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

Percentage Accurate: 70.3% → 97.6%

Time: 11.8s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.3% 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: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-272}:\\ \;\;\;\;y \cdot \left(\frac{\sqrt{\left(-z\right) - x}}{\sqrt{-y}} \cdot \left(-2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6.2e-272)
   (* y (* (/ (sqrt (- (- z) x)) (sqrt (- y))) (- 2.0)))
   (* (* 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 <= -6.2e-272) {
		tmp = y * ((sqrt((-z - x)) / sqrt(-y)) * -2.0);
	} 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 <= (-6.2d-272)) then
        tmp = y * ((sqrt((-z - x)) / sqrt(-y)) * -2.0d0)
    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 <= -6.2e-272) {
		tmp = y * ((Math.sqrt((-z - x)) / Math.sqrt(-y)) * -2.0);
	} 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 <= -6.2e-272:
		tmp = y * ((math.sqrt((-z - x)) / math.sqrt(-y)) * -2.0)
	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 <= -6.2e-272)
		tmp = Float64(y * Float64(Float64(sqrt(Float64(Float64(-z) - x)) / sqrt(Float64(-y))) * Float64(-2.0)));
	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 <= -6.2e-272)
		tmp = y * ((sqrt((-z - x)) / sqrt(-y)) * -2.0);
	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, -6.2e-272], N[(y * N[(N[(N[Sqrt[N[((-z) - x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-y)], $MachinePrecision]), $MachinePrecision] * (-2.0)), $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 < -6.20000000000000059e-272

   1. Initial program 66.8%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. Simplified 0.9%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.3

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

   8. Simplified 67.3%

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

      1. lift-+.f64 N/A

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

      2. frac-2neg N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-neg.f64 72.3

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

   10. Applied egg-rr 72.3%

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

   if -6.20000000000000059e-272 < y

   1. Initial program 72.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 53.2

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

   5. Simplified 53.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 48.0

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

   7. Applied egg-rr 48.0%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-272}:\\ \;\;\;\;y \cdot \left(\frac{\sqrt{\left(-z\right) - x}}{\sqrt{-y}} \cdot \left(-2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+64}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-194}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-271}:\\ \;\;\;\;x \cdot \left(-2 \cdot \sqrt{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+64)
   (* y (* -2.0 (sqrt (/ x y))))
   (if (<= y -2.5e-194)
     (* 2.0 (sqrt (* x (+ y z))))
     (if (<= y -3.1e-271)
       (* x (* -2.0 (sqrt (/ y 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 <= -2.1e+64) {
		tmp = y * (-2.0 * sqrt((x / y)));
	} else if (y <= -2.5e-194) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -3.1e-271) {
		tmp = x * (-2.0 * sqrt((y / x)));
	} 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 <= (-2.1d+64)) then
        tmp = y * ((-2.0d0) * sqrt((x / y)))
    else if (y <= (-2.5d-194)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-3.1d-271)) then
        tmp = x * ((-2.0d0) * sqrt((y / x)))
    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 <= -2.1e+64) {
		tmp = y * (-2.0 * Math.sqrt((x / y)));
	} else if (y <= -2.5e-194) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -3.1e-271) {
		tmp = x * (-2.0 * Math.sqrt((y / x)));
	} 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 <= -2.1e+64:
		tmp = y * (-2.0 * math.sqrt((x / y)))
	elif y <= -2.5e-194:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -3.1e-271:
		tmp = x * (-2.0 * math.sqrt((y / x)))
	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 <= -2.1e+64)
		tmp = Float64(y * Float64(-2.0 * sqrt(Float64(x / y))));
	elseif (y <= -2.5e-194)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -3.1e-271)
		tmp = Float64(x * Float64(-2.0 * sqrt(Float64(y / x))));
	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 <= -2.1e+64)
		tmp = y * (-2.0 * sqrt((x / y)));
	elseif (y <= -2.5e-194)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -3.1e-271)
		tmp = x * (-2.0 * sqrt((y / x)));
	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, -2.1e+64], N[(y * N[(-2.0 * N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.5e-194], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.1e-271], N[(x * N[(-2.0 * N[Sqrt[N[(y / 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 4 regimes

2. if y < -2.1e64

   1. Initial program 47.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. Simplified 0.8%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in z around 0

      \[\leadsto y \cdot \color{blue}{\left(-2 \cdot \sqrt{\frac{x}{y}}\right)} \]
   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 37.3

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

   11. Simplified 37.3%

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

   if -2.1e64 < y < -2.5000000000000001e-194

   1. Initial program 86.4%

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

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

      2. lower-+.f64 60.9

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

   5. Simplified 60.9%

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

   if -2.5000000000000001e-194 < y < -3.0999999999999999e-271

   1. Initial program 60.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. Simplified 0.8%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 28.6

