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

Percentage Accurate: 70.6% → 95.0%

Time: 18.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.6% 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: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-217}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          2.0
          (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0))))
   (if (<= y -1.02e+35)
     t_0
     (if (<= y -1.55e-217)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y 1.1e-291) t_0 (* 2.0 (* (sqrt z) (sqrt y))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -1.02e+35) {
		tmp = t_0;
	} else if (y <= -1.55e-217) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= 1.1e-291) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (exp((0.25d0 * (log((-y - z)) - log(((-1.0d0) / x))))) ** 2.0d0)
    if (y <= (-1.02d+35)) then
        tmp = t_0
    else if (y <= (-1.55d-217)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= 1.1d-291) then
        tmp = t_0
    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 t_0 = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((-y - z)) - Math.log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -1.02e+35) {
		tmp = t_0;
	} else if (y <= -1.55e-217) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= 1.1e-291) {
		tmp = t_0;
	} 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):
	t_0 = 2.0 * math.pow(math.exp((0.25 * (math.log((-y - z)) - math.log((-1.0 / x))))), 2.0)
	tmp = 0
	if y <= -1.02e+35:
		tmp = t_0
	elif y <= -1.55e-217:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= 1.1e-291:
		tmp = t_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)
	t_0 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0))
	tmp = 0.0
	if (y <= -1.02e+35)
		tmp = t_0;
	elseif (y <= -1.55e-217)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= 1.1e-291)
		tmp = t_0;
	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)
	t_0 = 2.0 * (exp((0.25 * (log((-y - z)) - log((-1.0 / x))))) ^ 2.0);
	tmp = 0.0;
	if (y <= -1.02e+35)
		tmp = t_0;
	elseif (y <= -1.55e-217)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= 1.1e-291)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.02e+35], t$95$0, If[LessEqual[y, -1.55e-217], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-291], t$95$0, N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.02000000000000007e35 or -1.5499999999999999e-217 < y < 1.10000000000000001e-291

   1. Initial program 61.5%

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

      1. associate-+l+ 61.5%

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

      2. +-commutative 61.5%

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

      3. distribute-rgt-in 61.6%

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

   3. Simplified 61.6%

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

      1. distribute-rgt-in 61.5%

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

      2. associate-+r+ 61.5%

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

      3. *-commutative 61.5%

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

      4. distribute-lft-in 61.6%

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

      5. +-commutative 61.6%

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

      6. fma-udef 61.9%

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

      7. add-sqr-sqrt 61.4%

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

      8. pow2 61.4%

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

      9. pow1/2 61.4%

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

      10. sqrt-pow1 61.4%

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

      11. fma-udef 61.1%

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

      12. *-commutative 61.1%

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

      13. +-commutative 61.1%

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

      14. distribute-rgt-in 61.1%

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

      15. associate-+l+ 61.1%

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

      16. distribute-lft-in 61.1%

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

      17. fma-def 61.3%

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

      18. metadata-eval 61.3%

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

   6. Applied egg-rr 61.3%

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

   7. Taylor expanded in x around -inf 40.4%

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

   if -1.02000000000000007e35 < y < -1.5499999999999999e-217

   1. Initial program 87.4%

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

      1. associate-+l+ 87.4%

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

      2. +-commutative 87.4%

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

      3. distribute-rgt-in 87.4%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in x around inf 55.9%

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

   if 1.10000000000000001e-291 < y

   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 0 28.3%

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

      1. sqrt-prod 39.2%

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

   7. Applied egg-rr 39.2%

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

      1. *-commutative 39.2%

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

   9. Simplified 39.2%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+35}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-217}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-291}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+38}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+32}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6e+38)
   (* 2.0 (pow (exp (* 0.25 (- (log (- y)) (log (/ -1.0 x))))) 2.0))
   (if (<= y 2.5e+32)
     (* 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 <= -6e+38) {
		tmp = 2.0 * pow(exp((0.25 * (log(-y) - log((-1.0 / x))))), 2.0);
	} else if (y <= 2.5e+32) {
		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 <= -6e+38)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(-y)) - log(Float64(-1.0 / x))))) ^ 2.0));
	elseif (y <= 2.5e+32)
		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, -6e+38], 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, 2.5e+32], 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 < -6.0000000000000002e38

