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

Percentage Accurate: 70.7% → 89.2%

Time: 12.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.7% 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: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+47}:\\ \;\;\;\;2 \cdot e^{0.25 \cdot \left(\log \left(z \cdot z + y \cdot \left(y + 2 \cdot z\right)\right) + -2 \cdot \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-257}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot \left(\left(z + x\right) + \frac{z \cdot x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot {\left(y + x\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3e+47)
   (*
    2.0
    (exp
     (*
      0.25
      (+ (log (+ (* z z) (* y (+ y (* 2.0 z))))) (* -2.0 (log (/ -1.0 x)))))))
   (if (<= y 6.7e-257)
     (* 2.0 (sqrt (* y (+ (+ z x) (/ (* z x) y)))))
     (* (* 2.0 (sqrt z)) (pow (+ y x) 0.5)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e+47) {
		tmp = 2.0 * exp((0.25 * (log(((z * z) + (y * (y + (2.0 * z))))) + (-2.0 * log((-1.0 / x))))));
	} else if (y <= 6.7e-257) {
		tmp = 2.0 * sqrt((y * ((z + x) + ((z * x) / y))));
	} else {
		tmp = (2.0 * sqrt(z)) * pow((y + x), 0.5);
	}
	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 <= (-3d+47)) then
        tmp = 2.0d0 * exp((0.25d0 * (log(((z * z) + (y * (y + (2.0d0 * z))))) + ((-2.0d0) * log(((-1.0d0) / x))))))
    else if (y <= 6.7d-257) then
        tmp = 2.0d0 * sqrt((y * ((z + x) + ((z * x) / y))))
    else
        tmp = (2.0d0 * sqrt(z)) * ((y + x) ** 0.5d0)
    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 <= -3e+47) {
		tmp = 2.0 * Math.exp((0.25 * (Math.log(((z * z) + (y * (y + (2.0 * z))))) + (-2.0 * Math.log((-1.0 / x))))));
	} else if (y <= 6.7e-257) {
		tmp = 2.0 * Math.sqrt((y * ((z + x) + ((z * x) / y))));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.pow((y + x), 0.5);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -3e+47:
		tmp = 2.0 * math.exp((0.25 * (math.log(((z * z) + (y * (y + (2.0 * z))))) + (-2.0 * math.log((-1.0 / x))))))
	elif y <= 6.7e-257:
		tmp = 2.0 * math.sqrt((y * ((z + x) + ((z * x) / y))))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.pow((y + x), 0.5)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3e+47)
		tmp = Float64(2.0 * exp(Float64(0.25 * Float64(log(Float64(Float64(z * z) + Float64(y * Float64(y + Float64(2.0 * z))))) + Float64(-2.0 * log(Float64(-1.0 / x)))))));
	elseif (y <= 6.7e-257)
		tmp = Float64(2.0 * sqrt(Float64(y * Float64(Float64(z + x) + Float64(Float64(z * x) / y)))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * (Float64(y + x) ^ 0.5));
	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 <= -3e+47)
		tmp = 2.0 * exp((0.25 * (log(((z * z) + (y * (y + (2.0 * z))))) + (-2.0 * log((-1.0 / x))))));
	elseif (y <= 6.7e-257)
		tmp = 2.0 * sqrt((y * ((z + x) + ((z * x) / y))));
	else
		tmp = (2.0 * sqrt(z)) * ((y + x) ^ 0.5);
	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, -3e+47], N[(2.0 * N[Exp[N[(0.25 * N[(N[Log[N[(N[(z * z), $MachinePrecision] + N[(y * N[(y + N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.7e-257], N[(2.0 * N[Sqrt[N[(y * N[(N[(z + x), $MachinePrecision] + N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Power[N[(y + x), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0000000000000001e47

   1. Initial program 46.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 46.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 46.9%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \color{blue}{{\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\left(x \cdot y + z \cdot \left(x + y\right)\right) \cdot \left(x \cdot y + z \cdot \left(x + y\right)\right)\right)}^{\color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}}\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\left(x \cdot y + z \cdot \left(x + y\right)\right) \cdot \left(x \cdot y + z \cdot \left(x + y\right)\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x \cdot y + z \cdot \left(x + y\right)\right), \left(x \cdot y + z \cdot \left(x + y\right)\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(z \cdot \left(x + y\right)\right)\right), \left(x \cdot y + z \cdot \left(x + y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right), \left(x \cdot y + z \cdot \left(x + y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right), \left(x \cdot y + z \cdot \left(x + y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x \cdot y + z \cdot \left(x + y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x \cdot y\right), \left(z \cdot \left(x + y\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right) \]

