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

Percentage Accurate: 70.9% → 95.2%

Time: 10.9s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\\ \mathbf{if}\;y \leq -3 \cdot 10^{+25}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot t_0}\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-190}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot t_0}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log (- (- y) z)) (log (/ -1.0 x)))))
   (if (<= y -3e+25)
     (* 2.0 (exp (* 0.5 t_0)))
     (if (<= y -4.5e-190)
       (* 2.0 (sqrt (+ (* x (+ y z)) (* y z))))
       (if (<= y 8.2e-278)
         (* 2.0 (pow (exp (* 0.25 t_0)) 2.0))
         (* 2.0 (* (sqrt z) (sqrt y))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = log((-y - z)) - log((-1.0 / x));
	double tmp;
	if (y <= -3e+25) {
		tmp = 2.0 * exp((0.5 * t_0));
	} else if (y <= -4.5e-190) {
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	} else if (y <= 8.2e-278) {
		tmp = 2.0 * pow(exp((0.25 * t_0)), 2.0);
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log((-y - z)) - log(((-1.0d0) / x))
    if (y <= (-3d+25)) then
        tmp = 2.0d0 * exp((0.5d0 * t_0))
    else if (y <= (-4.5d-190)) then
        tmp = 2.0d0 * sqrt(((x * (y + z)) + (y * z)))
    else if (y <= 8.2d-278) then
        tmp = 2.0d0 * (exp((0.25d0 * t_0)) ** 2.0d0)
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = Math.log((-y - z)) - Math.log((-1.0 / x));
	double tmp;
	if (y <= -3e+25) {
		tmp = 2.0 * Math.exp((0.5 * t_0));
	} else if (y <= -4.5e-190) {
		tmp = 2.0 * Math.sqrt(((x * (y + z)) + (y * z)));
	} else if (y <= 8.2e-278) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * t_0)), 2.0);
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = math.log((-y - z)) - math.log((-1.0 / x))
	tmp = 0
	if y <= -3e+25:
		tmp = 2.0 * math.exp((0.5 * t_0))
	elif y <= -4.5e-190:
		tmp = 2.0 * math.sqrt(((x * (y + z)) + (y * z)))
	elif y <= 8.2e-278:
		tmp = 2.0 * math.pow(math.exp((0.25 * t_0)), 2.0)
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x)))
	tmp = 0.0
	if (y <= -3e+25)
		tmp = Float64(2.0 * exp(Float64(0.5 * t_0)));
	elseif (y <= -4.5e-190)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * Float64(y + z)) + Float64(y * z))));
	elseif (y <= 8.2e-278)
		tmp = Float64(2.0 * (exp(Float64(0.25 * t_0)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = log((-y - z)) - log((-1.0 / x));
	tmp = 0.0;
	if (y <= -3e+25)
		tmp = 2.0 * exp((0.5 * t_0));
	elseif (y <= -4.5e-190)
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	elseif (y <= 8.2e-278)
		tmp = 2.0 * (exp((0.25 * t_0)) ^ 2.0);
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e+25], N[(2.0 * N[Exp[N[(0.5 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4.5e-190], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-278], N[(2.0 * N[Power[N[Exp[N[(0.25 * t$95$0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.00000000000000006e25

   1. Initial program 55.7%

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

      1. distribute-lft-out 55.7%

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

   3. Simplified 55.7%

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

      1. distribute-lft-in 55.7%

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

      2. associate-+r+ 55.7%

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

      3. distribute-rgt-in 55.9%

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

      4. +-commutative 55.9%

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

      5. fma-udef 56.3%

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

      6. add-cbrt-cube 34.3%

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

      7. pow1/3 31.9%

         \[\leadsto 2 \cdot \color{blue}{{\left(\left(\sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)} \cdot \sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}\right) \cdot \sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}\right)}^{0.3333333333333333}} \]

   5. Applied egg-rr 32.4%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 34.7%

         \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{1.5}}} \]