   1. Initial program 57.3%

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

      1. associate-+l+ 57.3%

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

      2. +-commutative 57.3%

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

      3. distribute-rgt-in 57.4%

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

   3. Simplified 57.4%

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

      1. distribute-rgt-in 57.3%

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

      2. associate-+r+ 57.3%

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

      3. *-commutative 57.3%

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

      4. distribute-lft-in 57.4%

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

      5. +-commutative 57.4%

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

      6. fma-udef 57.8%

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

      7. add-sqr-sqrt 57.3%

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

      8. pow2 57.3%

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

      9. pow1/2 57.3%

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

      10. sqrt-pow1 57.3%

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

      11. fma-udef 57.0%

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

      12. *-commutative 57.0%

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

      13. +-commutative 57.0%

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

      14. distribute-rgt-in 56.9%

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

      15. associate-+l+ 56.9%

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

      16. distribute-lft-in 57.0%

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

      17. fma-def 57.2%

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

      18. metadata-eval 57.2%

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

   6. Applied egg-rr 57.2%

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

   7. Taylor expanded in z around 0 28.4%

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

   8. Taylor expanded in x around -inf 37.6%

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

   if -6.0000000000000002e38 < y < 2.4999999999999999e32

   1. Initial program 85.1%

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

      1. associate-+l+ 85.1%

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

      2. *-commutative 85.1%

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

      3. *-commutative 85.1%

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

      4. *-commutative 85.1%

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

      5. +-commutative 85.1%

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

      6. +-commutative 85.1%

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

      7. associate-+l+ 85.1%

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

      8. *-commutative 85.1%

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

      9. *-commutative 85.1%

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

      10. +-commutative 85.1%

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

      11. +-commutative 85.1%

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

      12. *-commutative 85.1%

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

      13. associate-+r+ 85.1%

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

      14. *-commutative 85.1%

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

      15. *-commutative 85.1%

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

      16. +-commutative 85.1%

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

      17. fma-def 85.1%

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

   3. Simplified 85.2%

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

   if 2.4999999999999999e32 < y

   1. Initial program 60.1%

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

      1. associate-+l+ 60.1%

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

      2. +-commutative 60.1%

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

      3. distribute-rgt-in 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in x around 0 32.8%

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

      1. sqrt-prod 54.8%

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

   7. Applied egg-rr 54.8%

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

      1. *-commutative 54.8%

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

   9. Simplified 54.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+38}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+32}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{+32}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.36e+32)
   (* 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 <= 1.36e+32) {
		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 <= 1.36e+32)
		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, 1.36e+32], 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 2 regimes

2. if y < 1.3599999999999999e32

   1. Initial program 77.0%

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

      1. associate-+l+ 77.0%

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

      2. *-commutative 77.0%

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

      3. *-commutative 77.0%

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

      4. *-commutative 77.0%

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

      5. +-commutative 77.0%

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

      6. +-commutative 77.0%

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

      7. associate-+l+ 77.0%

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

      8. *-commutative 77.0%

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

      9. *-commutative 77.0%

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

      10. +-commutative 77.0%

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

      11. +-commutative 77.0%

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

      12. *-commutative 77.0%

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

      13. associate-+r+ 77.0%

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

      14. *-commutative 77.0%

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

      15. *-commutative 77.0%

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

      16. +-commutative 77.0%

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

      17. fma-def 77.0%

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

   3. Simplified 77.1%

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

   if 1.3599999999999999e32 < y

   1. Initial program 60.1%

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

      1. associate-+l+ 60.1%

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

      2. +-commutative 60.1%

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

      3. distribute-rgt-in 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in x around 0 32.8%

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

      1. sqrt-prod 54.8%

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

   7. Applied egg-rr 54.8%

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

      1. *-commutative 54.8%

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

   9. Simplified 54.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{+32}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+33}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.9e+33)
   (* 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 <= 3.9e+33) {
		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 <= 3.9e+33)
		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, 3.9e+33], 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 < 3.9000000000000002e33