      14. metadata-eval 25.9%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

   6. Applied egg-rr 25.9%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\color{blue}{\left(y \cdot \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right) + y \cdot {\left(x + z\right)}^{2}\right) + {x}^{2} \cdot {z}^{2}\right)}, \frac{1}{4}\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left({x}^{2} \cdot {z}^{2} + y \cdot \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right) + y \cdot {\left(x + z\right)}^{2}\right)\right), \frac{1}{4}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left({x}^{2} \cdot {z}^{2}\right), \left(y \cdot \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right) + y \cdot {\left(x + z\right)}^{2}\right)\right)\right), \frac{1}{4}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left({z}^{2} \cdot {x}^{2}\right), \left(y \cdot \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right) + y \cdot {\left(x + z\right)}^{2}\right)\right)\right), \frac{1}{4}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({z}^{2}\right), \left({x}^{2}\right)\right), \left(y \cdot \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right) + y \cdot {\left(x + z\right)}^{2}\right)\right)\right), \frac{1}{4}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(z \cdot z\right), \left({x}^{2}\right)\right), \left(y \cdot \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right) + y \cdot {\left(x + z\right)}^{2}\right)\right)\right), \frac{1}{4}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left({x}^{2}\right)\right), \left(y \cdot \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right) + y \cdot {\left(x + z\right)}^{2}\right)\right)\right), \frac{1}{4}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), \left(y \cdot \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right) + y \cdot {\left(x + z\right)}^{2}\right)\right)\right), \frac{1}{4}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \left(y \cdot \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right) + y \cdot {\left(x + z\right)}^{2}\right)\right)\right), \frac{1}{4}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right) + y \cdot {\left(x + z\right)}^{2}\right)\right)\right), \frac{1}{4}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot {\left(x + z\right)}^{2} + 2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot {\left(x + z\right)}^{2}\right), \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left({\left(x + z\right)}^{2}\right)\right), \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\left(x + z\right) \cdot \left(x + z\right)\right)\right), \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x + z\right), \left(x + z\right)\right)\right), \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(z + x\right), \left(x + z\right)\right)\right), \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \left(x + z\right)\right)\right), \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \left(z + x\right)\right)\right), \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{+.f64}\left(z, x\right)\right)\right), \left(2 \cdot \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{+.f64}\left(z, x\right)\right)\right), \mathsf{*.f64}\left(2, \left(x \cdot \left(z \cdot \left(x + z\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      20. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{+.f64}\left(z, x\right)\right)\right), \mathsf{*.f64}\left(2, \left(\left(x \cdot z\right) \cdot \left(x + z\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{+.f64}\left(z, x\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot z\right), \left(x + z\right)\right)\right)\right)\right)\right), \frac{1}{4}\right)\right) \]

   9. Simplified 19.6%

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

   10. Taylor expanded in x around -inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(e^{\frac{1}{4} \cdot \left(\log \left(y \cdot \left(y + 2 \cdot z\right) + {z}^{2}\right) + -2 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}\right) \]

       2. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\left(\frac{1}{4} \cdot \left(\log \left(y \cdot \left(y + 2 \cdot z\right) + {z}^{2}\right) + -2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\log \left(y \cdot \left(y + 2 \cdot z\right) + {z}^{2}\right) + -2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\log \left(y \cdot \left(y + 2 \cdot z\right) + {z}^{2}\right), \left(-2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right) \]

       5. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(y \cdot \left(y + 2 \cdot z\right) + {z}^{2}\right)\right), \left(-2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left({z}^{2} + y \cdot \left(y + 2 \cdot z\right)\right)\right), \left(-2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left({z}^{2}\right), \left(y \cdot \left(y + 2 \cdot z\right)\right)\right)\right), \left(-2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(z \cdot z\right), \left(y \cdot \left(y + 2 \cdot z\right)\right)\right)\right), \left(-2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(y \cdot \left(y + 2 \cdot z\right)\right)\right)\right), \left(-2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, \left(y + 2 \cdot z\right)\right)\right)\right), \left(-2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \left(2 \cdot z\right)\right)\right)\right)\right), \left(-2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(2, z\right)\right)\right)\right)\right), \left(-2 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(2, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-2, \log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right) \]

       14. log-lowering-log.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(2, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-2, \mathsf{log.f64}\left(\left(\frac{-1}{x}\right)\right)\right)\right)\right)\right)\right) \]

