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

Percentage Accurate: 70.7% → 95.5%

Time: 14.8s

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: 70.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-302}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y \cdot \left(1 + \frac{z}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3e+49)
   (* 2.0 (exp (* 0.5 (- (log (- (- x) z)) (log (/ -1.0 y))))))
   (if (<= y 3.8e-302)
     (* 2.0 (sqrt (* x (* y (+ 1.0 (/ z y))))))
     (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e+49) {
		tmp = 2.0 * exp((0.5 * (log((-x - z)) - log((-1.0 / y)))));
	} else if (y <= 3.8e-302) {
		tmp = 2.0 * sqrt((x * (y * (1.0 + (z / y)))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3d+49)) then
        tmp = 2.0d0 * exp((0.5d0 * (log((-x - z)) - log(((-1.0d0) / y)))))
    else if (y <= 3.8d-302) then
        tmp = 2.0d0 * sqrt((x * (y * (1.0d0 + (z / y)))))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e+49) {
		tmp = 2.0 * Math.exp((0.5 * (Math.log((-x - z)) - Math.log((-1.0 / y)))));
	} else if (y <= 3.8e-302) {
		tmp = 2.0 * Math.sqrt((x * (y * (1.0 + (z / y)))));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -3e+49:
		tmp = 2.0 * math.exp((0.5 * (math.log((-x - z)) - math.log((-1.0 / y)))))
	elif y <= 3.8e-302:
		tmp = 2.0 * math.sqrt((x * (y * (1.0 + (z / y)))))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3e+49)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(Float64(-x) - z)) - log(Float64(-1.0 / y))))));
	elseif (y <= 3.8e-302)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y * Float64(1.0 + Float64(z / y))))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3e+49)
		tmp = 2.0 * exp((0.5 * (log((-x - z)) - log((-1.0 / y)))));
	elseif (y <= 3.8e-302)
		tmp = 2.0 * sqrt((x * (y * (1.0 + (z / y)))));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -3e+49], N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[N[((-x) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-302], N[(2.0 * N[Sqrt[N[(x * N[(y * N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0000000000000002e49

   1. Initial program 58.2%

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

      1. associate-+l+ 58.2%

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

      2. *-commutative 58.2%

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

      3. *-commutative 58.2%

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

      4. +-commutative 58.2%

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

      5. fma-define 58.3%

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

      6. +-commutative 58.3%

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

      7. distribute-lft-out 58.6%

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

   3. Simplified 58.6%

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

      1. pow1/2 58.9%

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

      2. fma-undefine 58.4%

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

      3. distribute-rgt-in 58.2%

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

      4. associate-+l+ 58.2%

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

      5. pow-to-exp 54.1%

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

      6. associate-+l+ 54.1%

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

      7. distribute-rgt-in 54.3%

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

      8. fma-undefine 54.5%

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

   6. Applied egg-rr 54.5%

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

   7. Taylor expanded in y around -inf 82.4%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot x + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. distribute-lft-out 82.4%

         \[\leadsto 2 \cdot e^{\left(\log \color{blue}{\left(-1 \cdot \left(x + z\right)\right)} + -1 \cdot \log \left(\frac{-1}{y}\right)\right) \cdot 0.5} \]

      2. +-commutative 82.4%

         \[\leadsto 2 \cdot e^{\left(\log \left(-1 \cdot \color{blue}{\left(z + x\right)}\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right) \cdot 0.5} \]

      3. neg-mul-1 82.4%

         \[\leadsto 2 \cdot e^{\left(\log \color{blue}{\left(-\left(z + x\right)\right)} + -1 \cdot \log \left(\frac{-1}{y}\right)\right) \cdot 0.5} \]

      4. +-commutative 82.4%

         \[\leadsto 2 \cdot e^{\left(\log \left(-\color{blue}{\left(x + z\right)}\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right) \cdot 0.5} \]

      5. mul-1-neg 82.4%

         \[\leadsto 2 \cdot e^{\left(\log \left(-\left(x + z\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{y}\right)\right)}\right) \cdot 0.5} \]

