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

Percentage Accurate: 69.6% → 94.0%

Time: 10.3s

Alternatives: 11

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.6% 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: 94.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+15}:\\ \;\;\;\;{\left({\left(e^{0.25}\right)}^{\left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} \cdot 2\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x + y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e+15)
   (* (pow (pow (exp 0.25) (- (log (- (- z) y)) (log (/ -1.0 x)))) 2.0) 2.0)
   (if (<= y 2.35e-269)
     (* (sqrt (* (+ z y) x)) 2.0)
     (* (/ (* (sqrt (+ x y)) 2.0) (sqrt z)) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+15) {
		tmp = pow(pow(exp(0.25), (log((-z - y)) - log((-1.0 / x)))), 2.0) * 2.0;
	} else if (y <= 2.35e-269) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = ((sqrt((x + y)) * 2.0) / sqrt(z)) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.2d+15)) then
        tmp = ((exp(0.25d0) ** (log((-z - y)) - log(((-1.0d0) / x)))) ** 2.0d0) * 2.0d0
    else if (y <= 2.35d-269) then
        tmp = sqrt(((z + y) * x)) * 2.0d0
    else
        tmp = ((sqrt((x + y)) * 2.0d0) / sqrt(z)) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+15) {
		tmp = Math.pow(Math.pow(Math.exp(0.25), (Math.log((-z - y)) - Math.log((-1.0 / x)))), 2.0) * 2.0;
	} else if (y <= 2.35e-269) {
		tmp = Math.sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = ((Math.sqrt((x + y)) * 2.0) / Math.sqrt(z)) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -3.2e+15:
		tmp = math.pow(math.pow(math.exp(0.25), (math.log((-z - y)) - math.log((-1.0 / x)))), 2.0) * 2.0
	elif y <= 2.35e-269:
		tmp = math.sqrt(((z + y) * x)) * 2.0
	else:
		tmp = ((math.sqrt((x + y)) * 2.0) / math.sqrt(z)) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e+15)
		tmp = Float64(((exp(0.25) ^ Float64(log(Float64(Float64(-z) - y)) - log(Float64(-1.0 / x)))) ^ 2.0) * 2.0);
	elseif (y <= 2.35e-269)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(x + y)) * 2.0) / sqrt(z)) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.2e+15)
		tmp = ((exp(0.25) ^ (log((-z - y)) - log((-1.0 / x)))) ^ 2.0) * 2.0;
	elseif (y <= 2.35e-269)
		tmp = sqrt(((z + y) * x)) * 2.0;
	else
		tmp = ((sqrt((x + y)) * 2.0) / sqrt(z)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -3.2e+15], N[(N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[((-z) - y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[y, 2.35e-269], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(x + y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2e15

   1. Initial program 53.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in x around -inf

      \[\leadsto 2 \cdot {\color{blue}{\left(e^{\frac{1}{4} \cdot \left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}}^{2} \]
   6. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}}^{2} \]

      2. lower-pow.f64 N/A

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}}^{2} \]

      3. lower-exp.f64 N/A

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(e^{\frac{1}{4}}\right)}}^{\left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} \]

      4. mul-1-neg N/A

         \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \left(-1 \cdot y + -1 \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{-1}{x}\right)\right)\right)}\right)}\right)}^{2} \]

      5. unsub-neg N/A

         \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\color{blue}{\left(\log \left(-1 \cdot y + -1 \cdot z\right) - \log \left(\frac{-1}{x}\right)\right)}}\right)}^{2} \]

      6. lower--.f64 N/A

         \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\color{blue}{\left(\log \left(-1 \cdot y + -1 \cdot z\right) - \log \left(\frac{-1}{x}\right)\right)}}\right)}^{2} \]

      7. lower-log.f64 N/A

         \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\color{blue}{\log \left(-1 \cdot y + -1 \cdot z\right)} - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \color{blue}{\left(-1 \cdot z + -1 \cdot y\right)} - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} \]

      9. mul-1-neg N/A

         \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} \]

      10. unsub-neg N/A

          \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \color{blue}{\left(-1 \cdot z - y\right)} - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} \]

      11. lower--.f64 N/A

          \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \color{blue}{\left(-1 \cdot z - y\right)} - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} \]

      12. mul-1-neg N/A

          \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} \]