       15. /-lowering-/.f64 15.7%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(2, z\right)\right)\right)\right)\right), \mathsf{*.f64}\left(-2, \mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 15.7%

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

   if -3.0000000000000001e47 < y < 6.69999999999999977e-257

   1. Initial program 84.5%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 84.5%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 84.5%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\color{blue}{\left(y \cdot \left(x + \left(z + \frac{x \cdot z}{y}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(z + \frac{x \cdot z}{y}\right)\right)\right)\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \left(\left(x + z\right) + \frac{x \cdot z}{y}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x + z\right), \left(\frac{x \cdot z}{y}\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(z + x\right), \left(\frac{x \cdot z}{y}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \left(\frac{x \cdot z}{y}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{/.f64}\left(\left(x \cdot z\right), y\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{/.f64}\left(\left(z \cdot x\right), y\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 68.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, x\right), y\right)\right)\right)\right)\right) \]

   7. Simplified 68.9%

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

   if 6.69999999999999977e-257 < y

   1. Initial program 71.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 71.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 71.3%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 47.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 47.0%

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

      1. +-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \sqrt{z}\right), \color{blue}{\left({\left(x + y\right)}^{\frac{1}{2}}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{z}\right)\right), \left({\color{blue}{\left(x + y\right)}}^{\frac{1}{2}}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \left({\left(x + \color{blue}{y}\right)}^{\frac{1}{2}}\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \mathsf{pow.f64}\left(\left(x + y\right), \color{blue}{\frac{1}{2}}\right)\right) \]

      9. +-lowering-+.f64 49.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{1}{2}\right)\right) \]

   9. Applied egg-rr 49.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{z}\right) \cdot {\left(x + y\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+47}:\\ \;\;\;\;2 \cdot e^{0.25 \cdot \left(\log \left(z \cdot z + y \cdot \left(y + 2 \cdot z\right)\right) + -2 \cdot \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-257}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot \left(\left(z + x\right) + \frac{z \cdot x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot {\left(y + x\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot {\left(y + x\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.12e-268)
   (* 2.0 (sqrt (* x (+ y z))))
   (* (* 2.0 (sqrt z)) (pow (+ y x) 0.5))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.12e-268) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = (2.0 * sqrt(z)) * pow((y + x), 0.5);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.12d-268) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = (2.0d0 * sqrt(z)) * ((y + x) ** 0.5d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.12e-268) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.pow((y + x), 0.5);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.12e-268:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.pow((y + x), 0.5)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.12e-268)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * (Float64(y + x) ^ 0.5));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.12e-268)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = (2.0 * sqrt(z)) * ((y + x) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.12e-268], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Power[N[(y + x), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.11999999999999998e-268

   1. Initial program 69.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 69.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 69.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot \left(y + z\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot \left(y + z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(y + z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 50.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, z\right)\right)\right)\right) \]

   7. Simplified 50.3%

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

   if 1.11999999999999998e-268 < y

   1. Initial program 71.5%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 71.6%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 71.6%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 47.5%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 47.5%

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

      1. +-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \sqrt{z}\right), \color{blue}{\left({\left(x + y\right)}^{\frac{1}{2}}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{z}\right)\right), \left({\color{blue}{\left(x + y\right)}}^{\frac{1}{2}}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \left({\left(x + \color{blue}{y}\right)}^{\frac{1}{2}}\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \mathsf{pow.f64}\left(\left(x + y\right), \color{blue}{\frac{1}{2}}\right)\right) \]

      9. +-lowering-+.f64 50.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{1}{2}\right)\right) \]

   9. Applied egg-rr 50.3%

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{z}\right) \cdot {\left(x + y\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot {\left(y + x\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e-268)
   (* 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 <= 2.1e-268) {
		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 <= 2.1d-268) 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 <= 2.1e-268) {
		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 <= 2.1e-268:
		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 <= 2.1e-268)
		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 <= 2.1e-268)
		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, 2.1e-268], 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 < 2.09999999999999998e-268

   1. Initial program 69.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 69.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 69.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot \left(y + z\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot \left(y + z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(y + z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 50.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, z\right)\right)\right)\right) \]

   7. Simplified 50.3%

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

   if 2.09999999999999998e-268 < y

   1. Initial program 71.5%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 71.6%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 71.6%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 47.5%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 47.5%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{y}\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 29.2%