      2. fma-def 33.8%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(x \cdot \left(y + z\right) + y \cdot z\right)}}^{1.5}} \]

      3. +-commutative 33.8%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(y \cdot z + x \cdot \left(y + z\right)\right)}}^{1.5}} \]

      4. fma-def 34.7%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}}^{1.5}} \]

      5. *-commutative 34.7%

         \[\leadsto 2 \cdot \sqrt[3]{{\left(\mathsf{fma}\left(y, z, \color{blue}{\left(y + z\right) \cdot x}\right)\right)}^{1.5}} \]

   7. Simplified 34.7%

      \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)}^{1.5}}} \]
   8. Step-by-step derivation

      1. pow1/3 32.4%

         \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      2. pow-to-exp 32.3%

         \[\leadsto 2 \cdot {\color{blue}{\left(e^{\log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right) \cdot 1.5}\right)}}^{0.3333333333333333} \]

      3. pow-exp 51.8%

         \[\leadsto 2 \cdot \color{blue}{e^{\left(\log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right) \cdot 1.5\right) \cdot 0.3333333333333333}} \]

      4. add-log-exp 32.2%

         \[\leadsto 2 \cdot e^{\color{blue}{\log \left(e^{\log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right) \cdot 1.5}\right)} \cdot 0.3333333333333333} \]

      5. pow-to-exp 32.7%

         \[\leadsto 2 \cdot e^{\log \color{blue}{\left({\left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)}^{1.5}\right)} \cdot 0.3333333333333333} \]

      6. log-pow 51.8%

         \[\leadsto 2 \cdot e^{\color{blue}{\left(1.5 \cdot \log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)\right)} \cdot 0.3333333333333333} \]

      7. *-commutative 51.8%

         \[\leadsto 2 \cdot e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(y, z, \color{blue}{x \cdot \left(y + z\right)}\right)\right)\right) \cdot 0.3333333333333333} \]

   9. Applied egg-rr 51.8%

      \[\leadsto 2 \cdot \color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)\right) \cdot 0.3333333333333333}} \]

   10. Taylor expanded in x around -inf 46.9%

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

       1. +-commutative 46.9%

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

       2. mul-1-neg 46.9%

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

       3. unsub-neg 46.9%

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

       4. distribute-lft-in 46.9%

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

       5. mul-1-neg 46.9%

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

       6. unsub-neg 46.9%

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

       7. mul-1-neg 46.9%

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

   12. Simplified 46.9%

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

   if -3.00000000000000006e25 < y < -4.50000000000000021e-190

   1. Initial program 79.8%

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

      1. distribute-lft-out 79.8%

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

   3. Simplified 79.8%

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

   if -4.50000000000000021e-190 < y < 8.20000000000000002e-278

   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. distribute-lft-out 70.4%

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

   3. Simplified 70.4%

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

      1. add-sqr-sqrt 70.1%

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

      2. pow2 70.1%

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

      3. pow1/2 70.1%

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

      4. sqrt-pow1 70.1%

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

      5. fma-def 70.1%

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

      6. metadata-eval 70.1%

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

   5. Applied egg-rr 70.1%

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

   6. Taylor expanded in x around -inf 58.3%

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

   if 8.20000000000000002e-278 < y

   1. Initial program 69.9%

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

      1. distribute-lft-out 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in x around 0 24.6%

