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

Percentage Accurate: 69.8% → 93.0%

Time: 16.7s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -25000000000000:\\ \;\;\;\;y \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{\left(z + x\right) \cdot {y}^{3}}} - 2 \cdot \sqrt{\frac{z + x}{y}}\right)\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(2 \cdot \sqrt{\frac{y + x}{z}} + y \cdot \sqrt{\frac{x}{{z}^{3}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -25000000000000.0)
   (*
    y
    (-
     (* (* z x) (sqrt (/ 1.0 (* (+ z x) (pow y 3.0)))))
     (* 2.0 (sqrt (/ (+ z x) y)))))
   (if (<= y 9.2e+30)
     (* 2.0 (sqrt (+ (* x (+ y z)) (* y z))))
     (* z (+ (* 2.0 (sqrt (/ (+ y x) z))) (* y (sqrt (/ x (pow z 3.0)))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -25000000000000.0) {
		tmp = y * (((z * x) * sqrt((1.0 / ((z + x) * pow(y, 3.0))))) - (2.0 * sqrt(((z + x) / y))));
	} else if (y <= 9.2e+30) {
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = z * ((2.0 * sqrt(((y + x) / z))) + (y * sqrt((x / pow(z, 3.0)))));
	}
	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 <= (-25000000000000.0d0)) then
        tmp = y * (((z * x) * sqrt((1.0d0 / ((z + x) * (y ** 3.0d0))))) - (2.0d0 * sqrt(((z + x) / y))))
    else if (y <= 9.2d+30) then
        tmp = 2.0d0 * sqrt(((x * (y + z)) + (y * z)))
    else
        tmp = z * ((2.0d0 * sqrt(((y + x) / z))) + (y * sqrt((x / (z ** 3.0d0)))))
    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 <= -25000000000000.0) {
		tmp = y * (((z * x) * Math.sqrt((1.0 / ((z + x) * Math.pow(y, 3.0))))) - (2.0 * Math.sqrt(((z + x) / y))));
	} else if (y <= 9.2e+30) {
		tmp = 2.0 * Math.sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = z * ((2.0 * Math.sqrt(((y + x) / z))) + (y * Math.sqrt((x / Math.pow(z, 3.0)))));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -25000000000000.0:
		tmp = y * (((z * x) * math.sqrt((1.0 / ((z + x) * math.pow(y, 3.0))))) - (2.0 * math.sqrt(((z + x) / y))))
	elif y <= 9.2e+30:
		tmp = 2.0 * math.sqrt(((x * (y + z)) + (y * z)))
	else:
		tmp = z * ((2.0 * math.sqrt(((y + x) / z))) + (y * math.sqrt((x / math.pow(z, 3.0)))))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -25000000000000.0)
		tmp = Float64(y * Float64(Float64(Float64(z * x) * sqrt(Float64(1.0 / Float64(Float64(z + x) * (y ^ 3.0))))) - Float64(2.0 * sqrt(Float64(Float64(z + x) / y)))));
	elseif (y <= 9.2e+30)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * Float64(y + z)) + Float64(y * z))));
	else
		tmp = Float64(z * Float64(Float64(2.0 * sqrt(Float64(Float64(y + x) / z))) + Float64(y * sqrt(Float64(x / (z ^ 3.0))))));
	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 <= -25000000000000.0)
		tmp = y * (((z * x) * sqrt((1.0 / ((z + x) * (y ^ 3.0))))) - (2.0 * sqrt(((z + x) / y))));
	elseif (y <= 9.2e+30)
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	else
		tmp = z * ((2.0 * sqrt(((y + x) / z))) + (y * sqrt((x / (z ^ 3.0)))));
	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, -25000000000000.0], N[(y * N[(N[(N[(z * x), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(z + x), $MachinePrecision] * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[Sqrt[N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e+30], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(2.0 * N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(y * N[Sqrt[N[(x / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5e13

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

      2. associate-+r+ 61.8%

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

      3. *-commutative 61.8%

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

      4. +-commutative 61.8%

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

      5. +-commutative 61.8%

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

      6. *-commutative 61.8%

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

      7. *-commutative 61.8%

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

      8. associate-+l+ 61.8%

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

      9. +-commutative 61.8%

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

      10. *-commutative 61.8%

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

      11. associate-+l+ 61.8%

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

      12. *-commutative 61.8%

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

      13. *-commutative 61.8%

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

      14. +-commutative 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in y around inf 1.0%