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

       1. pow1/2 N/A

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

       2. unpow-prod-down N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot {z}^{\frac{1}{2}}\right), \color{blue}{\left({y}^{\frac{1}{2}}\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({z}^{\frac{1}{2}}\right)\right), \left({\color{blue}{y}}^{\frac{1}{2}}\right)\right) \]

       6. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{z}\right)\right), \left({y}^{\frac{1}{2}}\right)\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \left({y}^{\frac{1}{2}}\right)\right) \]

       8. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \left(\sqrt{y}\right)\right) \]

       9. sqrt-lowering-sqrt.f64 36.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \mathsf{sqrt.f64}\left(y\right)\right) \]

   12. Applied egg-rr 36.1%

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

Alternative 4: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot \left(\left(z + x\right) + \frac{z \cdot x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot \sqrt{\frac{z}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.6e+16)
   (* 2.0 (sqrt (* y (+ (+ z x) (/ (* z x) y)))))
   (* y (* 2.0 (sqrt (/ z y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.6e+16) {
		tmp = 2.0 * sqrt((y * ((z + x) + ((z * x) / y))));
	} else {
		tmp = y * (2.0 * sqrt((z / 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 <= 3.6d+16) then
        tmp = 2.0d0 * sqrt((y * ((z + x) + ((z * x) / y))))
    else
        tmp = y * (2.0d0 * sqrt((z / 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 <= 3.6e+16) {
		tmp = 2.0 * Math.sqrt((y * ((z + x) + ((z * x) / y))));
	} else {
		tmp = y * (2.0 * Math.sqrt((z / y)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 3.6e+16:
		tmp = 2.0 * math.sqrt((y * ((z + x) + ((z * x) / y))))
	else:
		tmp = y * (2.0 * math.sqrt((z / y)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 3.6e+16)
		tmp = Float64(2.0 * sqrt(Float64(y * Float64(Float64(z + x) + Float64(Float64(z * x) / y)))));
	else
		tmp = Float64(y * Float64(2.0 * sqrt(Float64(z / 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 <= 3.6e+16)
		tmp = 2.0 * sqrt((y * ((z + x) + ((z * x) / y))));
	else
		tmp = y * (2.0 * sqrt((z / 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, 3.6e+16], N[(2.0 * N[Sqrt[N[(y * N[(N[(z + x), $MachinePrecision] + N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * N[(2.0 * N[Sqrt[N[(z / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6e16

   1. Initial program 72.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 72.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\color{blue}{\left(y \cdot \left(x + \left(z + \frac{x \cdot z}{y}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \left(x + \left(z + \frac{x \cdot z}{y}\right)\right)\right)\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \left(\left(x + z\right) + \frac{x \cdot z}{y}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x + z\right), \left(\frac{x \cdot z}{y}\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(z + x\right), \left(\frac{x \cdot z}{y}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \left(\frac{x \cdot z}{y}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{/.f64}\left(\left(x \cdot z\right), y\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{/.f64}\left(\left(z \cdot x\right), y\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 62.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{+.f64}\left(z, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, x\right), y\right)\right)\right)\right)\right) \]

   7. Simplified 62.0%

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

   if 3.6e16 < y

   1. Initial program 63.2%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 63.3%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 31.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 31.2%

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

   8. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(2 \cdot \sqrt{\frac{z}{y}} + x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(2 \cdot \sqrt{\frac{z}{y}}\right), \color{blue}{\left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{\frac{z}{y}}\right)\right), \left(\color{blue}{x} \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\frac{z}{y}\right)\right)\right), \left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\sqrt{\frac{z}{{y}^{3}}}\right)}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\frac{z}{{y}^{3}}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left({y}^{3}\right)\right)\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left(y \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \left({y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 38.1%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 38.1%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(2 \cdot \sqrt{\frac{z}{y}}\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\frac{z}{y}}\right)}\right)\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\frac{z}{y}\right)\right)\right)\right) \]

       3. /-lowering-/.f64 38.1%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right)\right) \]