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

      1. sqrt-prod 34.5%

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

   6. Applied egg-rr 34.5%

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

      1. *-commutative 34.5%

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

   8. Simplified 34.5%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+25}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-190}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2: 95.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-187}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 2.0 (exp (* 0.5 (- (log (- (- y) z)) (log (/ -1.0 x))))))))
   (if (<= y -5.4e+20)
     t_0
     (if (<= y -1.8e-187)
       (* 2.0 (sqrt (+ (* x (+ y z)) (* y z))))
       (if (<= y 1.38e-285) t_0 (* 2.0 (* (sqrt z) (sqrt y))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 2.0 * exp((0.5 * (log((-y - z)) - log((-1.0 / x)))));
	double tmp;
	if (y <= -5.4e+20) {
		tmp = t_0;
	} else if (y <= -1.8e-187) {
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	} else if (y <= 1.38e-285) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * exp((0.5d0 * (log((-y - z)) - log(((-1.0d0) / x)))))
    if (y <= (-5.4d+20)) then
        tmp = t_0
    else if (y <= (-1.8d-187)) then
        tmp = 2.0d0 * sqrt(((x * (y + z)) + (y * z)))
    else if (y <= 1.38d-285) then
        tmp = t_0
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 2.0 * Math.exp((0.5 * (Math.log((-y - z)) - Math.log((-1.0 / x)))));
	double tmp;
	if (y <= -5.4e+20) {
		tmp = t_0;
	} else if (y <= -1.8e-187) {
		tmp = 2.0 * Math.sqrt(((x * (y + z)) + (y * z)));
	} else if (y <= 1.38e-285) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 2.0 * math.exp((0.5 * (math.log((-y - z)) - math.log((-1.0 / x)))))
	tmp = 0
	if y <= -5.4e+20:
		tmp = t_0
	elif y <= -1.8e-187:
		tmp = 2.0 * math.sqrt(((x * (y + z)) + (y * z)))
	elif y <= 1.38e-285:
		tmp = t_0
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))))
	tmp = 0.0
	if (y <= -5.4e+20)
		tmp = t_0;
	elseif (y <= -1.8e-187)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * Float64(y + z)) + Float64(y * z))));
	elseif (y <= 1.38e-285)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 2.0 * exp((0.5 * (log((-y - z)) - log((-1.0 / x)))));
	tmp = 0.0;
	if (y <= -5.4e+20)
		tmp = t_0;
	elseif (y <= -1.8e-187)
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	elseif (y <= 1.38e-285)
		tmp = t_0;
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.4e+20], t$95$0, If[LessEqual[y, -1.8e-187], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.38e-285], t$95$0, N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4e20 or -1.79999999999999997e-187 < y < 1.38000000000000009e-285

   1. Initial program 59.9%

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

      1. distribute-lft-out 60.0%

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

   3. Simplified 60.0%

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

      1. distribute-lft-in 59.9%

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

      2. associate-+r+ 59.9%

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

      3. distribute-rgt-in 60.1%

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

      4. +-commutative 60.1%

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

      5. fma-udef 60.4%

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

      6. add-cbrt-cube 38.9%

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

      7. pow1/3 36.3%

         \[\leadsto 2 \cdot \color{blue}{{\left(\left(\sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)} \cdot \sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}\right) \cdot \sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}\right)}^{0.3333333333333333}} \]

   5. Applied egg-rr 36.6%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 39.2%

         \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{1.5}}} \]

      2. fma-def 38.6%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(x \cdot \left(y + z\right) + y \cdot z\right)}}^{1.5}} \]

      3. +-commutative 38.6%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(y \cdot z + x \cdot \left(y + z\right)\right)}}^{1.5}} \]

      4. fma-def 39.2%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}}^{1.5}} \]

      5. *-commutative 39.2%

         \[\leadsto 2 \cdot \sqrt[3]{{\left(\mathsf{fma}\left(y, z, \color{blue}{\left(y + z\right) \cdot x}\right)\right)}^{1.5}} \]

   7. Simplified 39.2%

      \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)}^{1.5}}} \]
   8. Step-by-step derivation

      1. pow1/3 36.6%

         \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      2. pow-to-exp 36.7%

         \[\leadsto 2 \cdot {\color{blue}{\left(e^{\log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right) \cdot 1.5}\right)}}^{0.3333333333333333} \]

      3. pow-exp 55.6%

         \[\leadsto 2 \cdot \color{blue}{e^{\left(\log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right) \cdot 1.5\right) \cdot 0.3333333333333333}} \]