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

   6. Taylor expanded in y around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 77.0%

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

   8. Simplified 77.0%

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

   if -2.5e13 < y < 9.2e30

   1. Initial program 79.9%

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

      1. distribute-lft-out 79.9%

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

      2. *-commutative 79.9%

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

   4. Applied egg-rr 79.9%

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

   if 9.2e30 < y

   1. Initial program 54.0%

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

      1. +-commutative 54.0%

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

      2. associate-+r+ 54.0%

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

      3. *-commutative 54.0%

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

      4. +-commutative 54.0%

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

      5. +-commutative 54.0%

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

      6. *-commutative 54.0%

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

      7. *-commutative 54.0%

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

      8. associate-+l+ 54.0%

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

      9. +-commutative 54.0%

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

      10. *-commutative 54.0%

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

      11. associate-+l+ 54.0%

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

      12. *-commutative 54.0%

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

      13. *-commutative 54.0%

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

      14. +-commutative 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in z around inf 35.5%

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

   6. Taylor expanded in x around inf 37.6%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -25000000000000:\\ \;\;\;\;y \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{\left(z + x\right) \cdot {y}^{3}}} - 2 \cdot \sqrt{\frac{z + x}{y}}\right)\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(2 \cdot \sqrt{\frac{y + x}{z}} + y \cdot \sqrt{\frac{x}{{z}^{3}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-284}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.1e-284)
   (* 2.0 (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e-284) {
		tmp = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	} 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) :: tmp
    if (y <= 1.1d-284) then
        tmp = 2.0d0 * (exp((0.25d0 * (log((-y - z)) - log(((-1.0d0) / x))))) ** 2.0d0)
    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 <= 1.1e-284) {
		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(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.1e-284:
		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(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.1e-284)
		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(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 <= 1.1e-284)
		tmp = 2.0 * (exp((0.25 * (log((-y - z)) - log((-1.0 / x))))) ^ 2.0);
	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_] := If[LessEqual[y, 1.1e-284], 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[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e-284

   1. Initial program 70.1%

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

      1. +-commutative 70.1%

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

      2. associate-+r+ 70.1%

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

      3. *-commutative 70.1%

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

      4. +-commutative 70.1%

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

      5. associate-+l+ 70.1%

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

      6. *-commutative 70.1%

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

      7. *-commutative 70.1%

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

      8. *-commutative 70.1%

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

      9. fma-define 70.1%

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

      10. +-commutative 70.1%

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

      11. distribute-lft-out 70.2%

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

   3. Simplified 70.2%

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

      1. fma-undefine 70.1%

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

      2. distribute-rgt-in 70.1%

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

      3. associate-+l+ 70.1%

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

      4. add-sqr-sqrt 69.7%

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

      5. pow2 69.7%

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

      6. pow1/2 69.7%

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

      7. sqrt-pow1 69.7%

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

      8. associate-+l+ 69.7%

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

      9. distribute-rgt-in 69.7%

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

      10. fma-undefine 69.8%

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

      11. metadata-eval 69.8%

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

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in x around -inf 48.1%

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

   if 1.1e-284 < y

   1. Initial program 66.7%

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

      1. +-commutative 66.7%

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

      2. associate-+r+ 66.7%

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

      3. *-commutative 66.7%

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

      4. +-commutative 66.7%

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

      5. +-commutative 66.7%

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

      6. *-commutative 66.7%

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

      7. *-commutative 66.7%

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

      8. associate-+l+ 66.7%

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

      9. +-commutative 66.7%

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

      10. *-commutative 66.7%

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

      11. associate-+l+ 66.7%

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

      12. *-commutative 66.7%

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

      13. *-commutative 66.7%

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

      14. +-commutative 66.7%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in x around 0 25.5%

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

      1. *-commutative 25.5%

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

   7. Simplified 25.5%

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

      1. sqrt-prod 34.6%

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

   9. Applied egg-rr 34.6%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-284}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(2 \cdot \sqrt{\frac{y + x}{z}} + y \cdot \sqrt{\frac{x}{{z}^{3}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.9e+30)
   (* 2.0 (sqrt (fma x z (* y (+ z x)))))
   (* z (+ (* 2.0 (sqrt (/ (+ y x) z))) (* y (sqrt (/ x (pow z 3.0))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e+30) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = z * ((2.0 * sqrt(((y + x) / z))) + (y * sqrt((x / pow(z, 3.0)))));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.9e+30)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(z * Float64(Float64(2.0 * sqrt(Float64(Float64(y + x) / z))) + Float64(y * sqrt(Float64(x / (z ^ 3.0))))));
	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, 1.9e+30], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(2.0 * N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(y * N[Sqrt[N[(x / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9000000000000001e30