      13. lower-neg.f64 N/A

          \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \left(\color{blue}{\left(-z\right)} - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} \]

      14. lower-log.f64 N/A

          \[\leadsto 2 \cdot {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \left(\left(-z\right) - y\right) - \color{blue}{\log \left(\frac{-1}{x}\right)}\right)}\right)}^{2} \]

      15. lower-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   if -3.2e15 < y < 2.3499999999999999e-269

   1. Initial program 84.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if 2.3499999999999999e-269 < y

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 51.4%

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

   10. Applied rewrites 53.7%

       \[\leadsto \frac{\sqrt{x + y} \cdot 2}{\sqrt{z}} \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+15}:\\ \;\;\;\;{\left({\left(e^{0.25}\right)}^{\left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} \cdot 2\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x + y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-268}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{{y}^{-1}}{x}}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x + y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e-268)
   (* (/ 1.0 (sqrt (/ (pow y -1.0) x))) 2.0)
   (* (/ (* (sqrt (+ x y)) 2.0) (sqrt z)) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e-268) {
		tmp = (1.0 / sqrt((pow(y, -1.0) / x))) * 2.0;
	} else {
		tmp = ((sqrt((x + y)) * 2.0) / sqrt(z)) * 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 <= (-2.7d-268)) then
        tmp = (1.0d0 / sqrt(((y ** (-1.0d0)) / x))) * 2.0d0
    else
        tmp = ((sqrt((x + y)) * 2.0d0) / sqrt(z)) * 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 <= -2.7e-268) {
		tmp = (1.0 / Math.sqrt((Math.pow(y, -1.0) / x))) * 2.0;
	} else {
		tmp = ((Math.sqrt((x + y)) * 2.0) / Math.sqrt(z)) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2.7e-268:
		tmp = (1.0 / math.sqrt((math.pow(y, -1.0) / x))) * 2.0
	else:
		tmp = ((math.sqrt((x + y)) * 2.0) / math.sqrt(z)) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.7e-268)
		tmp = Float64(Float64(1.0 / sqrt(Float64((y ^ -1.0) / x))) * 2.0);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(x + y)) * 2.0) / sqrt(z)) * 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 <= -2.7e-268)
		tmp = (1.0 / sqrt(((y ^ -1.0) / x))) * 2.0;
	else
		tmp = ((sqrt((x + y)) * 2.0) / sqrt(z)) * 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, -2.7e-268], N[(N[(1.0 / N[Sqrt[N[(N[Power[y, -1.0], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(x + y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7000000000000001e-268

   1. Initial program 72.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \frac{1}{\sqrt{\color{blue}{\frac{1}{x \cdot y}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{1}{\sqrt{\color{blue}{\frac{1}{x \cdot y}}}} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \frac{1}{\sqrt{\frac{1}{\color{blue}{y \cdot x}}}} \]

      3. lower-*.f64 24.1

         \[\leadsto 2 \cdot \frac{1}{\sqrt{\frac{1}{\color{blue}{y \cdot x}}}} \]

   7. Applied rewrites 24.1%

      \[\leadsto 2 \cdot \frac{1}{\sqrt{\color{blue}{\frac{1}{y \cdot x}}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 25.4%

      \[\leadsto 2 \cdot \frac{1}{\sqrt{\frac{{y}^{-1}}{\color{blue}{x}}}} \]

   if -2.7000000000000001e-268 < y

   1. Initial program 70.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 53.1%

       \[\leadsto \frac{\sqrt{x + y} \cdot 2}{\sqrt{z}} \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-268}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{{y}^{-1}}{x}}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x + y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x + y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.35e-269)
   (* (sqrt (* (+ z y) x)) 2.0)
   (* (/ (* (sqrt (+ x y)) 2.0) (sqrt z)) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.35e-269) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = ((sqrt((x + y)) * 2.0) / sqrt(z)) * 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 <= 2.35d-269) then
        tmp = sqrt(((z + y) * x)) * 2.0d0
    else
        tmp = ((sqrt((x + y)) * 2.0d0) / sqrt(z)) * 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 <= 2.35e-269) {
		tmp = Math.sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = ((Math.sqrt((x + y)) * 2.0) / Math.sqrt(z)) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.35e-269:
		tmp = math.sqrt(((z + y) * x)) * 2.0
	else:
		tmp = ((math.sqrt((x + y)) * 2.0) / math.sqrt(z)) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.35e-269)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(x + y)) * 2.0) / sqrt(z)) * 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 <= 2.35e-269)
		tmp = sqrt(((z + y) * x)) * 2.0;
	else
		tmp = ((sqrt((x + y)) * 2.0) / sqrt(z)) * 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, 2.35e-269], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(x + y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3499999999999999e-269