   9. Simplified 82.4%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(-\left(x + z\right)\right) + \left(-\log \left(\frac{-1}{y}\right)\right)\right)} \cdot 0.5} \]

   if -3.0000000000000002e49 < y < 3.8e-302

   1. Initial program 84.4%

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

      1. distribute-lft-out 84.4%

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

      2. *-commutative 84.4%

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

   4. Applied egg-rr 84.4%

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

   5. Taylor expanded in y around inf 66.8%

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

   6. Taylor expanded in x around inf 44.3%

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

   if 3.8e-302 < y

   1. Initial program 67.9%

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

      1. associate-+l+ 67.9%

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

      2. *-commutative 67.9%

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

      3. *-commutative 67.9%

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

      4. *-commutative 67.9%

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

      5. +-commutative 67.9%

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

      6. +-commutative 67.9%

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

      7. associate-+l+ 67.9%

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

      8. *-commutative 67.9%

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

      9. *-commutative 67.9%

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

      10. +-commutative 67.9%

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

      11. +-commutative 67.9%

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

      12. *-commutative 67.9%

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

      13. associate-+l+ 67.9%

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

      14. +-commutative 67.9%

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

      15. distribute-rgt-in 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in z around inf 47.0%

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

      1. *-commutative 47.0%

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

      2. sqrt-prod 50.9%

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

      3. +-commutative 50.9%

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

   7. Applied egg-rr 50.9%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-302}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y \cdot \left(1 + \frac{z}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-299}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e-299)
   (* 2.0 (exp (* (- (log (- (- z) y)) (log (/ -1.0 x))) 0.5)))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e-299) {
		tmp = 2.0 * exp(((log((-z - y)) - log((-1.0 / x))) * 0.5));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-7.2d-299)) then
        tmp = 2.0d0 * exp(((log((-z - y)) - log(((-1.0d0) / x))) * 0.5d0))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e-299) {
		tmp = 2.0 * Math.exp(((Math.log((-z - y)) - Math.log((-1.0 / x))) * 0.5));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -7.2e-299:
		tmp = 2.0 * math.exp(((math.log((-z - y)) - math.log((-1.0 / x))) * 0.5))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e-299)
		tmp = Float64(2.0 * exp(Float64(Float64(log(Float64(Float64(-z) - y)) - log(Float64(-1.0 / x))) * 0.5)));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.2e-299)
		tmp = 2.0 * exp(((log((-z - y)) - log((-1.0 / x))) * 0.5));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -7.2e-299], N[(2.0 * N[Exp[N[(N[(N[Log[N[((-z) - y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.2e-299

   1. Initial program 71.6%

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

      1. associate-+l+ 71.6%

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

      2. *-commutative 71.6%

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

      3. *-commutative 71.6%

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

      4. +-commutative 71.6%

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

      5. fma-define 71.7%

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

      6. +-commutative 71.7%

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

      7. distribute-lft-out 71.9%

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

   3. Simplified 71.9%

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

      1. pow1/2 72.0%

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

      2. fma-undefine 71.7%

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

      3. distribute-rgt-in 71.6%

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

      4. associate-+l+ 71.6%

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

      5. pow-to-exp 66.7%

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

      6. associate-+l+ 66.7%

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

      7. distribute-rgt-in 66.8%

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

      8. fma-undefine 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in x around -inf 45.4%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. distribute-lft-out 45.4%

         \[\leadsto 2 \cdot e^{\left(\log \color{blue}{\left(-1 \cdot \left(y + z\right)\right)} + -1 \cdot \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]

      2. +-commutative 45.4%

         \[\leadsto 2 \cdot e^{\left(\log \left(-1 \cdot \color{blue}{\left(z + y\right)}\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]

      3. mul-1-neg 45.4%

         \[\leadsto 2 \cdot e^{\left(\log \left(-1 \cdot \left(z + y\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{x}\right)\right)}\right) \cdot 0.5} \]