   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%

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

      3. *-commutative 73.2%

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

      4. *-commutative 73.2%

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

      5. +-commutative 73.2%

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

      6. +-commutative 73.2%

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

      7. +-commutative 73.2%

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

      8. *-commutative 73.2%

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

      9. *-commutative 73.2%

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

      10. associate-+l+ 73.2%

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

      11. +-commutative 73.2%

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

      12. fma-define 73.2%

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

      13. distribute-lft-out 73.2%

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

   3. Simplified 73.2%

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

   if 1.9000000000000001e30 < y

   1. Initial program 54.0%

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

      1. +-commutative 54.0%

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

      2. associate-+r+ 54.0%

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

      3. *-commutative 54.0%

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

      4. +-commutative 54.0%

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

      5. +-commutative 54.0%

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

      6. *-commutative 54.0%

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

      7. *-commutative 54.0%

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

      8. associate-+l+ 54.0%

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

      9. +-commutative 54.0%

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

      10. *-commutative 54.0%

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

      11. associate-+l+ 54.0%

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

      12. *-commutative 54.0%

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

      13. *-commutative 54.0%

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

      14. +-commutative 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in z around inf 35.5%

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

   6. Taylor expanded in x around inf 37.6%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(2 \cdot \sqrt{\frac{y + x}{z}} + y \cdot \sqrt{\frac{x}{{z}^{3}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot \sqrt{\frac{z + x}{y}} + x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e+30)
   (* 2.0 (sqrt (fma x z (* y (+ z x)))))
   (* y (+ (* 2.0 (sqrt (/ (+ z x) y))) (* x (sqrt (/ z (pow y 3.0))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+30) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = y * ((2.0 * sqrt(((z + x) / y))) + (x * sqrt((z / pow(y, 3.0)))));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e+30)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(y * Float64(Float64(2.0 * sqrt(Float64(Float64(z + x) / y))) + Float64(x * sqrt(Float64(z / (y ^ 3.0))))));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 6.5e+30], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(2.0 * N[Sqrt[N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(x * N[Sqrt[N[(z / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5e30

   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%

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

      3. *-commutative 73.2%

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

      4. *-commutative 73.2%

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

      5. +-commutative 73.2%

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

      6. +-commutative 73.2%

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

      7. +-commutative 73.2%

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

      8. *-commutative 73.2%

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

      9. *-commutative 73.2%

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

      10. associate-+l+ 73.2%

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

      11. +-commutative 73.2%

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

      12. fma-define 73.2%

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

      13. distribute-lft-out 73.2%

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

   3. Simplified 73.2%

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

   if 6.5e30 < y

   1. Initial program 54.0%

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

      1. +-commutative 54.0%

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

      2. associate-+r+ 54.0%

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

      3. *-commutative 54.0%

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

      4. +-commutative 54.0%

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

      5. +-commutative 54.0%

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

      6. *-commutative 54.0%

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

      7. *-commutative 54.0%

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

      8. associate-+l+ 54.0%

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

      9. +-commutative 54.0%

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

      10. *-commutative 54.0%

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

      11. associate-+l+ 54.0%

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

      12. *-commutative 54.0%

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

      13. *-commutative 54.0%

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

      14. +-commutative 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in y around inf 70.6%

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

   6. Taylor expanded in x around 0 76.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot \sqrt{\frac{z + x}{y}} + x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+38}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.05e+38)
   (* 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 <= 1.05e+38) {
		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 <= 1.05e+38)
		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, 1.05e+38], 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 < 1.05e38

