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

Percentage Accurate: 70.9% → 95.6%

Time: 15.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.9% 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: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -126000:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -126000.0)
   (* 2.0 (pow (exp (* 0.25 (- (log (- (- z) x)) (log (/ -1.0 y))))) 2.0))
   (if (<= y -5.5e-277)
     (* 2.0 (sqrt (fma x y (* z (+ y x)))))
     (* 2.0 (* (sqrt z) (sqrt (fma x (/ y z) (+ y x))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -126000.0) {
		tmp = 2.0 * pow(exp((0.25 * (log((-z - x)) - log((-1.0 / y))))), 2.0);
	} else if (y <= -5.5e-277) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(fma(x, (y / z), (y + x))));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -126000.0)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-z) - x)) - log(Float64(-1.0 / y))))) ^ 2.0));
	elseif (y <= -5.5e-277)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(fma(x, Float64(y / z), Float64(y + x)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -126000.0], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[((-z) - x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.5e-277], N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[N[(x * N[(y / z), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -126000

   1. Initial program 64.3%

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

      1. +-commutative 64.3%

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

      2. *-commutative 64.3%

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

      3. +-commutative 64.3%

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

      4. *-commutative 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-+l+ 64.3%

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

      7. +-commutative 64.3%

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

      8. *-commutative 64.3%

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

      9. fma-define 64.3%

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

      10. +-commutative 64.3%

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

      11. distribute-lft-out 64.8%

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

   3. Simplified 64.8%

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

      1. fma-undefine 64.4%

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

      2. +-commutative 64.4%

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

      3. +-commutative 64.4%

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

      4. +-commutative 64.4%

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

      5. distribute-rgt-in 64.3%

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

      6. associate-+l+ 64.3%

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

      7. *-commutative 64.3%

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

      8. distribute-lft-in 64.3%

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

      9. +-commutative 64.3%

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

      10. fma-undefine 64.8%

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

      11. add-sqr-sqrt 64.2%

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

      12. pow2 64.2%

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

      13. pow1/2 64.2%

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

      14. sqrt-pow1 64.3%

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

      15. metadata-eval 64.3%

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

   6. Applied egg-rr 64.3%

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

   7. Taylor expanded in y around -inf 80.9%

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

   if -126000 < y < -5.49999999999999952e-277

   1. Initial program 85.3%

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

      1. +-commutative 85.3%

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

      2. *-commutative 85.3%

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

      3. +-commutative 85.3%

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

      4. *-commutative 85.3%

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

      5. *-commutative 85.3%

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

      6. associate-+l+ 85.3%

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

      7. +-commutative 85.3%

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

      8. *-commutative 85.3%

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

      9. fma-define 85.3%

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

      10. +-commutative 85.3%

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

      11. distribute-lft-out 85.4%

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

   3. Simplified 85.4%

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

   if -5.49999999999999952e-277 < 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 73.3%

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

   3. Simplified 73.3%

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

      1. add-cbrt-cube 40.6%

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

      2. pow3 40.7%

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

      3. +-commutative 40.7%

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

   6. Applied egg-rr 40.7%

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

   7. Taylor expanded in z around inf 65.3%

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

      1. associate-+r+ 65.3%

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

      2. +-commutative 65.3%

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

      3. associate-/l* 59.6%

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

   9. Simplified 59.6%

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

       1. sqrt-prod 51.5%

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

       2. *-commutative 51.5%

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

       3. +-commutative 51.5%

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

       4. fma-define 51.5%

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

       5. +-commutative 51.5%

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

   11. Applied egg-rr 51.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -126000:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-291}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.15e-291)
   (* 2.0 (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.15e-291) {
		tmp = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(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 <= (-3.15d-291)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log((-y - z)) - log(((-1.0d0) / x))))) ** 2.0d0)
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(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 <= -3.15e-291) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((-y - z)) - Math.log((-1.0 / x))))), 2.0);
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -3.15e-291:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log((-y - z)) - math.log((-1.0 / x))))), 2.0)
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3.15e-291)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(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 <= -3.15e-291)
		tmp = 2.0 * (exp((0.25 * (log((-y - z)) - log((-1.0 / x))))) ^ 2.0);
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(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, -3.15e-291], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.14999999999999996e-291