   9. Simplified 45.4%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot \left(z + y\right)\right) + \left(-\log \left(\frac{-1}{x}\right)\right)\right)} \cdot 0.5} \]

   if -7.2e-299 < y

   1. Initial program 67.7%

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

      1. associate-+l+ 67.7%

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

      2. *-commutative 67.7%

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

      3. *-commutative 67.7%

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

      4. *-commutative 67.7%

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

      5. +-commutative 67.7%

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

      6. +-commutative 67.7%

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

      7. associate-+l+ 67.7%

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

      8. *-commutative 67.7%

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

      9. *-commutative 67.7%

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

      10. +-commutative 67.7%

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

      11. +-commutative 67.7%

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

      12. *-commutative 67.7%

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

      13. associate-+l+ 67.7%

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

      14. +-commutative 67.7%

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

      15. distribute-rgt-in 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in z around inf 47.1%

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

      1. *-commutative 47.1%

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

      2. sqrt-prod 50.9%

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

      3. +-commutative 50.9%

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

   7. Applied egg-rr 50.9%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-299}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-248}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6e-248)
   (* 2.0 (sqrt (fma x z (* y (+ z x)))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e-248) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6e-248)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 6e-248], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.00000000000000027e-248

   1. Initial program 71.3%

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

      1. associate-+l+ 71.3%

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

      2. *-commutative 71.3%

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

      3. *-commutative 71.3%

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

      4. *-commutative 71.3%

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

      5. +-commutative 71.3%

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

      6. +-commutative 71.3%

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

      7. +-commutative 71.3%

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

      8. *-commutative 71.3%

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

      9. *-commutative 71.3%

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

      10. associate-+l+ 71.3%

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

      11. +-commutative 71.3%

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

      12. fma-define 71.3%

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

      13. distribute-lft-out 71.4%

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

   3. Simplified 71.4%

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

   if 6.00000000000000027e-248 < y

   1. Initial program 67.6%

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

      1. associate-+l+ 67.6%

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

      2. *-commutative 67.6%

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

      3. *-commutative 67.6%

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

      4. *-commutative 67.6%

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

      5. +-commutative 67.6%

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

      6. +-commutative 67.6%

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

      7. associate-+l+ 67.6%

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

      8. *-commutative 67.6%

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

      9. *-commutative 67.6%

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

      10. +-commutative 67.6%

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

      11. +-commutative 67.6%

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

      12. *-commutative 67.6%

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

      13. associate-+l+ 67.6%

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

      14. +-commutative 67.6%

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

      15. distribute-rgt-in 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in z around inf 44.7%

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

      1. *-commutative 44.7%

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

      2. sqrt-prod 51.4%

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

      3. +-commutative 51.4%

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

   7. Applied egg-rr 51.4%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-248}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y \cdot \left(1 + \frac{z}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.4e-279)
   (* 2.0 (sqrt (* x (* y (+ 1.0 (/ z y))))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.4e-279) {
		tmp = 2.0 * sqrt((x * (y * (1.0 + (z / y)))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.4d-279) then
        tmp = 2.0d0 * sqrt((x * (y * (1.0d0 + (z / y)))))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.4e-279) {
		tmp = 2.0 * Math.sqrt((x * (y * (1.0 + (z / y)))));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.4e-279:
		tmp = 2.0 * math.sqrt((x * (y * (1.0 + (z / y)))))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.4e-279)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y * Float64(1.0 + Float64(z / y))))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.4e-279)
		tmp = 2.0 * sqrt((x * (y * (1.0 + (z / y)))));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.4e-279], N[(2.0 * N[Sqrt[N[(x * N[(y * N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4e-279

   1. Initial program 70.9%

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

      1. distribute-lft-out 70.9%

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

      2. *-commutative 70.9%

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

   4. Applied egg-rr 70.9%

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

   5. Taylor expanded in y around inf 61.9%

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

   6. Taylor expanded in x around inf 35.8%

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

   if 1.4e-279 < 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. associate-+l+ 68.1%

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

      2. *-commutative 68.1%

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

      3. *-commutative 68.1%

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

      4. *-commutative 68.1%

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

      5. +-commutative 68.1%

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

      6. +-commutative 68.1%

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

      7. associate-+l+ 68.1%

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

      8. *-commutative 68.1%

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

      9. *-commutative 68.1%

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

      10. +-commutative 68.1%

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

      11. +-commutative 68.1%

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

      12. *-commutative 68.1%

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

      13. associate-+l+ 68.1%

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

      14. +-commutative 68.1%

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

      15. distribute-rgt-in 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in z around inf 46.0%

