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

Percentage Accurate: 71.6% → 94.8%

Time: 17.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.6% 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: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+76}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-232}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+76)
   (* 2.0 (pow (exp (* 0.25 (- (log (- y)) (log (/ -1.0 x))))) 2.0))
   (if (<= y -2e-232)
     (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
     (* 2.0 (* (sqrt (fma x (/ y z) (+ y x))) (sqrt z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+76) {
		tmp = 2.0 * pow(exp((0.25 * (log(-y) - log((-1.0 / x))))), 2.0);
	} else if (y <= -2e-232) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(fma(x, (y / z), (y + x))) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e+76)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(-y)) - log(Float64(-1.0 / x))))) ^ 2.0));
	elseif (y <= -2e-232)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(fma(x, Float64(y / z), Float64(y + x))) * sqrt(z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -2.1e+76], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[(-y)], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2e-232], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(x * N[(y / z), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.10000000000000007e76

   1. Initial program 56.1%

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

      1. associate-+l+ 56.1%

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

      2. +-commutative 56.1%

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

      3. distribute-rgt-in 56.1%

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

   3. Simplified 56.1%

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

      1. distribute-rgt-in 56.1%

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

      2. associate-+r+ 56.1%

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

      3. *-commutative 56.1%

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

      4. distribute-lft-in 56.1%

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

      5. +-commutative 56.1%

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

      6. fma-undefine 56.2%

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

      7. add-sqr-sqrt 55.9%

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

      8. pow2 55.9%

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

      9. pow1/2 55.9%

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

      10. sqrt-pow1 55.9%

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

      11. metadata-eval 55.9%

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

   6. Applied egg-rr 55.9%

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

   7. Taylor expanded in z around 0 28.2%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(1 \cdot \left(x \cdot y\right)\right)}^{0.25}\right)}}^{2} \]
   8. Step-by-step derivation

      1. *-lft-identity 28.2%

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

   9. Simplified 28.2%

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

   10. Taylor expanded in x around -inf 48.8%

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

   if -2.10000000000000007e76 < y < -2.00000000000000005e-232

   1. Initial program 88.6%

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

      1. associate-+l+ 88.6%

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

      2. +-commutative 88.6%

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

      3. distribute-rgt-in 88.6%

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

   3. Simplified 88.6%

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

   if -2.00000000000000005e-232 < y

   1. Initial program 73.2%

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

      1. associate-+l+ 73.2%

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

      2. +-commutative 73.2%

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

      3. distribute-rgt-in 73.2%

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

   3. Simplified 73.2%

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

      1. distribute-rgt-in 73.2%

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

      2. associate-+r+ 73.2%

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

      3. *-commutative 73.2%

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

      4. distribute-lft-in 73.2%

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

      5. +-commutative 73.2%

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

      6. fma-undefine 73.3%

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

      7. add-sqr-sqrt 73.3%

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

      8. add-sqr-sqrt 73.1%

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

      9. associate-*l* 73.1%

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

      10. pow1/2 73.1%

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

      11. sqrt-pow1 73.2%

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

      12. metadata-eval 73.2%

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

      13. pow1/2 73.2%

          \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.5}}} \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\right)} \]

      14. sqrt-pow1 73.1%

          \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \left(\color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\right)} \]

      15. metadata-eval 73.1%

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

   6. Applied egg-rr 73.1%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \left({\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\right)}} \]
   7. Step-by-step derivation

      1. pow1/2 73.2%

         \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \left({\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.5}}\right)} \]

      2. pow-prod-up 73.3%

         \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{\left(0.25 + 0.5\right)}}} \]

      3. metadata-eval 73.3%

         \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{\color{blue}{0.75}}} \]

   8. Applied egg-rr 73.3%

      \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.75}}} \]
   9. Step-by-step derivation

      1. +-commutative 73.3%

         \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(x, z, y \cdot \color{blue}{\left(z + x\right)}\right)\right)}^{0.75}} \]

   10. Simplified 73.3%

       \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)\right)}^{0.75}}} \]