   13. Simplified 38.1%

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

Alternative 5: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 10^{+15}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot \sqrt{\frac{z}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-285)
   (* 2.0 (sqrt (* x (+ y z))))
   (if (<= y 1e+15)
     (* 2.0 (sqrt (* z (+ y x))))
     (* y (* 2.0 (sqrt (/ z y)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-285) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= 1e+15) {
		tmp = 2.0 * sqrt((z * (y + x)));
	} else {
		tmp = y * (2.0 * sqrt((z / 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 <= (-5d-285)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= 1d+15) then
        tmp = 2.0d0 * sqrt((z * (y + x)))
    else
        tmp = y * (2.0d0 * sqrt((z / 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 <= -5e-285) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= 1e+15) {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	} else {
		tmp = y * (2.0 * Math.sqrt((z / y)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -5e-285:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= 1e+15:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	else:
		tmp = y * (2.0 * math.sqrt((z / y)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-285)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= 1e+15)
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	else
		tmp = Float64(y * Float64(2.0 * sqrt(Float64(z / 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 <= -5e-285)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= 1e+15)
		tmp = 2.0 * sqrt((z * (y + x)));
	else
		tmp = y * (2.0 * sqrt((z / 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, -5e-285], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+15], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * N[(2.0 * N[Sqrt[N[(z / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.00000000000000018e-285

   1. Initial program 68.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 68.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 68.3%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot \left(y + z\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot \left(y + z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(y + z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 47.4%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, z\right)\right)\right)\right) \]

   7. Simplified 47.4%

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

   if -5.00000000000000018e-285 < y < 1e15

   1. Initial program 78.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 78.6%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 78.6%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 65.1%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 65.1%

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

   if 1e15 < y

   1. Initial program 63.2%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 63.3%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 31.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 31.2%

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

   8. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(2 \cdot \sqrt{\frac{z}{y}} + x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(2 \cdot \sqrt{\frac{z}{y}}\right), \color{blue}{\left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{\frac{z}{y}}\right)\right), \left(\color{blue}{x} \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\frac{z}{y}\right)\right)\right), \left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\sqrt{\frac{z}{{y}^{3}}}\right)}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\frac{z}{{y}^{3}}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left({y}^{3}\right)\right)\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left(y \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \left({y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 38.1%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 38.1%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(2 \cdot \sqrt{\frac{z}{y}}\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\frac{z}{y}}\right)}\right)\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\frac{z}{y}\right)\right)\right)\right) \]

       3. /-lowering-/.f64 38.1%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right)\right) \]

   13. Simplified 38.1%

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

Alternative 6: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+15}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot \sqrt{\frac{z}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.02e+15)
   (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
   (* y (* 2.0 (sqrt (/ z y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.02e+15) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = y * (2.0 * sqrt((z / 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 <= 1.02d+15) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    else
        tmp = y * (2.0d0 * sqrt((z / 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 <= 1.02e+15) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = y * (2.0 * Math.sqrt((z / y)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.02e+15:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	else:
		tmp = y * (2.0 * math.sqrt((z / y)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.02e+15)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(y * Float64(2.0 * sqrt(Float64(z / 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 <= 1.02e+15)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	else
		tmp = y * (2.0 * sqrt((z / 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, 1.02e+15], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * N[(2.0 * N[Sqrt[N[(z / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e15

   1. Initial program 72.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 72.0%

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

   if 1.02e15 < y

   1. Initial program 63.2%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 63.3%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 31.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 31.2%

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

   8. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(2 \cdot \sqrt{\frac{z}{y}} + x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(2 \cdot \sqrt{\frac{z}{y}}\right), \color{blue}{\left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{\frac{z}{y}}\right)\right), \left(\color{blue}{x} \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\frac{z}{y}\right)\right)\right), \left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\sqrt{\frac{z}{{y}^{3}}}\right)}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\frac{z}{{y}^{3}}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left({y}^{3}\right)\right)\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left(y \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \left({y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 38.1%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 38.1%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(2 \cdot \sqrt{\frac{z}{y}}\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\frac{z}{y}}\right)}\right)\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\frac{z}{y}\right)\right)\right)\right) \]

       3. /-lowering-/.f64 38.1%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right)\right) \]

   13. Simplified 38.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+15}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot \sqrt{\frac{z}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-281) (* 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 <= -2e-281) {
		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 <= (-2d-281)) 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 <= -2e-281) {
		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 <= -2e-281:
		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 <= -2e-281)
		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 <= -2e-281)
		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, -2e-281], 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 < -2e-281

   1. Initial program 68.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 68.1%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot \left(y + z\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot \left(y + z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(y + z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 47.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, z\right)\right)\right)\right) \]

   7. Simplified 47.0%

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

   if -2e-281 < y

   1. Initial program 72.1%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 72.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 72.2%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 50.7%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 50.7%

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

Alternative 8: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-286}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6e-286) (* 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 <= -6e-286) {
		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 <= (-6d-286)) 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 <= -6e-286) {
		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 <= -6e-286:
		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 <= -6e-286)
		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 <= -6e-286)
		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, -6e-286], 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 < -6.0000000000000001e-286

