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

Percentage Accurate: 70.5% → 94.2%

Time: 15.0s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(2 \cdot \sqrt{\frac{y + x}{z}} + x \cdot \sqrt{\frac{y}{{z}^{3}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.95e+30)
   (* 2.0 (exp (* (- (log (- (- x) z)) (log (/ -1.0 y))) 0.5)))
   (if (<= y 3.5e-58)
     (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
     (* z (+ (* 2.0 (sqrt (/ (+ y x) z))) (* x (sqrt (/ y (pow z 3.0)))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.95e+30) {
		tmp = 2.0 * exp(((log((-x - z)) - log((-1.0 / y))) * 0.5));
	} else if (y <= 3.5e-58) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = z * ((2.0 * sqrt(((y + x) / z))) + (x * sqrt((y / 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 <= (-1.95d+30)) then
        tmp = 2.0d0 * exp(((log((-x - z)) - log(((-1.0d0) / y))) * 0.5d0))
    else if (y <= 3.5d-58) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    else
        tmp = z * ((2.0d0 * sqrt(((y + x) / z))) + (x * sqrt((y / (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 <= -1.95e+30) {
		tmp = 2.0 * Math.exp(((Math.log((-x - z)) - Math.log((-1.0 / y))) * 0.5));
	} else if (y <= 3.5e-58) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = z * ((2.0 * Math.sqrt(((y + x) / z))) + (x * Math.sqrt((y / Math.pow(z, 3.0)))));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.95e+30:
		tmp = 2.0 * math.exp(((math.log((-x - z)) - math.log((-1.0 / y))) * 0.5))
	elif y <= 3.5e-58:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	else:
		tmp = z * ((2.0 * math.sqrt(((y + x) / z))) + (x * math.sqrt((y / 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 <= -1.95e+30)
		tmp = Float64(2.0 * exp(Float64(Float64(log(Float64(Float64(-x) - z)) - log(Float64(-1.0 / y))) * 0.5)));
	elseif (y <= 3.5e-58)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(z * Float64(Float64(2.0 * sqrt(Float64(Float64(y + x) / z))) + Float64(x * sqrt(Float64(y / (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 <= -1.95e+30)
		tmp = 2.0 * exp(((log((-x - z)) - log((-1.0 / y))) * 0.5));
	elseif (y <= 3.5e-58)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	else
		tmp = z * ((2.0 * sqrt(((y + x) / z))) + (x * sqrt((y / (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, -1.95e+30], N[(2.0 * N[Exp[N[(N[(N[Log[N[((-x) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e-58], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(2.0 * N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(x * N[Sqrt[N[(y / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.95000000000000005e30

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. associate-+r+ 41.7%

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

      3. *-commutative 41.7%

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

      4. +-commutative 41.7%

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

      5. associate-+l+ 41.7%

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

      6. *-commutative 41.7%

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

      7. *-commutative 41.7%

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

      8. +-commutative 41.7%

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

      9. +-commutative 41.7%

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

      10. *-commutative 41.7%

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

      11. *-commutative 41.7%

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

      12. associate-+l+ 41.7%

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

      13. +-commutative 41.7%

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

      14. fma-define 41.7%

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

      15. distribute-lft-out 41.9%

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

   3. Simplified 41.9%

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

      1. pow1/2 41.9%

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

      2. pow-to-exp 38.9%

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

   6. Applied egg-rr 38.9%

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

   7. Taylor expanded in y around -inf 80.1%

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

      1. mul-1-neg 80.1%

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

      2. +-commutative 80.1%

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

      3. mul-1-neg 80.1%

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

   9. Simplified 80.1%

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

   if -1.95000000000000005e30 < y < 3.4999999999999999e-58

   1. Initial program 87.6%

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

      1. associate-+l+ 87.6%

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

      2. +-commutative 87.6%

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

      3. distribute-rgt-in 87.6%

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

   3. Simplified 87.6%

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

   if 3.4999999999999999e-58 < y

   1. Initial program 56.5%

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

      1. associate-+l+ 56.5%

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

      2. +-commutative 56.5%

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

      3. distribute-rgt-in 56.5%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in z around inf 41.1%