   11. Taylor expanded in z around inf 67.6%

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

       1. associate-+r+ 67.6%

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

       2. +-commutative 67.6%

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

       3. associate-/l* 64.1%

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

   13. Simplified 64.1%

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

       1. sqrt-prod 49.6%

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

       2. *-commutative 49.6%

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

       3. +-commutative 49.6%

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

       4. fma-define 49.6%

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

   15. Applied egg-rr 49.6%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+76}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-232}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{+62}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.75e+62)
   (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
   (* 2.0 (* (sqrt (fma x (/ y z) (+ y x))) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.75e+62) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(fma(x, (y / z), (y + x))) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 1.75e+62)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(fma(x, Float64(y / z), Float64(y + x))) * sqrt(z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 1.75e+62], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(x * N[(y / z), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.74999999999999992e62

   1. Initial program 77.1%

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

      1. associate-+l+ 77.1%

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

      2. +-commutative 77.1%

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

      3. distribute-rgt-in 77.2%

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

   3. Simplified 77.2%

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

   if 1.74999999999999992e62 < z

   1. Initial program 51.4%

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

      1. associate-+l+ 51.4%

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

      2. +-commutative 51.4%

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

      3. distribute-rgt-in 51.4%

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

   3. Simplified 51.4%

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

      1. distribute-rgt-in 51.4%

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

      2. associate-+r+ 51.4%

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

      3. *-commutative 51.4%

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

      4. distribute-lft-in 51.4%

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

      5. +-commutative 51.4%

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

      6. fma-undefine 51.6%

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

      7. add-sqr-sqrt 51.6%

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

      8. add-sqr-sqrt 51.5%

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

      9. associate-*l* 51.5%

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

      10. pow1/2 51.5%

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

      11. sqrt-pow1 51.5%

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

      12. metadata-eval 51.5%

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

      13. pow1/2 51.5%

          \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.5}}} \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\right)} \]

      14. sqrt-pow1 51.5%

          \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \left(\color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\right)} \]

      15. metadata-eval 51.5%

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

   6. Applied egg-rr 51.5%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \left({\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\right)}} \]
   7. Step-by-step derivation

      1. pow1/2 51.6%

         \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \left({\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.5}}\right)} \]

      2. pow-prod-up 51.7%

         \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{\left(0.25 + 0.5\right)}}} \]

      3. metadata-eval 51.7%

         \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{\color{blue}{0.75}}} \]

   8. Applied egg-rr 51.7%

      \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.75}}} \]
   9. Step-by-step derivation

      1. +-commutative 51.7%

         \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(x, z, y \cdot \color{blue}{\left(z + x\right)}\right)\right)}^{0.75}} \]

   10. Simplified 51.7%

       \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.25} \cdot \color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)\right)}^{0.75}}} \]

   11. Taylor expanded in z around inf 51.4%

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

       1. associate-+r+ 51.4%

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

       2. +-commutative 51.4%

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

       3. associate-/l* 51.8%

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

   13. Simplified 51.8%

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

       1. sqrt-prod 99.6%

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

       2. *-commutative 99.6%

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

       3. +-commutative 99.6%

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

       4. fma-define 99.6%

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

   15. Applied egg-rr 99.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{+62}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + \left(z + y \cdot \frac{z}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y + x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5e-289)
   (* 2.0 (sqrt (* x (+ y (+ z (* y (/ z x)))))))
   (* 2.0 (* (sqrt z) (sqrt (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e-289) {
		tmp = 2.0 * sqrt((x * (y + (z + (y * (z / x))))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt((y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.5d-289) then
        tmp = 2.0d0 * sqrt((x * (y + (z + (y * (z / x))))))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt((y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e-289) {
		tmp = 2.0 * Math.sqrt((x * (y + (z + (y * (z / x))))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt((y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.5e-289:
		tmp = 2.0 * math.sqrt((x * (y + (z + (y * (z / x))))))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt((y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5e-289)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + Float64(z + Float64(y * Float64(z / x)))))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5e-289)
		tmp = 2.0 * sqrt((x * (y + (z + (y * (z / x))))));
	else
		tmp = 2.0 * (sqrt(z) * sqrt((y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2.5e-289], N[(2.0 * N[Sqrt[N[(x * N[(y + N[(z + N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.50000000000000014e-289

   1. Initial program 74.0%

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

      1. associate-+l+ 74.0%

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

      2. +-commutative 74.0%

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

      3. distribute-rgt-in 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in x around inf 64.7%