   1. Initial program 68.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 68.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 68.3%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot \left(y + z\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot \left(y + z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(y + z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 47.4%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, z\right)\right)\right)\right) \]

   7. Simplified 47.4%

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

   if -6.0000000000000001e-286 < y

   1. Initial program 71.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 72.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot z\right)\right)\right) \]

      3. *-lowering-*.f64 26.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, z\right)\right)\right) \]

   7. Simplified 26.2%

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

Alternative 9: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-286}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6e-286) (* 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 <= -6e-286) {
		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 <= (-6d-286)) 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 <= -6e-286) {
		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 <= -6e-286:
		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 <= -6e-286)
		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 <= -6e-286)
		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, -6e-286], 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 < -6.0000000000000001e-286

   1. Initial program 68.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 68.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 68.3%

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

   5. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot y}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 29.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, x\right)\right)\right) \]

   7. Simplified 29.0%

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

   if -6.0000000000000001e-286 < y

   1. Initial program 71.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 72.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot z\right)\right)\right) \]

      3. *-lowering-*.f64 26.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, z\right)\right)\right) \]

   7. Simplified 26.2%

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

Alternative 10: 35.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f64 70.1%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

3. Simplified 70.1%

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

5. Taylor expanded in z around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot y}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 26.7%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, x\right)\right)\right) \]

7. Simplified 26.7%

   \[\leadsto \color{blue}{2 \cdot \sqrt{y \cdot x}} \]
8. Add Preprocessing

Developer Target 1: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\\ \mathbf{if}\;z < 7.636950090573675 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z)))
          (* (pow z 0.25) (pow y 0.25)))))
   (if (< z 7.636950090573675e+176)
     (* 2.0 (sqrt (+ (* (+ x y) z) (* x y))))
     (* (* t_0 t_0) 2.0))))

C

double code(double x, double y, double z) {
	double t_0 = (0.25 * ((pow(y, -0.75) * (pow(z, -0.75) * x)) * (y + z))) + (pow(z, 0.25) * pow(y, 0.25));
	double tmp;
	if (z < 7.636950090573675e+176) {
		tmp = 2.0 * sqrt((((x + y) * z) + (x * y)));
	} else {
		tmp = (t_0 * t_0) * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.25d0 * (((y ** (-0.75d0)) * ((z ** (-0.75d0)) * x)) * (y + z))) + ((z ** 0.25d0) * (y ** 0.25d0))
    if (z < 7.636950090573675d+176) then
        tmp = 2.0d0 * sqrt((((x + y) * z) + (x * y)))
    else
        tmp = (t_0 * t_0) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.25 * ((Math.pow(y, -0.75) * (Math.pow(z, -0.75) * x)) * (y + z))) + (Math.pow(z, 0.25) * Math.pow(y, 0.25));
	double tmp;
	if (z < 7.636950090573675e+176) {
		tmp = 2.0 * Math.sqrt((((x + y) * z) + (x * y)));
	} else {
		tmp = (t_0 * t_0) * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.25 * ((math.pow(y, -0.75) * (math.pow(z, -0.75) * x)) * (y + z))) + (math.pow(z, 0.25) * math.pow(y, 0.25))
	tmp = 0
	if z < 7.636950090573675e+176:
		tmp = 2.0 * math.sqrt((((x + y) * z) + (x * y)))
	else:
		tmp = (t_0 * t_0) * 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.25 * Float64(Float64((y ^ -0.75) * Float64((z ^ -0.75) * x)) * Float64(y + z))) + Float64((z ^ 0.25) * (y ^ 0.25)))
	tmp = 0.0
	if (z < 7.636950090573675e+176)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x + y) * z) + Float64(x * y))));
	else
		tmp = Float64(Float64(t_0 * t_0) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.25 * (((y ^ -0.75) * ((z ^ -0.75) * x)) * (y + z))) + ((z ^ 0.25) * (y ^ 0.25));
	tmp = 0.0;
	if (z < 7.636950090573675e+176)
		tmp = 2.0 * sqrt((((x + y) * z) + (x * y)));
	else
		tmp = (t_0 * t_0) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.25 * N[(N[(N[Power[y, -0.75], $MachinePrecision] * N[(N[Power[z, -0.75], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[z, 0.25], $MachinePrecision] * N[Power[y, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, 7.636950090573675e+176], N[(2.0 * N[Sqrt[N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 2.0), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024150 
(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)))))