   1. Initial program 72.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 48.7

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

   5. Applied rewrites 48.7%

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

   if 2.3499999999999999e-269 < y

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 51.4%

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

   10. Applied rewrites 53.7%

       \[\leadsto \frac{\sqrt{x + y} \cdot 2}{\sqrt{z}} \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x + y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e-269)
   (* (sqrt (* (+ z y) x)) 2.0)
   (* (/ (* (sqrt y) 2.0) (sqrt z)) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e-269) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = ((sqrt(y) * 2.0) / sqrt(z)) * 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 <= 2.4d-269) then
        tmp = sqrt(((z + y) * x)) * 2.0d0
    else
        tmp = ((sqrt(y) * 2.0d0) / sqrt(z)) * 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 <= 2.4e-269) {
		tmp = Math.sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = ((Math.sqrt(y) * 2.0) / Math.sqrt(z)) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.4e-269:
		tmp = math.sqrt(((z + y) * x)) * 2.0
	else:
		tmp = ((math.sqrt(y) * 2.0) / math.sqrt(z)) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e-269)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	else
		tmp = Float64(Float64(Float64(sqrt(y) * 2.0) / sqrt(z)) * 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 <= 2.4e-269)
		tmp = sqrt(((z + y) * x)) * 2.0;
	else
		tmp = ((sqrt(y) * 2.0) / sqrt(z)) * 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, 2.4e-269], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4000000000000001e-269

   1. Initial program 72.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 48.7

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

   5. Applied rewrites 48.7%

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

   if 2.4000000000000001e-269 < y

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 51.4%

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

   10. Applied rewrites 53.7%

       \[\leadsto \frac{\sqrt{x + y} \cdot 2}{\sqrt{z}} \cdot z \]

   11. Taylor expanded in y around inf

       \[\leadsto \frac{\sqrt{y} \cdot 2}{\sqrt{z}} \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 37.0%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x + y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-308)
   (* (sqrt (* (+ z y) x)) 2.0)
   (if (<= y 2.3e-6)
     (* (sqrt (* (+ x y) z)) 2.0)
     (* (* (sqrt (/ (+ x y) z)) 2.0) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-308) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else if (y <= 2.3e-6) {
		tmp = sqrt(((x + y) * z)) * 2.0;
	} else {
		tmp = (sqrt(((x + y) / z)) * 2.0) * 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 <= (-2d-308)) then
        tmp = sqrt(((z + y) * x)) * 2.0d0
    else if (y <= 2.3d-6) then
        tmp = sqrt(((x + y) * z)) * 2.0d0
    else
        tmp = (sqrt(((x + y) / z)) * 2.0d0) * 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 <= -2e-308) {
		tmp = Math.sqrt(((z + y) * x)) * 2.0;
	} else if (y <= 2.3e-6) {
		tmp = Math.sqrt(((x + y) * z)) * 2.0;
	} else {
		tmp = (Math.sqrt(((x + y) / z)) * 2.0) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2e-308:
		tmp = math.sqrt(((z + y) * x)) * 2.0
	elif y <= 2.3e-6:
		tmp = math.sqrt(((x + y) * z)) * 2.0
	else:
		tmp = (math.sqrt(((x + y) / z)) * 2.0) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-308)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	elseif (y <= 2.3e-6)
		tmp = Float64(sqrt(Float64(Float64(x + y) * z)) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(x + y) / z)) * 2.0) * 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 <= -2e-308)
		tmp = sqrt(((z + y) * x)) * 2.0;
	elseif (y <= 2.3e-6)
		tmp = sqrt(((x + y) * z)) * 2.0;
	else
		tmp = (sqrt(((x + y) / z)) * 2.0) * 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, -2e-308], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[y, 2.3e-6], N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9999999999999998e-308