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

   6. Taylor expanded in x around 0 41.4%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(2 \cdot \sqrt{\frac{y + x}{z}} + x \cdot \sqrt{\frac{y}{{z}^{3}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+25}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-241}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + \left(\frac{y}{x} + 1\right) \cdot \left(z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4e+25)
   (* 2.0 (exp (* (- (log (- (- x) z)) (log (/ -1.0 y))) 0.5)))
   (if (<= y 8.6e-241)
     (* 2.0 (sqrt (+ (* y x) (* (+ (/ y x) 1.0) (* 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 <= -4e+25) {
		tmp = 2.0 * exp(((log((-x - z)) - log((-1.0 / y))) * 0.5));
	} else if (y <= 8.6e-241) {
		tmp = 2.0 * sqrt(((y * x) + (((y / x) + 1.0) * (z * x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4d+25)) then
        tmp = 2.0d0 * exp(((log((-x - z)) - log(((-1.0d0) / y))) * 0.5d0))
    else if (y <= 8.6d-241) then
        tmp = 2.0d0 * sqrt(((y * x) + (((y / x) + 1.0d0) * (z * x))))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+25) {
		tmp = 2.0 * Math.exp(((Math.log((-x - z)) - Math.log((-1.0 / y))) * 0.5));
	} else if (y <= 8.6e-241) {
		tmp = 2.0 * Math.sqrt(((y * x) + (((y / x) + 1.0) * (z * x))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -4e+25:
		tmp = 2.0 * math.exp(((math.log((-x - z)) - math.log((-1.0 / y))) * 0.5))
	elif y <= 8.6e-241:
		tmp = 2.0 * math.sqrt(((y * x) + (((y / x) + 1.0) * (z * x))))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -4e+25)
		tmp = Float64(2.0 * exp(Float64(Float64(log(Float64(Float64(-x) - z)) - log(Float64(-1.0 / y))) * 0.5)));
	elseif (y <= 8.6e-241)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(Float64(Float64(y / x) + 1.0) * Float64(z * x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4e+25)
		tmp = 2.0 * exp(((log((-x - z)) - log((-1.0 / y))) * 0.5));
	elseif (y <= 8.6e-241)
		tmp = 2.0 * sqrt(((y * x) + (((y / x) + 1.0) * (z * x))));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -4e+25], N[(2.0 * N[Exp[N[(N[(N[Log[N[((-x) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e-241], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(N[(N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision] * 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 3 regimes

2. if y < -4.00000000000000036e25

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. associate-+r+ 41.7%

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

      3. *-commutative 41.7%

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

      4. +-commutative 41.7%

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

      5. associate-+l+ 41.7%

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

      6. *-commutative 41.7%

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

      7. *-commutative 41.7%

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

      8. +-commutative 41.7%

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

      9. +-commutative 41.7%

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

      10. *-commutative 41.7%

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

      11. *-commutative 41.7%

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

      12. associate-+l+ 41.7%

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

      13. +-commutative 41.7%

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

      14. fma-define 41.7%

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

      15. distribute-lft-out 41.9%

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

   3. Simplified 41.9%

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

      1. pow1/2 41.9%

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

      2. pow-to-exp 38.9%

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

   6. Applied egg-rr 38.9%

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

   7. Taylor expanded in y around -inf 80.1%

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

      1. mul-1-neg 80.1%

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

      2. +-commutative 80.1%

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

      3. mul-1-neg 80.1%

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

   9. Simplified 80.1%

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

   if -4.00000000000000036e25 < y < 8.5999999999999997e-241

   1. Initial program 87.5%

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

      1. associate-+l+ 87.5%

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

      2. +-commutative 87.5%

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

      3. distribute-rgt-in 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in x around inf 87.6%

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

      1. associate-*r* 86.5%

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

      2. +-commutative 86.5%

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

      3. distribute-rgt-in 86.2%

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

      4. *-un-lft-identity 86.2%

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

   7. Applied egg-rr 86.2%

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

      1. distribute-lft1-in 86.5%

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

      2. *-commutative 86.5%

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

   9. Applied egg-rr 86.5%

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

   if 8.5999999999999997e-241 < y

   1. Initial program 68.0%

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

      1. associate-+l+ 68.0%

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

      2. +-commutative 68.0%

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

      3. distribute-rgt-in 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in x around 0 22.1%