      4. add-log-exp 36.6%

         \[\leadsto 2 \cdot e^{\color{blue}{\log \left(e^{\log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right) \cdot 1.5}\right)} \cdot 0.3333333333333333} \]

      5. pow-to-exp 36.9%

         \[\leadsto 2 \cdot e^{\log \color{blue}{\left({\left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)}^{1.5}\right)} \cdot 0.3333333333333333} \]

      6. log-pow 55.6%

         \[\leadsto 2 \cdot e^{\color{blue}{\left(1.5 \cdot \log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)\right)} \cdot 0.3333333333333333} \]

      7. *-commutative 55.6%

         \[\leadsto 2 \cdot e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(y, z, \color{blue}{x \cdot \left(y + z\right)}\right)\right)\right) \cdot 0.3333333333333333} \]

   9. Applied egg-rr 55.6%

      \[\leadsto 2 \cdot \color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)\right) \cdot 0.3333333333333333}} \]

   10. Taylor expanded in x around -inf 49.1%

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

       1. +-commutative 49.1%

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

       2. mul-1-neg 49.1%

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

       3. unsub-neg 49.1%

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

       4. distribute-lft-in 49.1%

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

       5. mul-1-neg 49.1%

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

       6. unsub-neg 49.1%

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

       7. mul-1-neg 49.1%

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

   12. Simplified 49.1%

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

   if -5.4e20 < y < -1.79999999999999997e-187

   1. Initial program 79.3%

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

      1. distribute-lft-out 79.3%

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

   3. Simplified 79.3%

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

   if 1.38000000000000009e-285 < 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. distribute-lft-out 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in x around 0 24.4%

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

      1. sqrt-prod 34.2%

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

   6. Applied egg-rr 34.2%

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

      1. *-commutative 34.2%

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

   8. Simplified 34.2%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-187}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3: 94.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -8.5e+20)
   (* 2.0 (exp (* 0.5 (- (log (- y)) (log (/ -1.0 x))))))
   (if (<= y 2.55e+45)
     (* 2.0 (sqrt (+ (* x (+ y z)) (* y z))))
     (* 2.0 (* (sqrt z) (sqrt y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.5e+20) {
		tmp = 2.0 * exp((0.5 * (log(-y) - log((-1.0 / x)))));
	} else if (y <= 2.55e+45) {
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-8.5d+20)) then
        tmp = 2.0d0 * exp((0.5d0 * (log(-y) - log(((-1.0d0) / x)))))
    else if (y <= 2.55d+45) then
        tmp = 2.0d0 * sqrt(((x * (y + z)) + (y * z)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.5e+20) {
		tmp = 2.0 * Math.exp((0.5 * (Math.log(-y) - Math.log((-1.0 / x)))));
	} else if (y <= 2.55e+45) {
		tmp = 2.0 * Math.sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -8.5e+20:
		tmp = 2.0 * math.exp((0.5 * (math.log(-y) - math.log((-1.0 / x)))))
	elif y <= 2.55e+45:
		tmp = 2.0 * math.sqrt(((x * (y + z)) + (y * z)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -8.5e+20)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(-y)) - log(Float64(-1.0 / x))))));
	elseif (y <= 2.55e+45)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * Float64(y + z)) + Float64(y * z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.5e+20)
		tmp = 2.0 * exp((0.5 * (log(-y) - log((-1.0 / x)))));
	elseif (y <= 2.55e+45)
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -8.5e+20], N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[(-y)], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.55e+45], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5e20

   1. Initial program 56.4%

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

      1. distribute-lft-out 56.5%

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

   3. Simplified 56.5%

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

      1. distribute-lft-in 56.4%

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

      2. associate-+r+ 56.4%

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

      3. distribute-rgt-in 56.6%

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

      4. +-commutative 56.6%

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

      5. fma-udef 57.0%

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

      6. add-cbrt-cube 35.4%

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

      7. pow1/3 32.9%

         \[\leadsto 2 \cdot \color{blue}{{\left(\left(\sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)} \cdot \sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}\right) \cdot \sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}\right)}^{0.3333333333333333}} \]