   1. Initial program 72.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 47.3

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

   5. Applied rewrites 47.3%

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

   if -1.9999999999999998e-308 < y < 2.3e-6

   1. Initial program 83.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if 2.3e-6 < y

   1. Initial program 51.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 54.1%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x + y}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{elif}\;y \leq 0.0075:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-308)
   (* (sqrt (* (+ z y) x)) 2.0)
   (if (<= y 0.0075)
     (* (sqrt (* (+ x y) z)) 2.0)
     (* (* (sqrt (/ y z)) 2.0) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-308) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else if (y <= 0.0075) {
		tmp = sqrt(((x + y) * z)) * 2.0;
	} else {
		tmp = (sqrt((y / z)) * 2.0) * 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 <= (-2d-308)) then
        tmp = sqrt(((z + y) * x)) * 2.0d0
    else if (y <= 0.0075d0) then
        tmp = sqrt(((x + y) * z)) * 2.0d0
    else
        tmp = (sqrt((y / z)) * 2.0d0) * 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 <= -2e-308) {
		tmp = Math.sqrt(((z + y) * x)) * 2.0;
	} else if (y <= 0.0075) {
		tmp = Math.sqrt(((x + y) * z)) * 2.0;
	} else {
		tmp = (Math.sqrt((y / z)) * 2.0) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2e-308:
		tmp = math.sqrt(((z + y) * x)) * 2.0
	elif y <= 0.0075:
		tmp = math.sqrt(((x + y) * z)) * 2.0
	else:
		tmp = (math.sqrt((y / z)) * 2.0) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-308)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	elseif (y <= 0.0075)
		tmp = Float64(sqrt(Float64(Float64(x + y) * z)) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(y / z)) * 2.0) * 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 <= -2e-308)
		tmp = sqrt(((z + y) * x)) * 2.0;
	elseif (y <= 0.0075)
		tmp = sqrt(((x + y) * z)) * 2.0;
	else
		tmp = (sqrt((y / z)) * 2.0) * 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, -2e-308], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[y, 0.0075], N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9999999999999998e-308

   1. Initial program 72.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 47.3

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

   5. Applied rewrites 47.3%

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

   if -1.9999999999999998e-308 < y < 0.0074999999999999997

   1. Initial program 83.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if 0.0074999999999999997 < y

   1. Initial program 51.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.5%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{elif}\;y \leq 0.0075:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{z \cdot y + \left(x \cdot z + x \cdot y\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x + y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.3e-6)
   (* (sqrt (+ (* z y) (+ (* x z) (* x y)))) 2.0)
   (* (* (sqrt (/ (+ x y) z)) 2.0) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e-6) {
		tmp = sqrt(((z * y) + ((x * z) + (x * y)))) * 2.0;
	} else {
		tmp = (sqrt(((x + y) / z)) * 2.0) * 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 <= 2.3d-6) then
        tmp = sqrt(((z * y) + ((x * z) + (x * y)))) * 2.0d0
    else
        tmp = (sqrt(((x + y) / z)) * 2.0d0) * 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 <= 2.3e-6) {
		tmp = Math.sqrt(((z * y) + ((x * z) + (x * y)))) * 2.0;
	} else {
		tmp = (Math.sqrt(((x + y) / z)) * 2.0) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.3e-6:
		tmp = math.sqrt(((z * y) + ((x * z) + (x * y)))) * 2.0
	else:
		tmp = (math.sqrt(((x + y) / z)) * 2.0) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.3e-6)
		tmp = Float64(sqrt(Float64(Float64(z * y) + Float64(Float64(x * z) + Float64(x * y)))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(x + y) / z)) * 2.0) * 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 <= 2.3e-6)
		tmp = sqrt(((z * y) + ((x * z) + (x * y)))) * 2.0;
	else
		tmp = (sqrt(((x + y) / z)) * 2.0) * 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, 2.3e-6], N[(N[Sqrt[N[(N[(z * y), $MachinePrecision] + N[(N[(x * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3e-6

   1. Initial program 76.4%

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

   if 2.3e-6 < y

   1. Initial program 51.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 54.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{z \cdot y + \left(x \cdot z + x \cdot y\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x + y}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \end{array} \end{array} \]

