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

Percentage Accurate: 70.2% → 98.0%

Time: 12.2s

Alternatives: 10

Speedup: 1.2×

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: 70.2% 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: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{-290}:\\ \;\;\;\;y \cdot \left(2 \cdot \frac{\sqrt{\left(-z\right) - x}}{-\sqrt{-y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.45e-290)
   (* y (* 2.0 (/ (sqrt (- (- z) x)) (- (sqrt (- y))))))
   (* (* 2.0 (sqrt z)) (sqrt (+ y x)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.45e-290) {
		tmp = y * (2.0 * (sqrt((-z - x)) / -sqrt(-y)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((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 <= (-2.45d-290)) then
        tmp = y * (2.0d0 * (sqrt((-z - x)) / -sqrt(-y)))
    else
        tmp = (2.0d0 * sqrt(z)) * sqrt((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 <= -2.45e-290) {
		tmp = y * (2.0 * (Math.sqrt((-z - x)) / -Math.sqrt(-y)));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.sqrt((y + x));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2.45e-290:
		tmp = y * (2.0 * (math.sqrt((-z - x)) / -math.sqrt(-y)))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.sqrt((y + x))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.45e-290)
		tmp = Float64(y * Float64(2.0 * Float64(sqrt(Float64(Float64(-z) - x)) / Float64(-sqrt(Float64(-y))))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(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 <= -2.45e-290)
		tmp = y * (2.0 * (sqrt((-z - x)) / -sqrt(-y)));
	else
		tmp = (2.0 * sqrt(z)) * sqrt((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, -2.45e-290], N[(y * N[(2.0 * N[(N[Sqrt[N[((-z) - x), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[(-y)], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.45e-290

   1. Initial program 74.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 58.9

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

   8. Applied rewrites 58.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. sqrt-div N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-neg.f64 70.8

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

   10. Applied rewrites 70.8%

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

   if -2.45e-290 < y

   1. Initial program 68.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 43.6

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

   5. Applied rewrites 43.6%

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

      1. lift-+.f64 N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 47.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 47.8

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

   7. Applied rewrites 47.8%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-292}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e+25)
   (* y (* -2.0 (sqrt (/ x y))))
   (if (<= y 6.5e-292)
     (* 2.0 (sqrt (fma y (+ z x) (* z x))))
     (* (* 2.0 (sqrt z)) (sqrt (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+25) {
		tmp = y * (-2.0 * sqrt((x / y)));
	} else if (y <= 6.5e-292) {
		tmp = 2.0 * sqrt(fma(y, (z + x), (z * x)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((y + x));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e+25)
		tmp = Float64(y * Float64(-2.0 * sqrt(Float64(x / y))));
	elseif (y <= 6.5e-292)
		tmp = Float64(2.0 * sqrt(fma(y, Float64(z + x), Float64(z * x))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(Float64(y + x)));
	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, -3.1e+25], N[(y * N[(-2.0 * N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e-292], N[(2.0 * N[Sqrt[N[(y * N[(z + x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0999999999999998e25

   1. Initial program 62.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 1.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 80.4

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 46.3

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

   11. Applied rewrites 46.3%

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

   if -3.0999999999999998e25 < y < 6.4999999999999997e-292

   1. Initial program 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 81.2

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

   4. Applied rewrites 81.2%

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

   if 6.4999999999999997e-292 < y

   1. Initial program 68.7%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 N/A

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

      2. lower-+.f64 42.9

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

   5. Applied rewrites 42.9%

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

      1. lift-+.f64 N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 48.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 48.1

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

   7. Applied rewrites 48.1%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-292}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e+25)
   (* y (* -2.0 (sqrt (/ x y))))
   (if (<= y 1.5e-266)
     (* 2.0 (sqrt (fma y (+ z x) (* z x))))
     (* (* 2.0 (sqrt z)) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+25) {
		tmp = y * (-2.0 * sqrt((x / y)));
	} else if (y <= 1.5e-266) {
		tmp = 2.0 * sqrt(fma(y, (z + x), (z * x)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt(y);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e+25)
		tmp = Float64(y * Float64(-2.0 * sqrt(Float64(x / y))));
	elseif (y <= 1.5e-266)
		tmp = Float64(2.0 * sqrt(fma(y, Float64(z + x), Float64(z * x))));
	else
		tmp = Float64(Float64(2.0 * 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, -3.1e+25], N[(y * N[(-2.0 * N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e-266], N[(2.0 * N[Sqrt[N[(y * N[(z + x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0999999999999998e25

   1. Initial program 62.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 1.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 80.4