   1. Initial program 73.4%

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

      1. associate-+l+ 73.4%

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

      2. *-commutative 73.4%

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

      3. *-commutative 73.4%

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

      4. *-commutative 73.4%

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

      5. +-commutative 73.4%

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

      6. +-commutative 73.4%

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

      7. +-commutative 73.4%

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

      8. *-commutative 73.4%

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

      9. *-commutative 73.4%

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

      10. associate-+l+ 73.4%

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

      11. +-commutative 73.4%

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

      12. fma-define 73.4%

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

      13. distribute-lft-out 73.5%

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

   3. Simplified 73.5%

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

   if 1.05e38 < y

   1. Initial program 52.4%

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

      1. +-commutative 52.4%

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

      2. associate-+r+ 52.4%

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

      3. *-commutative 52.4%

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

      4. +-commutative 52.4%

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

      5. +-commutative 52.4%

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

      6. *-commutative 52.4%

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

      7. *-commutative 52.4%

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

      8. associate-+l+ 52.4%

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

      9. +-commutative 52.4%

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

      10. *-commutative 52.4%

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

      11. associate-+l+ 52.4%

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

      12. *-commutative 52.4%

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

      13. *-commutative 52.4%

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

      14. +-commutative 52.4%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in x around 0 21.6%

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

      1. *-commutative 21.6%

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

   7. Simplified 21.6%

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

      1. sqrt-prod 39.8%

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

   9. Applied egg-rr 39.8%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+38}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+38}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.3e+38)
   (* 2.0 (sqrt (+ (* x (+ y z)) (* y z))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e+38) {
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.3d+38) then
        tmp = 2.0d0 * sqrt(((x * (y + z)) + (y * z)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e+38) {
		tmp = 2.0 * Math.sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.3e+38:
		tmp = 2.0 * math.sqrt(((x * (y + z)) + (y * z)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.3e+38)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * Float64(y + z)) + Float64(y * z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.3e+38)
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.3e+38], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3e38

   1. Initial program 73.4%

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

      1. distribute-lft-out 73.4%

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

      2. *-commutative 73.4%

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

   4. Applied egg-rr 73.4%

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

   if 1.3e38 < y

   1. Initial program 52.4%

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

      1. +-commutative 52.4%

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

      2. associate-+r+ 52.4%

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

      3. *-commutative 52.4%

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

      4. +-commutative 52.4%

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

      5. +-commutative 52.4%

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

      6. *-commutative 52.4%

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

      7. *-commutative 52.4%

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

      8. associate-+l+ 52.4%

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

      9. +-commutative 52.4%

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

      10. *-commutative 52.4%

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

      11. associate-+l+ 52.4%

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

      12. *-commutative 52.4%

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

      13. *-commutative 52.4%

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

      14. +-commutative 52.4%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in x around 0 21.6%

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

      1. *-commutative 21.6%

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

   7. Simplified 21.6%

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

      1. sqrt-prod 39.8%

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

   9. Applied egg-rr 39.8%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+38}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-252}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e-252)
   (* 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 <= -1.6e-252) {
		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 <= (-1.6d-252)) 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 <= -1.6e-252) {
		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 <= -1.6e-252:
		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 <= -1.6e-252)
		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 <= -1.6e-252)
		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, -1.6e-252], 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 < -1.6000000000000001e-252

   1. Initial program 68.8%

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

      1. +-commutative 68.8%

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

      2. associate-+r+ 68.8%

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

      3. *-commutative 68.8%

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

      4. +-commutative 68.8%

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

      5. +-commutative 68.8%

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

      6. *-commutative 68.8%

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

      7. *-commutative 68.8%

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

      8. associate-+l+ 68.8%

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

      9. +-commutative 68.8%

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

      10. *-commutative 68.8%

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

      11. associate-+l+ 68.8%

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

      12. *-commutative 68.8%

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

      13. *-commutative 68.8%

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

      14. +-commutative 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in x around inf 44.8%

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

      1. +-commutative 44.8%

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

   7. Simplified 44.8%

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

   if -1.6000000000000001e-252 < y

   1. Initial program 68.3%

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

      1. +-commutative 68.3%

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

      2. associate-+r+ 68.3%

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

      3. *-commutative 68.3%

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

      4. +-commutative 68.3%

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

      5. +-commutative 68.3%

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

      6. *-commutative 68.3%

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

      7. *-commutative 68.3%

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

      8. associate-+l+ 68.3%

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

      9. +-commutative 68.3%

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

      10. *-commutative 68.3%

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

      11. associate-+l+ 68.3%

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

      12. *-commutative 68.3%

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

      13. *-commutative 68.3%

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

      14. +-commutative 68.3%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in z around inf 44.7%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-252}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-252}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot \left(z \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e-252) (* 2.0 (sqrt (* x (+ y z)))) (sqrt (* y (* z 4.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e-252) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = sqrt((y * (z * 4.0)));
	}
	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 <= (-1.6d-252)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = sqrt((y * (z * 4.0d0)))
    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 <= -1.6e-252) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = Math.sqrt((y * (z * 4.0)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.6e-252:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = math.sqrt((y * (z * 4.0)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.6e-252)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = sqrt(Float64(y * Float64(z * 4.0)));
	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 <= -1.6e-252)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = sqrt((y * (z * 4.0)));
	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, -1.6e-252], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(y * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6000000000000001e-252