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

      1. *-commutative 46.0%

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

      2. sqrt-prod 51.7%

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

      3. +-commutative 51.7%

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

   7. Applied egg-rr 51.7%

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

Alternative 5: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y \cdot \left(1 + \frac{z}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5e-266)
   (* 2.0 (sqrt (* x (* y (+ 1.0 (/ z y))))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e-266) {
		tmp = 2.0 * sqrt((x * (y * (1.0 + (z / y)))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.5d-266) then
        tmp = 2.0d0 * sqrt((x * (y * (1.0d0 + (z / y)))))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e-266) {
		tmp = 2.0 * Math.sqrt((x * (y * (1.0 + (z / y)))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.5e-266:
		tmp = 2.0 * math.sqrt((x * (y * (1.0 + (z / y)))))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5e-266)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y * Float64(1.0 + Float64(z / y))))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.5e-266)
		tmp = 2.0 * sqrt((x * (y * (1.0 + (z / y)))));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.5e-266], N[(2.0 * N[Sqrt[N[(x * N[(y * N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e-266

   1. Initial program 70.3%

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

      1. distribute-lft-out 70.4%

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

      2. *-commutative 70.4%

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

   4. Applied egg-rr 70.4%

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

   5. Taylor expanded in y around inf 61.4%

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

   6. Taylor expanded in x around inf 35.6%

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

   if 1.5e-266 < y

   1. Initial program 68.6%

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

      1. associate-+l+ 68.6%

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

      2. *-commutative 68.6%

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

      3. *-commutative 68.6%

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

      4. *-commutative 68.6%

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

      5. +-commutative 68.6%

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

      6. +-commutative 68.6%

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

      7. associate-+l+ 68.6%

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

      8. *-commutative 68.6%

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

      9. *-commutative 68.6%

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

      10. +-commutative 68.6%

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

      11. +-commutative 68.6%

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

      12. *-commutative 68.6%

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

      13. associate-+l+ 68.6%

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

      14. +-commutative 68.6%

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

      15. distribute-rgt-in 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in x around 0 27.7%

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

      1. *-commutative 27.7%

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

   7. Simplified 27.7%

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

      1. sqrt-prod 41.1%

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

   9. Applied egg-rr 41.1%

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

Alternative 6: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-274}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y \cdot \left(1 + \frac{z}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -8.8e-274)
   (* 2.0 (sqrt (* x (* y (+ 1.0 (/ z y))))))
   (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.8e-274) {
		tmp = 2.0 * sqrt((x * (y * (1.0 + (z / y)))));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-8.8d-274)) then
        tmp = 2.0d0 * sqrt((x * (y * (1.0d0 + (z / y)))))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.8e-274) {
		tmp = 2.0 * Math.sqrt((x * (y * (1.0 + (z / y)))));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -8.8e-274:
		tmp = 2.0 * math.sqrt((x * (y * (1.0 + (z / y)))))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -8.8e-274)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y * Float64(1.0 + Float64(z / y))))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.8e-274)
		tmp = 2.0 * sqrt((x * (y * (1.0 + (z / y)))));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -8.8e-274], N[(2.0 * N[Sqrt[N[(x * N[(y * N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.7999999999999998e-274