   1. Initial program 74.4%

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

      1. +-commutative 74.4%

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

      2. *-commutative 74.4%

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

      3. +-commutative 74.4%

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

      4. *-commutative 74.4%

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

      5. *-commutative 74.4%

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

      6. associate-+l+ 74.4%

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

      7. +-commutative 74.4%

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

      8. *-commutative 74.4%

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

      9. fma-define 74.4%

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

      10. +-commutative 74.4%

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

      11. distribute-lft-out 74.6%

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

   3. Simplified 74.6%

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

      1. fma-undefine 74.5%

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

      2. +-commutative 74.5%

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

      3. +-commutative 74.5%

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

      4. +-commutative 74.5%

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

      5. distribute-rgt-in 74.4%

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

      6. associate-+l+ 74.4%

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

      7. *-commutative 74.4%

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

      8. distribute-lft-in 74.4%

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

      9. +-commutative 74.4%

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

      10. fma-undefine 74.6%

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

      11. add-sqr-sqrt 74.1%

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

      12. pow2 74.1%

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

      13. pow1/2 74.1%

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

      14. sqrt-pow1 74.2%

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

      15. metadata-eval 74.2%

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

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in x around -inf 52.6%

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

   if -3.14999999999999996e-291 < y

   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 inf 50.2%

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

      1. *-commutative 50.2%

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

      2. sqrt-prod 50.9%

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

   7. Applied egg-rr 50.9%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-291}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(y \cdot x + z \cdot x\right) + y \cdot z\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+305}\right):\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \end{array} \end{array} \]

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 (+ (+ (* y x) (* z x)) (* y z))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+305)))
     (* 2.0 (* (sqrt z) (sqrt (fma x (/ y z) (+ y x)))))
     (* 2.0 (sqrt (fma x y (* z (+ y x))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = ((y * x) + (z * x)) + (y * z);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+305)) {
		tmp = 2.0 * (sqrt(z) * sqrt(fma(x, (y / z), (y + x))));
	} else {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(Float64(y * x) + Float64(z * x)) + Float64(y * z))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+305))
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(fma(x, Float64(y / z), Float64(y + x)))));
	else
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y * x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+305]], $MachinePrecision]], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[N[(x * N[(y / z), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)) < 0.0 or 1.9999999999999999e305 < (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))

   1. Initial program 6.3%

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

      1. associate-+l+ 6.3%

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

      2. +-commutative 6.3%

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

      3. distribute-rgt-in 6.5%

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

   3. Simplified 6.5%

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

      1. add-cbrt-cube 5.9%

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

      2. pow3 5.9%

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

      3. +-commutative 5.9%

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

   6. Applied egg-rr 5.9%

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

   7. Taylor expanded in z around inf 6.6%

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

      1. associate-+r+ 6.6%

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

      2. +-commutative 6.6%

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

      3. associate-/l* 7.1%

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

   9. Simplified 7.1%

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

       1. sqrt-prod 38.7%

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

       2. *-commutative 38.7%

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

       3. +-commutative 38.7%

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

       4. fma-define 38.7%

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

       5. +-commutative 38.7%

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

   11. Applied egg-rr 38.7%

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

   if 0.0 < (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)) < 1.9999999999999999e305

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. fma-define 99.8%

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

      10. +-commutative 99.8%

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

      11. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x + z \cdot x\right) + y \cdot z \leq 0 \lor \neg \left(\left(y \cdot x + z \cdot x\right) + y \cdot z \leq 2 \cdot 10^{+305}\right):\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e-278)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e-278) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(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 <= 1.35d-278) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(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 <= 1.35e-278) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.35e-278:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e-278)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(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 <= 1.35e-278)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(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, 1.35e-278], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3500000000000001e-278

   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 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in x around inf 57.9%

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

      1. +-commutative 57.9%

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

   7. Simplified 57.9%

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

   if 1.3500000000000001e-278 < y

   1. Initial program 74.4%

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

      1. associate-+l+ 74.4%

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

      2. +-commutative 74.4%

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

      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 47.8%

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

      1. *-commutative 47.8%

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

      2. sqrt-prod 50.7%

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

   7. Applied egg-rr 50.7%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.4e-278)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.4e-278) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.4d-278) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.4e-278) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.4e-278:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.4e-278)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.4e-278)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.4e-278], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.40000000000000004e-278

   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 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in x around inf 57.9%