   1. Initial program 77.0%

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

      1. associate-+l+ 77.0%

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

      2. *-commutative 77.0%

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

      3. *-commutative 77.0%

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

      4. *-commutative 77.0%

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

      5. +-commutative 77.0%

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

      6. +-commutative 77.0%

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

      7. +-commutative 77.0%

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

      8. *-commutative 77.0%

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

      9. *-commutative 77.0%

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

      10. associate-+l+ 77.0%

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

      11. +-commutative 77.0%

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

      12. fma-def 77.1%

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

      13. distribute-lft-out 77.2%

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

   3. Simplified 77.2%

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

   if 3.9000000000000002e33 < y

   1. Initial program 60.1%

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

      1. associate-+l+ 60.1%

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

      2. +-commutative 60.1%

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

      3. distribute-rgt-in 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in x around 0 32.8%

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

      1. sqrt-prod 54.8%

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

   7. Applied egg-rr 54.8%

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

      1. *-commutative 54.8%

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

   9. Simplified 54.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+33}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{+33}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5.8e+33)
   (* 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 <= 5.8e+33) {
		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 <= 5.8d+33) 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 <= 5.8e+33) {
		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 <= 5.8e+33:
		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 <= 5.8e+33)
		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 <= 5.8e+33)
		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, 5.8e+33], 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 < 5.80000000000000049e33

   1. Initial program 77.0%

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

      1. associate-+l+ 77.0%

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

      2. +-commutative 77.0%

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

      3. distribute-rgt-in 77.1%

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

   3. Simplified 77.1%

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

   if 5.80000000000000049e33 < y

   1. Initial program 60.1%

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

      1. associate-+l+ 60.1%

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

      2. +-commutative 60.1%

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

      3. distribute-rgt-in 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in x around 0 32.8%

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

      1. sqrt-prod 54.8%

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

   7. Applied egg-rr 54.8%

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

      1. *-commutative 54.8%

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

   9. Simplified 54.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{+33}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

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-279) (* 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-279) {
		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-279)) 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-279) {
		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-279:
		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-279)
		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-279)
		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-279], 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 < -4.99999999999999969e-279

   1. Initial program 72.0%

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

      1. associate-+l+ 72.0%

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

      2. +-commutative 72.0%

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

      3. 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 inf 45.2%

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

   if -4.99999999999999969e-279 < y

   1. Initial program 74.5%

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

      1. associate-+l+ 74.5%

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

      2. +-commutative 74.5%

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

      3. distribute-rgt-in 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in x around 0 26.9%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-279) (* 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 <= -5e-279) {
		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 <= (-5d-279)) 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 <= -5e-279) {
		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 <= -5e-279:
		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 <= -5e-279)
		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 <= -5e-279)
		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, -5e-279], 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 < -4.99999999999999969e-279

   1. Initial program 72.0%

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

      1. associate-+l+ 72.0%

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

      2. +-commutative 72.0%

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

      3. 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 inf 45.2%

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

   if -4.99999999999999969e-279 < y

   1. Initial program 74.5%

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

      1. associate-+l+ 74.5%

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

      2. +-commutative 74.5%

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

      3. distribute-rgt-in 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in z around inf 52.1%

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

      1. +-commutative 52.1%

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

   7. Simplified 52.1%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.7% 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 73.3%

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

   1. associate-+l+ 73.3%

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

   2. +-commutative 73.3%

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

   3. distribute-rgt-in 73.3%

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

3. Simplified 73.3%

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

5. Final simplification 73.3%

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

Alternative 9: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-279) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-279) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-279)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-279) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -5e-279:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-279)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-279)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -5e-279], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999969e-279

   1. Initial program 72.0%

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

      1. associate-+l+ 72.0%

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

      2. +-commutative 72.0%

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

      3. 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 z around 0 28.1%

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

      1. *-commutative 28.1%

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

   7. Simplified 28.1%

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

   if -4.99999999999999969e-279 < y

   1. Initial program 74.5%

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

      1. associate-+l+ 74.5%

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

      2. +-commutative 74.5%

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

      3. distribute-rgt-in 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in x around 0 26.9%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. associate-+l+ 73.3%

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

   2. +-commutative 73.3%

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

   3. distribute-rgt-in 73.3%

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

3. Simplified 73.3%

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

5. Taylor expanded in z around 0 26.1%

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

   1. *-commutative 26.1%

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

7. Simplified 26.1%

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

8. Final simplification 26.1%

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

Developer target: 83.0% 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 2024017 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

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

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