   5. Applied egg-rr 33.3%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 35.8%

         \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{1.5}}} \]

      2. fma-def 34.8%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(x \cdot \left(y + z\right) + y \cdot z\right)}}^{1.5}} \]

      3. +-commutative 34.8%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(y \cdot z + x \cdot \left(y + z\right)\right)}}^{1.5}} \]

      4. fma-def 35.8%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}}^{1.5}} \]

      5. *-commutative 35.8%

         \[\leadsto 2 \cdot \sqrt[3]{{\left(\mathsf{fma}\left(y, z, \color{blue}{\left(y + z\right) \cdot x}\right)\right)}^{1.5}} \]

   7. Simplified 35.8%

      \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)}^{1.5}}} \]
   8. Step-by-step derivation

      1. pow1/3 33.3%

         \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      2. pow-to-exp 33.2%

         \[\leadsto 2 \cdot {\color{blue}{\left(e^{\log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right) \cdot 1.5}\right)}}^{0.3333333333333333} \]

      3. pow-exp 52.4%

         \[\leadsto 2 \cdot \color{blue}{e^{\left(\log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right) \cdot 1.5\right) \cdot 0.3333333333333333}} \]

      4. add-log-exp 33.1%

         \[\leadsto 2 \cdot e^{\color{blue}{\log \left(e^{\log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right) \cdot 1.5}\right)} \cdot 0.3333333333333333} \]

      5. pow-to-exp 33.6%

         \[\leadsto 2 \cdot e^{\log \color{blue}{\left({\left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)}^{1.5}\right)} \cdot 0.3333333333333333} \]

      6. log-pow 52.4%

         \[\leadsto 2 \cdot e^{\color{blue}{\left(1.5 \cdot \log \left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)\right)} \cdot 0.3333333333333333} \]

      7. *-commutative 52.4%

         \[\leadsto 2 \cdot e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(y, z, \color{blue}{x \cdot \left(y + z\right)}\right)\right)\right) \cdot 0.3333333333333333} \]

   9. Applied egg-rr 52.4%

      \[\leadsto 2 \cdot \color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)\right) \cdot 0.3333333333333333}} \]

   10. Taylor expanded in z around 0 24.7%

       \[\leadsto 2 \cdot e^{\color{blue}{\left(1.5 \cdot \log \left(y \cdot x\right)\right)} \cdot 0.3333333333333333} \]
   11. Step-by-step derivation

       1. *-commutative 24.7%

          \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(y \cdot x\right) \cdot 1.5\right)} \cdot 0.3333333333333333} \]

   12. Simplified 24.7%

       \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(y \cdot x\right) \cdot 1.5\right)} \cdot 0.3333333333333333} \]

   13. Taylor expanded in x around -inf 44.9%

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

       1. +-commutative 44.9%

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

       2. mul-1-neg 44.9%

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

       3. unsub-neg 44.9%

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

       4. mul-1-neg 44.9%

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

   15. Simplified 44.9%

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

   if -8.5e20 < y < 2.5499999999999999e45

   1. Initial program 81.7%

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

      1. distribute-lft-out 81.7%

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

   3. Simplified 81.7%

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

   if 2.5499999999999999e45 < y

   1. Initial program 47.4%

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

      1. distribute-lft-out 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in x around 0 23.3%

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

      1. sqrt-prod 45.7%

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

   6. Applied egg-rr 45.7%

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

      1. *-commutative 45.7%

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

   8. Simplified 45.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y + x, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.55e+45)
   (* 2.0 (sqrt (fma z (+ y x) (* y x))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.55e+45) {
		tmp = 2.0 * sqrt(fma(z, (y + x), (y * x)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.55e+45)
		tmp = Float64(2.0 * sqrt(fma(z, Float64(y + x), Float64(y * x))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2.55e+45], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision] + N[(y * x), $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 < 2.5499999999999999e45