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

   8. Simplified 28.6%

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

   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. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. *-commutative N/A

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

       5. distribute-lft-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. cube-mult N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 N/A

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

       19. lower-/.f64 1.7

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

   11. Simplified 1.7%

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

   12. Taylor expanded in z around 0

       \[\leadsto x \cdot \color{blue}{\left(-2 \cdot \sqrt{\frac{y}{x}}\right)} \]
   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 10.6

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

   14. Simplified 10.6%

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

   if -3.0999999999999999e-271 < y

   1. Initial program 72.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 53.2

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

   5. Simplified 53.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 48.0

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

   7. Applied egg-rr 48.0%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+64}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-194}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-271}:\\ \;\;\;\;x \cdot \left(-2 \cdot \sqrt{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+64}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-194}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \left(-2 \cdot \sqrt{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+64)
   (* y (* -2.0 (sqrt (/ x y))))
   (if (<= y -2.5e-194)
     (* 2.0 (sqrt (* x (+ y z))))
     (if (<= y 7.6e-307)
       (* x (* -2.0 (sqrt (/ y x))))
       (* 2.0 (* (sqrt z) (sqrt y)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+64) {
		tmp = y * (-2.0 * sqrt((x / y)));
	} else if (y <= -2.5e-194) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= 7.6e-307) {
		tmp = x * (-2.0 * sqrt((y / x)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.1d+64)) then
        tmp = y * ((-2.0d0) * sqrt((x / y)))
    else if (y <= (-2.5d-194)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= 7.6d-307) then
        tmp = x * ((-2.0d0) * sqrt((y / x)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+64) {
		tmp = y * (-2.0 * Math.sqrt((x / y)));
	} else if (y <= -2.5e-194) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= 7.6e-307) {
		tmp = x * (-2.0 * Math.sqrt((y / x)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2.1e+64:
		tmp = y * (-2.0 * math.sqrt((x / y)))
	elif y <= -2.5e-194:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= 7.6e-307:
		tmp = x * (-2.0 * math.sqrt((y / x)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e+64)
		tmp = Float64(y * Float64(-2.0 * sqrt(Float64(x / y))));
	elseif (y <= -2.5e-194)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= 7.6e-307)
		tmp = Float64(x * Float64(-2.0 * sqrt(Float64(y / x))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.1e+64)
		tmp = y * (-2.0 * sqrt((x / y)));
	elseif (y <= -2.5e-194)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= 7.6e-307)
		tmp = x * (-2.0 * sqrt((y / x)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -2.1e+64], N[(y * N[(-2.0 * N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.5e-194], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.6e-307], N[(x * N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.1e64

   1. Initial program 47.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. Simplified 0.8%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 88.1

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

   8. Simplified 88.1%

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

   9. Taylor expanded in z around 0

      \[\leadsto y \cdot \color{blue}{\left(-2 \cdot \sqrt{\frac{x}{y}}\right)} \]
   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 37.3

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

   11. Simplified 37.3%

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

   if -2.1e64 < y < -2.5000000000000001e-194

   1. Initial program 86.4%

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

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

      2. lower-+.f64 60.9

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

   5. Simplified 60.9%

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

   if -2.5000000000000001e-194 < y < 7.59999999999999971e-307

   1. Initial program 64.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. Simplified 0.8%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 23.6

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

   8. Simplified 23.6%

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

   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. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. *-commutative N/A

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

       5. distribute-lft-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. cube-mult N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 N/A

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

       19. lower-/.f64 1.7

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

   11. Simplified 1.7%

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

   12. Taylor expanded in z around 0

       \[\leadsto x \cdot \color{blue}{\left(-2 \cdot \sqrt{\frac{y}{x}}\right)} \]
   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 9.1