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

      1. *-commutative 22.1%

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

   7. Simplified 22.1%

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

      1. sqrt-prod 32.2%

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

   9. Applied egg-rr 32.2%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+25}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-241}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + \left(\frac{y}{x} + 1\right) \cdot \left(z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-241}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 8.6e-241)
   (* 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 <= 8.6e-241) {
		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 <= 8.6e-241)
		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, 8.6e-241], 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 < 8.5999999999999997e-241

   1. Initial program 71.1%

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

      1. +-commutative 71.1%

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

      2. associate-+r+ 71.1%

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

      3. *-commutative 71.1%

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

      4. +-commutative 71.1%

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

      5. associate-+l+ 71.1%

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

      6. *-commutative 71.1%

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

      7. *-commutative 71.1%

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

      8. +-commutative 71.1%

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

      9. +-commutative 71.1%

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

      10. *-commutative 71.1%

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

      11. *-commutative 71.1%

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

      12. associate-+l+ 71.1%

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

      13. +-commutative 71.1%

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

      14. fma-define 71.1%

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

      15. distribute-lft-out 71.1%

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

   3. Simplified 71.1%

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

   if 8.5999999999999997e-241 < y

   1. Initial program 68.0%

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

      1. associate-+l+ 68.0%

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

      2. +-commutative 68.0%

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

      3. distribute-rgt-in 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in x around 0 22.1%

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

      1. *-commutative 22.1%

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

   7. Simplified 22.1%

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

      1. sqrt-prod 32.2%

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

   9. Applied egg-rr 32.2%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-241}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-241}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(z \cdot \frac{y}{x} + \left(y + z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 8.6e-241)
   (* 2.0 (sqrt (* x (+ (* z (/ y x)) (+ y z)))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.6e-241) {
		tmp = 2.0 * sqrt((x * ((z * (y / x)) + (y + z))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.6d-241) then
        tmp = 2.0d0 * sqrt((x * ((z * (y / x)) + (y + z))))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.6e-241) {
		tmp = 2.0 * Math.sqrt((x * ((z * (y / x)) + (y + z))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 8.6e-241:
		tmp = 2.0 * math.sqrt((x * ((z * (y / x)) + (y + z))))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 8.6e-241)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(Float64(z * Float64(y / x)) + Float64(y + z)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.6e-241)
		tmp = 2.0 * sqrt((x * ((z * (y / x)) + (y + z))));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 8.6e-241], N[(2.0 * N[Sqrt[N[(x * N[(N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(y + z), $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 < 8.5999999999999997e-241

   1. Initial program 71.1%

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

      1. +-commutative 71.1%

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

      2. associate-+r+ 71.1%

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

      3. *-commutative 71.1%

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

      4. +-commutative 71.1%

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

      5. associate-+l+ 71.1%

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

      6. *-commutative 71.1%

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

      7. *-commutative 71.1%

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

      8. +-commutative 71.1%

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

      9. +-commutative 71.1%

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

      10. *-commutative 71.1%

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

      11. *-commutative 71.1%

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

      12. associate-+l+ 71.1%

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

      13. +-commutative 71.1%

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

      14. fma-define 71.1%

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

      15. distribute-lft-out 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in x around inf 63.9%

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

      1. +-commutative 63.9%

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

      2. +-commutative 63.9%

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

      3. associate-+l+ 63.9%

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

      4. *-commutative 63.9%

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

      5. associate-/l* 61.9%

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

   7. Simplified 61.9%

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

   if 8.5999999999999997e-241 < y

   1. Initial program 68.0%

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

      1. associate-+l+ 68.0%

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

      2. +-commutative 68.0%

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

      3. distribute-rgt-in 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in x around 0 22.1%

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

      1. *-commutative 22.1%

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

   7. Simplified 22.1%

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

      1. sqrt-prod 32.2%

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

   9. Applied egg-rr 32.2%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-241}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(z \cdot \frac{y}{x} + \left(y + z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-298) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-298) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5d-298) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-298) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 5e-298:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-298)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e-298)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 5e-298], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e-298