   1. Initial program 73.8%

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

      1. associate-+l+ 73.8%

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

      2. +-commutative 73.8%

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

      3. distribute-rgt-out 73.9%

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

      4. fma-def 74.0%

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

   3. Simplified 74.0%

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

   if 2.5499999999999999e45 < y

   1. Initial program 47.4%

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

      1. distribute-lft-out 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in x around 0 23.3%

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

      1. sqrt-prod 45.7%

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

   6. Applied egg-rr 45.7%

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

      1. *-commutative 45.7%

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

   8. Simplified 45.7%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y + x, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 5: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.55e+45)
   (* 2.0 (sqrt (+ (* x (+ y z)) (* y z))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.55e+45) {
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.55d+45) then
        tmp = 2.0d0 * sqrt(((x * (y + z)) + (y * z)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.55e+45) {
		tmp = 2.0 * Math.sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.55e+45:
		tmp = 2.0 * math.sqrt(((x * (y + z)) + (y * z)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.55e+45)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * Float64(y + z)) + Float64(y * z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.55e+45)
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2.55e+45], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5499999999999999e45

   1. Initial program 73.8%

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

      1. distribute-lft-out 73.9%

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

   3. Simplified 73.9%

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

   if 2.5499999999999999e45 < y

   1. Initial program 47.4%

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

      1. distribute-lft-out 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in x around 0 23.3%

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

      1. sqrt-prod 45.7%

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

   6. Applied egg-rr 45.7%

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

      1. *-commutative 45.7%

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

   8. Simplified 45.7%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 6: 70.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

   1. distribute-lft-out 68.1%

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

3. Simplified 68.1%

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

4. Final simplification 68.1%

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

Alternative 7: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5.2e-275) (* 2.0 (sqrt (* y 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 <= 5.2e-275) {
		tmp = 2.0 * sqrt((y * 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 <= 5.2d-275) then
        tmp = 2.0d0 * sqrt((y * 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 <= 5.2e-275) {
		tmp = 2.0 * Math.sqrt((y * 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 <= 5.2e-275:
		tmp = 2.0 * math.sqrt((y * 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 <= 5.2e-275)
		tmp = Float64(2.0 * sqrt(Float64(y * 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 <= 5.2e-275)
		tmp = 2.0 * sqrt((y * 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, 5.2e-275], N[(2.0 * N[Sqrt[N[(y * 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 < 5.19999999999999985e-275

   1. Initial program 65.6%

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

      1. distribute-lft-out 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in z around 0 19.8%

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

   if 5.19999999999999985e-275 < 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. distribute-lft-out 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in z around inf 49.0%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 8: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5.2e-275)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-275) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5.2d-275) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-275) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 5.2e-275:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e-275)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.2e-275)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 5.2e-275], 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 < 5.19999999999999985e-275

   1. Initial program 65.6%

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

      1. distribute-lft-out 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in x around inf 38.2%

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

   if 5.19999999999999985e-275 < 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. distribute-lft-out 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in z around inf 49.0%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 9: 68.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5.2e-275) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-275) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5.2d-275) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-275) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 5.2e-275:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e-275)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.2e-275)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 5.2e-275], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.19999999999999985e-275

   1. Initial program 65.6%

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

      1. distribute-lft-out 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in z around 0 19.8%

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

   if 5.19999999999999985e-275 < 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. distribute-lft-out 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in x around 0 24.7%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 10: 35.6% 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 68.1%

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

   1. distribute-lft-out 68.1%

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

3. Simplified 68.1%

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

4. Taylor expanded in z around 0 21.8%

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

5. Final simplification 21.8%

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

Developer target: 83.7% 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 2023213 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

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

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