Initial program 68.0%

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

      1. associate-+l+ 68.0%

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

      2. *-commutative 68.0%

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

      3. *-commutative 68.0%

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

      4. *-commutative 68.0%

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

      5. +-commutative 68.0%

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

      6. +-commutative 68.0%

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

      7. associate-+l+ 68.0%

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

      8. *-commutative 68.0%

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

      9. *-commutative 68.0%

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

      10. +-commutative 68.0%

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

      11. +-commutative 68.0%

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

      12. *-commutative 68.0%

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

      13. *-commutative 68.0%

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

      14. associate-+l+ 68.0%

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

      15. +-commutative 68.0%

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

      16. *-commutative 68.0%

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

      17. fma-define 68.1%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in y around inf 62.2%

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

      1. associate-+r+ 62.2%

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

      2. +-commutative 62.2%

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

      3. associate-/l* 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around inf 44.8%

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

   if -2.4999999999999999e-281 < y

   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. +-commutative 72.1%

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

      2. associate-+r+ 72.1%

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

      3. *-commutative 72.1%

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

      4. +-commutative 72.1%

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

      5. associate-+l+ 72.1%

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

      6. *-commutative 72.1%

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

      7. distribute-rgt-in 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in z around inf 50.7%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y \cdot \left(1 + \frac{z}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.7% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-281) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-281) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2d-281)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-281) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2e-281:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-281)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-281)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -2e-281], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e-281

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. associate-+r+ 68.0%

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

      3. *-commutative 68.0%

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

      4. +-commutative 68.0%

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

      5. associate-+l+ 68.0%

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

      6. *-commutative 68.0%

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

      7. distribute-rgt-in 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in x around inf 47.0%

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

      1. +-commutative 47.0%

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

   7. Simplified 47.0%

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

   if -2e-281 < y

   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. +-commutative 72.1%

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

      2. associate-+r+ 72.1%

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

      3. *-commutative 72.1%

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

      4. +-commutative 72.1%

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

      5. associate-+l+ 72.1%

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

      6. *-commutative 72.1%

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

      7. distribute-rgt-in 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in z around inf 50.7%

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

4. Final simplification 48.9%

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

Alternative 8: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-286}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6e-286) (* 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 <= -6e-286) {
		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 <= (-6d-286)) 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 <= -6e-286) {
		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 <= -6e-286:
		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 <= -6e-286)
		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 <= -6e-286)
		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, -6e-286], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.0000000000000001e-286

   1. Initial program 68.3%

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

      1. +-commutative 68.3%

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

      2. associate-+r+ 68.3%

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

      3. *-commutative 68.3%

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

      4. +-commutative 68.3%

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

      5. associate-+l+ 68.3%

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

      6. *-commutative 68.3%

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

      7. distribute-rgt-in 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in x around inf 47.4%

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

      1. +-commutative 47.4%

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

   7. Simplified 47.4%

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

   if -6.0000000000000001e-286 < y

   1. Initial program 71.9%

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

      1. +-commutative 71.9%

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

      2. associate-+r+ 71.9%

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

      3. *-commutative 71.9%

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

      4. +-commutative 71.9%

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

      5. associate-+l+ 71.9%

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

      6. *-commutative 71.9%

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

      7. 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 26.2%

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

      1. *-commutative 26.2%

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

   7. Simplified 26.2%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-286}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 1.0× speedup

Math

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

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

C

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

Python

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

Julia

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

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 2.0 * sqrt(((z * (y + x)) + (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[(z * N[(y + x), $MachinePrecision]), $MachinePrecision] + N[(y * x), $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. +-commutative 70.1%

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

   2. associate-+r+ 70.1%

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

   3. *-commutative 70.1%

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

   4. +-commutative 70.1%

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

   5. associate-+l+ 70.1%

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

   6. *-commutative 70.1%

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

   7. 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%

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

Alternative 10: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-286}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6e-286) (* 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 <= -6e-286) {
		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 <= (-6d-286)) 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 <= -6e-286) {
		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 <= -6e-286:
		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 <= -6e-286)
		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 <= -6e-286)
		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, -6e-286], 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 < -6.0000000000000001e-286