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 46.3

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

   11. Applied rewrites 46.3%

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

   if -3.0999999999999998e25 < y < 1.5e-266

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 82.2

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

   4. Applied rewrites 82.2%

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

   if 1.5e-266 < y

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 67.3

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

   4. Applied rewrites 67.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 26.9

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

   7. Applied rewrites 26.9%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 39.6

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

   9. Applied rewrites 39.6%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \left(-2 \cdot \sqrt{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e+25)
   (* x (* -2.0 (sqrt (/ y x))))
   (if (<= y 1.5e-266)
     (* 2.0 (sqrt (fma y (+ z x) (* z x))))
     (* (* 2.0 (sqrt z)) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+25) {
		tmp = x * (-2.0 * sqrt((y / x)));
	} else if (y <= 1.5e-266) {
		tmp = 2.0 * sqrt(fma(y, (z + x), (z * x)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt(y);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e+25)
		tmp = Float64(x * Float64(-2.0 * sqrt(Float64(y / x))));
	elseif (y <= 1.5e-266)
		tmp = Float64(2.0 * sqrt(fma(y, Float64(z + x), Float64(z * x))));
	else
		tmp = Float64(Float64(2.0 * 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, -3.1e+25], N[(x * N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e-266], N[(2.0 * N[Sqrt[N[(y * N[(z + x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0999999999999998e25

   1. Initial program 62.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 1.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 80.4

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. cube-mult N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 40.0

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

   11. Applied rewrites 40.0%

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

   12. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 45.2

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

   14. Applied rewrites 45.2%

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

   if -3.0999999999999998e25 < y < 1.5e-266

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 82.2

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

   4. Applied rewrites 82.2%

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

   if 1.5e-266 < y

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 67.3

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

   4. Applied rewrites 67.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 26.9

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

   7. Applied rewrites 26.9%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 39.6

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

   9. Applied rewrites 39.6%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \left(-2 \cdot \sqrt{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-292}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e-292)
   (* 2.0 (sqrt (* x (+ y z))))
   (* (* 2.0 (sqrt z)) (sqrt y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e-292) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt(y);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6.5d-292) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = (2.0d0 * sqrt(z)) * sqrt(y)
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e-292) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.sqrt(y);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 6.5e-292:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.sqrt(y)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e-292)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(Float64(2.0 * 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 <= 6.5e-292)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = (2.0 * sqrt(z)) * sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 6.5e-292], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4999999999999997e-292

   1. Initial program 74.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 N/A

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

      2. lower-+.f64 53.3

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

   5. Applied rewrites 53.3%

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

   if 6.4999999999999997e-292 < y

   1. Initial program 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 25.4

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

   7. Applied rewrites 25.4%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 37.2

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

   9. Applied rewrites 37.2%

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

Alternative 6: 70.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-290}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e-290)
   (* 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 <= -2.7e-290) {
		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 <= (-2.7d-290)) 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 <= -2.7e-290) {
		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 <= -2.7e-290:
		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 <= -2.7e-290)
		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 <= -2.7e-290)
		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, -2.7e-290], 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 < -2.69999999999999999e-290

   1. Initial program 74.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 N/A

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

      2. lower-+.f64 52.9

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

   5. Applied rewrites 52.9%

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

   if -2.69999999999999999e-290 < y

   1. Initial program 68.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 43.6

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

   5. Applied rewrites 43.6%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-290}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-290}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.6e-290) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.6e-290) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} 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 <= (-2.6d-290)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    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 <= -2.6e-290) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} 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 <= -2.6e-290:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	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 <= -2.6e-290)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	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 <= -2.6e-290)
		tmp = 2.0 * sqrt((x * (y + z)));
	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, -2.6e-290], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.60000000000000001e-290

   1. Initial program 74.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 N/A

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

      2. lower-+.f64 52.9

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

   5. Applied rewrites 52.9%

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

   if -2.60000000000000001e-290 < y

   1. Initial program 68.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 24.6

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

   5. Applied rewrites 24.6%

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

Alternative 8: 70.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(2.0 * sqrt(fma(y, Float64(z + x), Float64(z * 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 * N[(z + x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. associate-+r+ N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-+.f64 71.4

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

4. Applied rewrites 71.4%

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

5. Final simplification 71.4%

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

Alternative 9: 68.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-290}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.6e-290) (* 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 <= -2.6e-290) {
		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 <= (-2.6d-290)) 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 <= -2.6e-290) {
		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 <= -2.6e-290:
		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 <= -2.6e-290)
		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 <= -2.6e-290)
		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, -2.6e-290], 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 < -2.60000000000000001e-290

   1. Initial program 74.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 32.5

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

   5. Applied rewrites 32.5%

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

   if -2.60000000000000001e-290 < y

   1. Initial program 68.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 24.6

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

   5. Applied rewrites 24.6%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-290}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.7% accurate, 1.8× 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 71.3%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
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. lower-*.f64 29.6

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

5. Applied rewrites 29.6%

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

6. Final simplification 29.6%

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

Developer Target 1: 83.1% 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 2024221 
(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)))))