   1. Initial program 70.8%

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

      1. distribute-lft-out 70.9%

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

      2. *-commutative 70.9%

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

   4. Applied egg-rr 70.9%

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

   5. Taylor expanded in y around inf 64.4%

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

   6. Taylor expanded in x around inf 35.2%

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

   if -8.7999999999999998e-274 < y

   1. Initial program 68.4%

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

      1. associate-+l+ 68.4%

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

      2. *-commutative 68.4%

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

      3. *-commutative 68.4%

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

      4. *-commutative 68.4%

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

      5. +-commutative 68.4%

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

      6. +-commutative 68.4%

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

      7. associate-+l+ 68.4%

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

      8. *-commutative 68.4%

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

      9. *-commutative 68.4%

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

      10. +-commutative 68.4%

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

      11. +-commutative 68.4%

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

      12. *-commutative 68.4%

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

      13. associate-+l+ 68.4%

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

      14. +-commutative 68.4%

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

      15. distribute-rgt-in 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in z around inf 48.1%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-274}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y \cdot \left(1 + \frac{z}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.15e-276)
   (* 2.0 (sqrt (* (+ y z) x)))
   (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.15e-276) {
		tmp = 2.0 * sqrt(((y + z) * x));
	} 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 <= (-1.15d-276)) then
        tmp = 2.0d0 * sqrt(((y + z) * x))
    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 <= -1.15e-276) {
		tmp = 2.0 * Math.sqrt(((y + z) * x));
	} 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 <= -1.15e-276:
		tmp = 2.0 * math.sqrt(((y + z) * x))
	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 <= -1.15e-276)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + z) * x)));
	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 <= -1.15e-276)
		tmp = 2.0 * sqrt(((y + z) * x));
	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, -1.15e-276], N[(2.0 * N[Sqrt[N[(N[(y + z), $MachinePrecision] * x), $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 < -1.14999999999999991e-276

   1. Initial program 70.8%

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

      1. associate-+l+ 70.8%

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

      2. *-commutative 70.8%

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

      3. *-commutative 70.8%

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

      4. *-commutative 70.8%

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

      5. +-commutative 70.8%

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

      6. +-commutative 70.8%

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

      7. associate-+l+ 70.8%

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

      8. *-commutative 70.8%

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

      9. *-commutative 70.8%

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

      10. +-commutative 70.8%

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

      11. +-commutative 70.8%

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

      12. *-commutative 70.8%

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

      13. associate-+l+ 70.8%

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

      14. +-commutative 70.8%

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

      15. distribute-rgt-in 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in x around inf 39.9%

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

      1. +-commutative 39.9%

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

   7. Simplified 39.9%

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

   if -1.14999999999999991e-276 < y

   1. Initial program 68.4%

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

      1. associate-+l+ 68.4%

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

      2. *-commutative 68.4%

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

      3. *-commutative 68.4%

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

      4. *-commutative 68.4%

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

      5. +-commutative 68.4%

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

      6. +-commutative 68.4%

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

      7. associate-+l+ 68.4%

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

      8. *-commutative 68.4%

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

      9. *-commutative 68.4%

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

      10. +-commutative 68.4%

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

      11. +-commutative 68.4%

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

      12. *-commutative 68.4%

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

      13. associate-+l+ 68.4%

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

      14. +-commutative 68.4%

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

      15. distribute-rgt-in 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in z around inf 48.1%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e-276) (* 2.0 (sqrt (* (+ y z) x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e-276) {
		tmp = 2.0 * sqrt(((y + z) * x));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.1d-276)) then
        tmp = 2.0d0 * sqrt(((y + z) * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e-276) {
		tmp = 2.0 * Math.sqrt(((y + z) * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.1e-276:
		tmp = 2.0 * math.sqrt(((y + z) * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.1e-276)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + z) * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.1e-276)
		tmp = 2.0 * sqrt(((y + z) * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.1e-276], N[(2.0 * N[Sqrt[N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0999999999999999e-276

   1. Initial program 70.8%

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

      1. associate-+l+ 70.8%

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

      2. *-commutative 70.8%

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

      3. *-commutative 70.8%

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

      4. *-commutative 70.8%

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

      5. +-commutative 70.8%

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

      6. +-commutative 70.8%

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

      7. associate-+l+ 70.8%

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

      8. *-commutative 70.8%

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

      9. *-commutative 70.8%

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

      10. +-commutative 70.8%

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

      11. +-commutative 70.8%

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

      12. *-commutative 70.8%

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

      13. associate-+l+ 70.8%

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

      14. +-commutative 70.8%

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

      15. distribute-rgt-in 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in x around inf 39.9%