   1. Initial program 69.2%

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

      1. associate-+l+ 69.2%

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

      2. +-commutative 69.2%

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

      3. distribute-rgt-in 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around inf 51.8%

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

      1. +-commutative 51.8%

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

   7. Simplified 51.8%

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

   if 5.0000000000000002e-298 < y

   1. Initial program 70.4%

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

      1. associate-+l+ 70.4%

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

      2. +-commutative 70.4%

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

      3. distribute-rgt-in 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in z around inf 49.2%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot \left(z \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.4e-293) (* 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 <= 3.4e-293) {
		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 <= 3.4d-293) 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 <= 3.4e-293) {
		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 <= 3.4e-293:
		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 <= 3.4e-293)
		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 <= 3.4e-293)
		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, 3.4e-293], 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 < 3.4e-293

   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%

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

      3. distribute-rgt-in 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in x around inf 52.2%

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

      1. +-commutative 52.2%

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

   7. Simplified 52.2%

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

   if 3.4e-293 < y

   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. associate-+l+ 70.1%

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

      2. +-commutative 70.1%

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

      3. distribute-rgt-in 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 20.9%

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

      1. *-commutative 20.9%

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

   7. Simplified 20.9%

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

      1. add-sqr-sqrt 20.8%

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

      2. sqrt-unprod 20.9%

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

      3. *-commutative 20.9%

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

      4. *-commutative 20.9%

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

      5. swap-sqr 20.9%

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

      6. add-sqr-sqrt 20.9%

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

      7. *-commutative 20.9%

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

      8. metadata-eval 20.9%

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

   9. Applied egg-rr 20.9%

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

       1. associate-*l* 20.9%

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

   11. Simplified 20.9%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot \left(z \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.5% 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 69.8%

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

   1. associate-+l+ 69.8%

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

   2. +-commutative 69.8%

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

   3. distribute-rgt-in 69.8%

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

3. Simplified 69.8%

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

5. Final simplification 69.8%

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

Alternative 8: 68.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-310) (* 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 <= -4e-310) {
		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 <= (-4d-310)) 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 <= -4e-310) {
		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 <= -4e-310:
		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 <= -4e-310)
		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 <= -4e-310)
		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, -4e-310], 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 < -3.999999999999988e-310

   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. associate-+l+ 68.5%

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

      2. +-commutative 68.5%

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

      3. distribute-rgt-in 68.5%

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

   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 z around 0 23.8%

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

   if -3.999999999999988e-310 < y

   1. Initial program 71.1%

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

      1. associate-+l+ 71.1%

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

      2. +-commutative 71.1%

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

      3. distribute-rgt-in 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in x around 0 20.3%

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

      1. *-commutative 20.3%

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

   7. Simplified 20.3%

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

      1. add-sqr-sqrt 20.2%

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

      2. sqrt-unprod 20.3%

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

      3. *-commutative 20.3%

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

      4. *-commutative 20.3%

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

      5. swap-sqr 20.3%

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

      6. add-sqr-sqrt 20.3%

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

      7. *-commutative 20.3%

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

      8. metadata-eval 20.3%

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

   9. Applied egg-rr 20.3%

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

       1. associate-*l* 20.4%

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

   11. Simplified 20.4%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot \left(z \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.1% 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 69.8%

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

   1. associate-+l+ 69.8%

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

   2. +-commutative 69.8%

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

   3. distribute-rgt-in 69.8%

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

3. Simplified 69.8%

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

5. Taylor expanded in x around 0 19.8%

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

   1. *-commutative 19.8%

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

7. Simplified 19.8%

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

   1. add-sqr-sqrt 19.7%

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

   2. sqrt-unprod 19.8%

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

   3. *-commutative 19.8%

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

   4. *-commutative 19.8%

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

   5. swap-sqr 19.8%

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

   6. add-sqr-sqrt 19.8%

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

   7. *-commutative 19.8%

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

   8. metadata-eval 19.8%

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

9. Applied egg-rr 19.8%

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

    1. associate-*l* 19.8%

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

11. Simplified 19.8%

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

Developer Target 1: 83.4% 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 2024144 
(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)))))