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

   14. Simplified 9.1%

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

   if 7.59999999999999971e-307 < y

   1. Initial program 71.6%

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

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. pow1/2 N/A

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

      7. sqr-pow N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

   4. Applied egg-rr 71.4%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 23.0

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

   7. Simplified 23.0%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 31.9

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

   9. Applied egg-rr 31.9%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+64}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-194}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \left(-2 \cdot \sqrt{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := x \cdot \left(-2 \cdot \sqrt{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-194}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-307}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* -2.0 (sqrt (/ y x))))))
   (if (<= y -1.32e+72)
     t_0
     (if (<= y -2.5e-194)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y 7.6e-307) t_0 (* 2.0 (* (sqrt z) (sqrt y))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = x * (-2.0 * sqrt((y / x)));
	double tmp;
	if (y <= -1.32e+72) {
		tmp = t_0;
	} else if (y <= -2.5e-194) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= 7.6e-307) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-2.0d0) * sqrt((y / x)))
    if (y <= (-1.32d+72)) then
        tmp = t_0
    else if (y <= (-2.5d-194)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= 7.6d-307) then
        tmp = t_0
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = x * (-2.0 * Math.sqrt((y / x)));
	double tmp;
	if (y <= -1.32e+72) {
		tmp = t_0;
	} else if (y <= -2.5e-194) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= 7.6e-307) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = x * (-2.0 * math.sqrt((y / x)))
	tmp = 0
	if y <= -1.32e+72:
		tmp = t_0
	elif y <= -2.5e-194:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= 7.6e-307:
		tmp = t_0
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(x * Float64(-2.0 * sqrt(Float64(y / x))))
	tmp = 0.0
	if (y <= -1.32e+72)
		tmp = t_0;
	elseif (y <= -2.5e-194)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= 7.6e-307)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = x * (-2.0 * sqrt((y / x)));
	tmp = 0.0;
	if (y <= -1.32e+72)
		tmp = t_0;
	elseif (y <= -2.5e-194)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= 7.6e-307)
		tmp = t_0;
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(-2.0 * N[Sqrt[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.32e+72], t$95$0, If[LessEqual[y, -2.5e-194], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.6e-307], t$95$0, N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3199999999999999e72 or -2.5000000000000001e-194 < y < 7.59999999999999971e-307

   1. Initial program 53.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. Simplified 0.8%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 63.2

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

   8. Simplified 63.2%

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

   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. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. *-commutative N/A

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

       5. distribute-lft-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. cube-mult N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 N/A

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

       19. lower-/.f64 21.8

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

   11. Simplified 21.8%

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

   12. Taylor expanded in z around 0

       \[\leadsto x \cdot \color{blue}{\left(-2 \cdot \sqrt{\frac{y}{x}}\right)} \]
   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 26.7

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

   14. Simplified 26.7%

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

   if -1.3199999999999999e72 < y < -2.5000000000000001e-194

   1. Initial program 86.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 59.8

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

   5. Simplified 59.8%

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

   if 7.59999999999999971e-307 < y

   1. Initial program 71.6%

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

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. pow1/2 N/A

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

      7. sqr-pow N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

   4. Applied egg-rr 71.4%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 23.0

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

   7. Simplified 23.0%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 31.9

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

   9. Applied egg-rr 31.9%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+72}:\\ \;\;\;\;x \cdot \left(-2 \cdot \sqrt{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-194}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \left(-2 \cdot \sqrt{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-299}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e-299)
   (* 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-299) {
		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-299) 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-299) {
		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-299:
		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-299)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.5e-299)
		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-299], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4999999999999997e-299

   1. Initial program 67.4%

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

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

      2. lower-+.f64 47.0

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

   5. Simplified 47.0%

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

   if 6.4999999999999997e-299 < y

   1. Initial program 71.7%

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

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. pow1/2 N/A

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

      7. sqr-pow N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

   4. Applied egg-rr 71.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 23.5

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

   7. Simplified 23.5%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 32.7

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

   9. Applied egg-rr 32.7%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-299}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6.4e-268)
   (* 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 <= -6.4e-268) {
		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 <= (-6.4d-268)) 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 <= -6.4e-268) {
		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 <= -6.4e-268:
		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 <= -6.4e-268)
		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 <= -6.4e-268)
		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, -6.4e-268], 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 < -6.3999999999999997e-268

   1. Initial program 66.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 44.6

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

   5. Simplified 44.6%

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

   if -6.3999999999999997e-268 < y

   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 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 53.5

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

   5. Simplified 53.5%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6.4e-268) (* 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 <= -6.4e-268) {
		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 <= (-6.4d-268)) 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 <= -6.4e-268) {
		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 <= -6.4e-268:
		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 <= -6.4e-268)
		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 <= -6.4e-268)
		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, -6.4e-268], 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 < -6.3999999999999997e-268

   1. Initial program 66.6%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 44.6

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

   5. Simplified 44.6%

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

   if -6.3999999999999997e-268 < y

   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 0

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

      1. lower-*.f64 22.0

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

   5. Simplified 22.0%

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

Alternative 8: 68.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e-271) (* 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 <= -3.2e-271) {
		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 <= (-3.2d-271)) 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 <= -3.2e-271) {
		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 <= -3.2e-271:
		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 <= -3.2e-271)
		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 <= -3.2e-271)
		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, -3.2e-271], 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 < -3.19999999999999978e-271

   1. Initial program 66.8%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 27.1

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

   5. Simplified 27.1%

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

   if -3.19999999999999978e-271 < y

   1. Initial program 72.0%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 22.2

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

   5. Simplified 22.2%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.4% 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 69.4%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
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. lower-*.f64 24.3

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

5. Simplified 24.3%

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

6. Final simplification 24.3%

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

Developer Target 1: 82.9% 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 2024208 
(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)))))