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

      1. associate-/l* 62.5%

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

   7. Simplified 62.5%

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

   if 2.50000000000000014e-289 < y

   1. Initial program 71.1%

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

      1. associate-+l+ 71.1%

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

      2. +-commutative 71.1%

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

      3. distribute-rgt-in 71.2%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in z around inf 46.9%

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

      1. +-commutative 46.9%

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

   7. Simplified 46.9%

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

      1. sqrt-prod 45.3%

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

      2. *-commutative 45.3%

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

      3. +-commutative 45.3%

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

   9. Applied egg-rr 45.3%

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

       1. +-commutative 45.3%

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

   11. Simplified 45.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + \left(z + y \cdot \frac{z}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y + x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.05e+22)
   (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.05e+22) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.05d+22) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.05e+22) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.05e+22:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.05e+22)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.05e+22)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.05e+22], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0499999999999999e22

   1. Initial program 77.5%

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

      1. associate-+l+ 77.5%

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

      2. +-commutative 77.5%

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

      3. distribute-rgt-in 77.5%

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

   3. Simplified 77.5%

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

   if 1.0499999999999999e22 < y

   1. Initial program 56.3%

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

      1. associate-+l+ 56.3%

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

      2. +-commutative 56.3%

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

      3. distribute-rgt-in 56.5%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in x around 0 24.4%

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

      1. *-commutative 24.4%

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

      2. sqrt-prod 41.8%

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

   7. Applied egg-rr 41.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-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 <= -1e-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 <= (-1d-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 <= -1e-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 <= -1e-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 <= -1e-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 <= -1e-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, -1e-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 < -1e-281

   1. Initial program 72.9%

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

      1. associate-+l+ 72.9%

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

      2. +-commutative 72.9%

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

      3. distribute-rgt-in 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in x around inf 48.0%

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

   if -1e-281 < y

   1. Initial program 72.3%

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

      1. associate-+l+ 72.3%

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

      2. +-commutative 72.3%

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

      3. distribute-rgt-in 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in z around inf 49.8%

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

      1. +-commutative 49.8%

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

   7. Simplified 49.8%

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

Alternative 6: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-261}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-261) (* 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 <= 5e-261) {
		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 <= 5d-261) 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 <= 5e-261) {
		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 <= 5e-261:
		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 <= 5e-261)
		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 <= 5e-261)
		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, 5e-261], 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 < 4.99999999999999981e-261

   1. Initial program 74.6%

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

      1. associate-+l+ 74.6%

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

      2. +-commutative 74.6%

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

      3. distribute-rgt-in 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around inf 52.5%

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

   if 4.99999999999999981e-261 < y

   1. Initial program 70.2%

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

      1. associate-+l+ 70.2%

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

      2. +-commutative 70.2%

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

      3. distribute-rgt-in 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in x around 0 25.5%

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

Alternative 7: 71.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

   1. associate-+l+ 72.6%

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

   2. +-commutative 72.6%

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

   3. distribute-rgt-in 72.6%

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

3. Simplified 72.6%

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

5. Final simplification 72.6%

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

Alternative 8: 69.5% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-310) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4d-310)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -4e-310:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-310)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4e-310)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -4e-310], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.999999999999988e-310

   1. Initial program 74.1%

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

      1. associate-+l+ 74.1%

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

      2. +-commutative 74.1%

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

      3. distribute-rgt-in 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in z around 0 29.1%

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

      1. *-commutative 29.1%

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

   7. Simplified 29.1%

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

   if -3.999999999999988e-310 < y

   1. Initial program 71.0%

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

      1. associate-+l+ 71.0%

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

      2. +-commutative 71.0%

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

      3. distribute-rgt-in 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in x around 0 23.7%

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

Alternative 9: 36.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((y * x));
end

Wolfram

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

Derivation

1. Initial program 72.6%

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

   1. associate-+l+ 72.6%

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

   2. +-commutative 72.6%

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

   3. distribute-rgt-in 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in z around 0 27.3%

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

   1. *-commutative 27.3%

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

7. Simplified 27.3%

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

Developer target: 83.8% 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 2024086 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

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

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