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

      1. +-commutative 39.9%

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

   7. Simplified 39.9%

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

   if -1.0999999999999999e-276 < y

   1. Initial program 68.4%

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

      1. associate-+l+ 68.4%

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

      2. *-commutative 68.4%

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

      3. *-commutative 68.4%

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

      4. *-commutative 68.4%

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

      5. +-commutative 68.4%

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

      6. +-commutative 68.4%

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

      7. associate-+l+ 68.4%

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

      8. *-commutative 68.4%

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

      9. *-commutative 68.4%

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

      10. +-commutative 68.4%

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

      11. +-commutative 68.4%

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

      12. *-commutative 68.4%

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

      13. associate-+l+ 68.4%

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

      14. +-commutative 68.4%

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

      15. distribute-rgt-in 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in x around 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%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.8% accurate, 1.0× speedup

Math

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

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 69.4%

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

   1. distribute-lft-out 69.4%

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

   2. *-commutative 69.4%

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

4. Applied egg-rr 69.4%

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

Alternative 10: 70.8% 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 69.4%

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

   1. associate-+l+ 69.4%

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

   2. *-commutative 69.4%

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

   3. *-commutative 69.4%

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

   4. *-commutative 69.4%

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

   5. +-commutative 69.4%

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

   6. +-commutative 69.4%

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

   7. associate-+l+ 69.4%

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

   8. *-commutative 69.4%

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

   9. *-commutative 69.4%

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

   10. +-commutative 69.4%

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

   11. +-commutative 69.4%

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

   12. *-commutative 69.4%

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

   13. associate-+l+ 69.4%

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

   14. +-commutative 69.4%

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

   15. distribute-rgt-in 69.4%

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

3. Simplified 69.4%

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

5. Final simplification 69.4%

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

Alternative 11: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e-276) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e-276) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.1d-276)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e-276) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.1e-276:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.1e-276)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.1e-276)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.1e-276], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0999999999999999e-276

   1. Initial program 70.8%

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

      1. associate-+l+ 70.8%

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

      2. *-commutative 70.8%

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

      3. *-commutative 70.8%

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

      4. *-commutative 70.8%

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

      5. +-commutative 70.8%

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

      6. +-commutative 70.8%

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

      7. associate-+l+ 70.8%

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

      8. *-commutative 70.8%

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

      9. *-commutative 70.8%

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

      10. +-commutative 70.8%

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

      11. +-commutative 70.8%

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

      12. *-commutative 70.8%

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

      13. associate-+l+ 70.8%

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

      14. +-commutative 70.8%

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

      15. distribute-rgt-in 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in z around 0 25.2%

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

   if -1.0999999999999999e-276 < y

   1. Initial program 68.4%

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

      1. associate-+l+ 68.4%

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

      2. *-commutative 68.4%

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

      3. *-commutative 68.4%

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

      4. *-commutative 68.4%

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

      5. +-commutative 68.4%

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

      6. +-commutative 68.4%

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

      7. associate-+l+ 68.4%

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

      8. *-commutative 68.4%

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

      9. *-commutative 68.4%

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

      10. +-commutative 68.4%

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

      11. +-commutative 68.4%

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

      12. *-commutative 68.4%

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

      13. associate-+l+ 68.4%

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

      14. +-commutative 68.4%

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

      15. distribute-rgt-in 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in x around 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%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

   1. associate-+l+ 69.4%

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

   2. *-commutative 69.4%

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

   3. *-commutative 69.4%

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

   4. *-commutative 69.4%

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

   5. +-commutative 69.4%

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

   6. +-commutative 69.4%

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

   7. associate-+l+ 69.4%

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

   8. *-commutative 69.4%

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

   9. *-commutative 69.4%

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

   10. +-commutative 69.4%

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

   11. +-commutative 69.4%

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

   12. *-commutative 69.4%

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

   13. associate-+l+ 69.4%

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

   14. +-commutative 69.4%

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

   15. distribute-rgt-in 69.4%

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

3. Simplified 69.4%

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

5. Taylor expanded in z around 0 24.0%

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

6. Final simplification 24.0%

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

Developer Target 1: 82.9% 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 2024135 
(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)))))