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-268)
   (* (sqrt (* (+ z y) x)) 2.0)
   (* (sqrt (* (+ x y) z)) 2.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e-268) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = sqrt(((x + y) * z)) * 2.0;
	}
	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-268)) then
        tmp = sqrt(((z + y) * x)) * 2.0d0
    else
        tmp = sqrt(((x + y) * z)) * 2.0d0
    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-268) {
		tmp = Math.sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = Math.sqrt(((x + y) * z)) * 2.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -3.4e-268:
		tmp = math.sqrt(((z + y) * x)) * 2.0
	else:
		tmp = math.sqrt(((x + y) * z)) * 2.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3.4e-268)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	else
		tmp = Float64(sqrt(Float64(Float64(x + y) * z)) * 2.0);
	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-268)
		tmp = sqrt(((z + y) * x)) * 2.0;
	else
		tmp = sqrt(((x + y) * z)) * 2.0;
	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-268], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e-268

   1. Initial program 72.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 44.9

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

   5. Applied rewrites 44.9%

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

   if -3.4e-268 < y

   1. Initial program 70.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 54.8

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

   5. Applied rewrites 54.8%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{x \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e-268) (* (sqrt (* x y)) 2.0) (* (sqrt (* (+ x y) z)) 2.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e-268) {
		tmp = sqrt((x * y)) * 2.0;
	} else {
		tmp = sqrt(((x + y) * z)) * 2.0;
	}
	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 <= (-2.7d-268)) then
        tmp = sqrt((x * y)) * 2.0d0
    else
        tmp = sqrt(((x + y) * z)) * 2.0d0
    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 <= -2.7e-268) {
		tmp = Math.sqrt((x * y)) * 2.0;
	} else {
		tmp = Math.sqrt(((x + y) * z)) * 2.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2.7e-268:
		tmp = math.sqrt((x * y)) * 2.0
	else:
		tmp = math.sqrt(((x + y) * z)) * 2.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.7e-268)
		tmp = Float64(sqrt(Float64(x * y)) * 2.0);
	else
		tmp = Float64(sqrt(Float64(Float64(x + y) * z)) * 2.0);
	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 <= -2.7e-268)
		tmp = sqrt((x * y)) * 2.0;
	else
		tmp = sqrt(((x + y) * z)) * 2.0;
	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, -2.7e-268], N[(N[Sqrt[N[(x * y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7000000000000001e-268

   1. Initial program 72.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.3

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

   5. Applied rewrites 24.3%

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

   if -2.7000000000000001e-268 < y

   1. Initial program 70.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 54.8

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

   5. Applied rewrites 54.8%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{x \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{x \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot y} \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.8e-268) (* (sqrt (* x y)) 2.0) (* (sqrt (* z y)) 2.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.8e-268) {
		tmp = sqrt((x * y)) * 2.0;
	} else {
		tmp = sqrt((z * y)) * 2.0;
	}
	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 <= (-2.8d-268)) then
        tmp = sqrt((x * y)) * 2.0d0
    else
        tmp = sqrt((z * y)) * 2.0d0
    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 <= -2.8e-268) {
		tmp = Math.sqrt((x * y)) * 2.0;
	} else {
		tmp = Math.sqrt((z * y)) * 2.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2.8e-268:
		tmp = math.sqrt((x * y)) * 2.0
	else:
		tmp = math.sqrt((z * y)) * 2.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.8e-268)
		tmp = Float64(sqrt(Float64(x * y)) * 2.0);
	else
		tmp = Float64(sqrt(Float64(z * y)) * 2.0);
	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 <= -2.8e-268)
		tmp = sqrt((x * y)) * 2.0;
	else
		tmp = sqrt((z * y)) * 2.0;
	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, -2.8e-268], N[(N[Sqrt[N[(x * y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Sqrt[N[(z * y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000015e-268

   1. Initial program 72.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.3

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

   5. Applied rewrites 24.3%

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

   if -2.80000000000000015e-268 < y

   1. Initial program 70.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.4

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

   5. Applied rewrites 24.4%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{x \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot y} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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((x * y)) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 21.3

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

5. Applied rewrites 21.3%

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

6. Final simplification 21.3%

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

Developer Target 1: 82.1% accurate, 0.0× 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 2024277 
(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)))))