   1. Initial program 68.8%

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

      1. +-commutative 68.8%

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

      2. associate-+r+ 68.8%

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

      3. *-commutative 68.8%

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

      4. +-commutative 68.8%

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

      5. +-commutative 68.8%

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

      6. *-commutative 68.8%

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

      7. *-commutative 68.8%

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

      8. associate-+l+ 68.8%

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

      9. +-commutative 68.8%

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

      10. *-commutative 68.8%

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

      11. associate-+l+ 68.8%

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

      12. *-commutative 68.8%

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

      13. *-commutative 68.8%

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

      14. +-commutative 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in x around inf 44.8%

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

      1. +-commutative 44.8%

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

   7. Simplified 44.8%

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

   if -1.6000000000000001e-252 < y

   1. Initial program 68.3%

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

      1. +-commutative 68.3%

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

      2. associate-+r+ 68.3%

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

      3. *-commutative 68.3%

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

      4. +-commutative 68.3%

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

      5. +-commutative 68.3%

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

      6. *-commutative 68.3%

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

      7. *-commutative 68.3%

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

      8. associate-+l+ 68.3%

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

      9. +-commutative 68.3%

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

      10. *-commutative 68.3%

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

      11. associate-+l+ 68.3%

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

      12. *-commutative 68.3%

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

      13. *-commutative 68.3%

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

      14. +-commutative 68.3%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in x around 0 22.5%

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

      1. *-commutative 22.5%

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

   7. Simplified 22.5%

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

      1. add-sqr-sqrt 22.4%

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

      2. sqrt-unprod 22.5%

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

      3. metadata-eval 22.5%

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

      4. metadata-eval 22.5%

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

      5. sqrt-pow2 22.3%

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

      6. *-commutative 22.3%

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

      7. metadata-eval 22.3%

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

      8. metadata-eval 22.3%

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

      9. sqrt-pow2 22.2%

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

      10. *-commutative 22.2%

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

      11. swap-sqr 22.2%

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

      12. add-sqr-sqrt 22.2%

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

      13. *-commutative 22.2%

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

      14. sqrt-pow2 22.3%

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

      15. metadata-eval 22.3%

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

      16. metadata-eval 22.3%

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

      17. sqrt-pow2 22.5%

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

      18. metadata-eval 22.5%

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

      19. metadata-eval 22.5%

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

      20. metadata-eval 22.5%

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

   9. Applied egg-rr 22.5%

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

       1. associate-*l* 22.5%

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

   11. Simplified 22.5%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-252}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot \left(z \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.8% accurate, 1.0× speedup

Math

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

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

C

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

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(((x * (y + z)) + (y * z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 68.5%

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

   1. distribute-lft-out 68.5%

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

   2. *-commutative 68.5%

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

4. Applied egg-rr 68.5%

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

5. Final simplification 68.5%

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

Alternative 10: 69.8% 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 68.5%

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

   1. +-commutative 68.5%

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

   2. associate-+r+ 68.5%

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

   3. *-commutative 68.5%

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

   4. +-commutative 68.5%

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

   5. +-commutative 68.5%

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

   6. *-commutative 68.5%

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

   7. *-commutative 68.5%

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

   8. associate-+l+ 68.5%

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

   9. +-commutative 68.5%

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

   10. *-commutative 68.5%

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

   11. associate-+l+ 68.5%

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

   12. *-commutative 68.5%

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

   13. *-commutative 68.5%

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

   14. +-commutative 68.5%

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

3. Simplified 68.5%

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

5. Final simplification 68.5%

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

Alternative 11: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot \left(z \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -7.6e-268) (* 2.0 (sqrt (* y x))) (sqrt (* y (* z 4.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.6e-268) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = sqrt((y * (z * 4.0)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-7.6d-268)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = sqrt((y * (z * 4.0d0)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.6e-268) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = Math.sqrt((y * (z * 4.0)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -7.6e-268:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = math.sqrt((y * (z * 4.0)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -7.6e-268)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = sqrt(Float64(y * Float64(z * 4.0)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.6e-268)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = sqrt((y * (z * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -7.6e-268], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(y * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.6000000000000005e-268