   1. Initial program 68.3%

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

      1. +-commutative 68.3%

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

      2. associate-+r+ 68.3%

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

      3. *-commutative 68.3%

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

      4. +-commutative 68.3%

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

      5. associate-+l+ 68.3%

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

      6. *-commutative 68.3%

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

      7. distribute-rgt-in 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in z around 0 29.0%

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

   if -6.0000000000000001e-286 < y

   1. Initial program 71.9%

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

      1. +-commutative 71.9%

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

      2. associate-+r+ 71.9%

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

      3. *-commutative 71.9%

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

      4. +-commutative 71.9%

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

      5. associate-+l+ 71.9%

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

      6. *-commutative 71.9%

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

      7. 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 26.2%

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

      1. *-commutative 26.2%

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

   7. Simplified 26.2%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-286}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.6% 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. +-commutative 70.1%

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

   2. associate-+r+ 70.1%

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

   3. *-commutative 70.1%

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

   4. +-commutative 70.1%

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

   5. associate-+l+ 70.1%

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

   6. *-commutative 70.1%

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

   7. 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 26.7%

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

6. Final simplification 26.7%

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

Developer Target 1: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\\ \mathbf{if}\;z < 7.636950090573675 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z)))
          (* (pow z 0.25) (pow y 0.25)))))
   (if (< z 7.636950090573675e+176)
     (* 2.0 (sqrt (+ (* (+ x y) z) (* x y))))
     (* (* t_0 t_0) 2.0))))

C

double code(double x, double y, double z) {
	double t_0 = (0.25 * ((pow(y, -0.75) * (pow(z, -0.75) * x)) * (y + z))) + (pow(z, 0.25) * pow(y, 0.25));
	double tmp;
	if (z < 7.636950090573675e+176) {
		tmp = 2.0 * sqrt((((x + y) * z) + (x * y)));
	} else {
		tmp = (t_0 * t_0) * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.25d0 * (((y ** (-0.75d0)) * ((z ** (-0.75d0)) * x)) * (y + z))) + ((z ** 0.25d0) * (y ** 0.25d0))
    if (z < 7.636950090573675d+176) then
        tmp = 2.0d0 * sqrt((((x + y) * z) + (x * y)))
    else
        tmp = (t_0 * t_0) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.25 * ((Math.pow(y, -0.75) * (Math.pow(z, -0.75) * x)) * (y + z))) + (Math.pow(z, 0.25) * Math.pow(y, 0.25));
	double tmp;
	if (z < 7.636950090573675e+176) {
		tmp = 2.0 * Math.sqrt((((x + y) * z) + (x * y)));
	} else {
		tmp = (t_0 * t_0) * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.25 * ((math.pow(y, -0.75) * (math.pow(z, -0.75) * x)) * (y + z))) + (math.pow(z, 0.25) * math.pow(y, 0.25))
	tmp = 0
	if z < 7.636950090573675e+176:
		tmp = 2.0 * math.sqrt((((x + y) * z) + (x * y)))
	else:
		tmp = (t_0 * t_0) * 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.25 * Float64(Float64((y ^ -0.75) * Float64((z ^ -0.75) * x)) * Float64(y + z))) + Float64((z ^ 0.25) * (y ^ 0.25)))
	tmp = 0.0
	if (z < 7.636950090573675e+176)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x + y) * z) + Float64(x * y))));
	else
		tmp = Float64(Float64(t_0 * t_0) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.25 * (((y ^ -0.75) * ((z ^ -0.75) * x)) * (y + z))) + ((z ^ 0.25) * (y ^ 0.25));
	tmp = 0.0;
	if (z < 7.636950090573675e+176)
		tmp = 2.0 * sqrt((((x + y) * z) + (x * y)));
	else
		tmp = (t_0 * t_0) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.25 * N[(N[(N[Power[y, -0.75], $MachinePrecision] * N[(N[Power[z, -0.75], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[z, 0.25], $MachinePrecision] * N[Power[y, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, 7.636950090573675e+176], N[(2.0 * N[Sqrt[N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 2.0), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024150 
(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)))))