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

      1. +-commutative 57.9%

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

   7. Simplified 57.9%

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

   if 1.40000000000000004e-278 < y

   1. Initial program 74.4%

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

      1. associate-+l+ 74.4%

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

      2. +-commutative 74.4%

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

      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 47.8%

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

      1. *-commutative 47.8%

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

      2. sqrt-prod 50.7%

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

   7. Applied egg-rr 50.7%

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

   8. Taylor expanded in x around 0 36.4%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y \cdot \left(1 + \frac{x}{y}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4.4e-281)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (sqrt (* z (* y (+ 1.0 (/ x y))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.4e-281) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y * (1.0 + (x / 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 <= (-4.4d-281)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y * (1.0d0 + (x / 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 <= -4.4e-281) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y * (1.0 + (x / y)))));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -4.4e-281:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y * (1.0 + (x / y)))))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -4.4e-281)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y * Float64(1.0 + Float64(x / 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 <= -4.4e-281)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y * (1.0 + (x / 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, -4.4e-281], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.40000000000000008e-281

   1. Initial program 74.8%

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

      1. associate-+l+ 74.8%

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

      2. +-commutative 74.8%

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

      3. distribute-rgt-in 74.8%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in x around inf 56.7%

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

      1. +-commutative 56.7%

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

   7. Simplified 56.7%

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

   if -4.40000000000000008e-281 < y

   1. Initial program 73.7%

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

      1. associate-+l+ 73.7%

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

      2. +-commutative 73.7%

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

      3. distribute-rgt-in 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in z around inf 50.2%

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

   6. Taylor expanded in y around inf 45.1%

      \[\leadsto 2 \cdot \sqrt{z \cdot \color{blue}{\left(y \cdot \left(1 + \frac{x}{y}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y \cdot \left(1 + \frac{x}{y}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5.5e-277)
   (* 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 <= -5.5e-277) {
		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 <= (-5.5d-277)) 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 <= -5.5e-277) {
		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 <= -5.5e-277:
		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 <= -5.5e-277)
		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 <= -5.5e-277)
		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, -5.5e-277], 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 < -5.49999999999999952e-277

   1. Initial program 75.3%

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

      1. associate-+l+ 75.3%

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

      2. +-commutative 75.3%

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

      3. distribute-rgt-in 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in x around inf 57.1%

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

      1. +-commutative 57.1%

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

   7. Simplified 57.1%

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

   if -5.49999999999999952e-277 < 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 73.3%

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

   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 inf 49.9%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5.5e-277) (* 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 <= -5.5e-277) {
		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 <= (-5.5d-277)) 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 <= -5.5e-277) {
		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 <= -5.5e-277:
		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 <= -5.5e-277)
		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 <= -5.5e-277)
		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, -5.5e-277], 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 < -5.49999999999999952e-277

   1. Initial program 75.3%

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

      1. associate-+l+ 75.3%

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

      2. +-commutative 75.3%

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

      3. distribute-rgt-in 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in x around inf 57.1%

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

      1. +-commutative 57.1%

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

   7. Simplified 57.1%

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

   if -5.49999999999999952e-277 < 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 73.3%

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

   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 x around 0 23.5%

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

      1. *-commutative 23.5%

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

   7. Simplified 23.5%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. associate-+l+ 74.2%

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

   2. +-commutative 74.2%

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

   3. distribute-rgt-in 74.3%

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

3. Simplified 74.3%

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

5. Final simplification 74.3%

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

Alternative 10: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5.5e-277) (* 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 <= -5.5e-277) {
		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 <= (-5.5d-277)) 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 <= -5.5e-277) {
		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 <= -5.5e-277:
		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 <= -5.5e-277)
		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 <= -5.5e-277)
		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, -5.5e-277], 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 < -5.49999999999999952e-277

   1. Initial program 75.3%

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

      1. associate-+l+ 75.3%

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

      2. +-commutative 75.3%

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

      3. distribute-rgt-in 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in z around 0 36.9%

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

   if -5.49999999999999952e-277 < 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 73.3%

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

   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 x around 0 23.5%

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

      1. *-commutative 23.5%

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

   7. Simplified 23.5%

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

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.7% 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 74.2%

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

   1. associate-+l+ 74.2%

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

   2. +-commutative 74.2%

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

   3. distribute-rgt-in 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in z around 0 30.8%

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

6. Final simplification 30.8%

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

Developer Target 1: 83.1% 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 2024132 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 763695009057367500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 2 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 1/4 (* (* (pow y -3/4) (* (pow z -3/4) x)) (+ y z))) (* (pow z 1/4) (pow y 1/4))) (+ (* 1/4 (* (* (pow y -3/4) (* (pow z -3/4) x)) (+ y z))) (* (pow z 1/4) (pow y 1/4)))) 2)))

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