   1. Initial program 69.0%

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

      1. +-commutative 69.0%

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

      2. associate-+r+ 69.0%

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

      3. *-commutative 69.0%

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

      4. +-commutative 69.0%

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

      5. +-commutative 69.0%

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

      6. *-commutative 69.0%

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

      7. *-commutative 69.0%

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

      8. associate-+l+ 69.0%

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

      9. +-commutative 69.0%

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

      10. *-commutative 69.0%

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

      11. associate-+l+ 69.0%

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

      12. *-commutative 69.0%

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

      13. *-commutative 69.0%

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

      14. +-commutative 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in z around 0 32.3%

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

   if -7.6000000000000005e-268 < y

   1. Initial program 68.1%

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

      1. +-commutative 68.1%

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

      2. associate-+r+ 68.1%

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

      3. *-commutative 68.1%

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

      4. +-commutative 68.1%

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

      5. +-commutative 68.1%

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

      6. *-commutative 68.1%

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

      7. *-commutative 68.1%

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

      8. associate-+l+ 68.1%

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

      9. +-commutative 68.1%

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

      10. *-commutative 68.1%

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

      11. associate-+l+ 68.1%

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

      12. *-commutative 68.1%

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

      13. *-commutative 68.1%

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

      14. +-commutative 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in x around 0 23.2%

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

      1. *-commutative 23.2%

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

   7. Simplified 23.2%

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

      1. add-sqr-sqrt 23.1%

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

      2. sqrt-unprod 23.2%

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

      3. metadata-eval 23.2%

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

      4. metadata-eval 23.2%

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

      5. sqrt-pow2 22.9%

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

      6. *-commutative 22.9%

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

      7. metadata-eval 22.9%

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

      8. metadata-eval 22.9%

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

      9. sqrt-pow2 22.8%

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

      10. *-commutative 22.8%

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

      11. swap-sqr 22.8%

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

      12. add-sqr-sqrt 22.8%

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

      13. *-commutative 22.8%

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

      14. sqrt-pow2 23.0%

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

      15. metadata-eval 23.0%

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

      16. metadata-eval 23.0%

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

      17. sqrt-pow2 23.2%

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

      18. metadata-eval 23.2%

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

      19. metadata-eval 23.2%

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

      20. metadata-eval 23.2%

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

   9. Applied egg-rr 23.2%

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

       1. associate-*l* 23.2%

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

   11. Simplified 23.2%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot \left(z \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt((y * (z * 4.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

   1. +-commutative 68.5%

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

   2. associate-+r+ 68.5%

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

   3. *-commutative 68.5%

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

   4. +-commutative 68.5%

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

   5. +-commutative 68.5%

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

   6. *-commutative 68.5%

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

   7. *-commutative 68.5%

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

   8. associate-+l+ 68.5%

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

   9. +-commutative 68.5%

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

   10. *-commutative 68.5%

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

   11. associate-+l+ 68.5%

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

   12. *-commutative 68.5%

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

   13. *-commutative 68.5%

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

   14. +-commutative 68.5%

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

3. Simplified 68.5%

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

5. Taylor expanded in x around 0 23.8%

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

   1. *-commutative 23.8%

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

7. Simplified 23.8%

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

   1. add-sqr-sqrt 23.6%

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

   2. sqrt-unprod 23.8%

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

   3. metadata-eval 23.8%

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

   4. metadata-eval 23.8%

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

   5. sqrt-pow2 23.5%

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

   6. *-commutative 23.5%

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

   7. metadata-eval 23.5%

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

   8. metadata-eval 23.5%

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

   9. sqrt-pow2 23.4%

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

   10. *-commutative 23.4%

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

   11. swap-sqr 23.4%

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

   12. add-sqr-sqrt 23.4%

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

   13. *-commutative 23.4%

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

   14. sqrt-pow2 23.5%

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

   15. metadata-eval 23.5%

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

   16. metadata-eval 23.5%

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

   17. sqrt-pow2 23.8%

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

   18. metadata-eval 23.8%

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

   19. metadata-eval 23.8%

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

   20. metadata-eval 23.8%

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

9. Applied egg-rr 23.8%

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

    1. associate-*l* 23.8%

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

11. Simplified 23.8%

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

Developer Target 1: 82.5% 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 2024165 
(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)))))