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

Percentage Accurate: 69.5% → 97.0%

Time: 13.9s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(0 - y\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -6800000000:\\ \;\;\;\;2 \cdot \left(\sqrt{\left(y + z\right) \cdot \left(y + z\right)} \cdot \sqrt{\frac{x}{y + z}}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.35e+154)
   (* 2.0 (exp (* (- (log (- 0.0 y)) (log (/ -1.0 x))) 0.5)))
   (if (<= y -6800000000.0)
     (* 2.0 (* (sqrt (* (+ y z) (+ y z))) (sqrt (/ x (+ y z)))))
     (if (<= y 5e-293)
       (* 2.0 (sqrt (+ (* x z) (* y x))))
       (* (* 2.0 (sqrt z)) (sqrt (+ y x)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+154) {
		tmp = 2.0 * exp(((log((0.0 - y)) - log((-1.0 / x))) * 0.5));
	} else if (y <= -6800000000.0) {
		tmp = 2.0 * (sqrt(((y + z) * (y + z))) * sqrt((x / (y + z))));
	} else if (y <= 5e-293) {
		tmp = 2.0 * sqrt(((x * z) + (y * x)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((y + x));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.35d+154)) then
        tmp = 2.0d0 * exp(((log((0.0d0 - y)) - log(((-1.0d0) / x))) * 0.5d0))
    else if (y <= (-6800000000.0d0)) then
        tmp = 2.0d0 * (sqrt(((y + z) * (y + z))) * sqrt((x / (y + z))))
    else if (y <= 5d-293) then
        tmp = 2.0d0 * sqrt(((x * z) + (y * x)))
    else
        tmp = (2.0d0 * sqrt(z)) * sqrt((y + x))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+154) {
		tmp = 2.0 * Math.exp(((Math.log((0.0 - y)) - Math.log((-1.0 / x))) * 0.5));
	} else if (y <= -6800000000.0) {
		tmp = 2.0 * (Math.sqrt(((y + z) * (y + z))) * Math.sqrt((x / (y + z))));
	} else if (y <= 5e-293) {
		tmp = 2.0 * Math.sqrt(((x * z) + (y * x)));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.sqrt((y + x));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.35e+154:
		tmp = 2.0 * math.exp(((math.log((0.0 - y)) - math.log((-1.0 / x))) * 0.5))
	elif y <= -6800000000.0:
		tmp = 2.0 * (math.sqrt(((y + z) * (y + z))) * math.sqrt((x / (y + z))))
	elif y <= 5e-293:
		tmp = 2.0 * math.sqrt(((x * z) + (y * x)))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.sqrt((y + x))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.35e+154)
		tmp = Float64(2.0 * exp(Float64(Float64(log(Float64(0.0 - y)) - log(Float64(-1.0 / x))) * 0.5)));
	elseif (y <= -6800000000.0)
		tmp = Float64(2.0 * Float64(sqrt(Float64(Float64(y + z) * Float64(y + z))) * sqrt(Float64(x / Float64(y + z)))));
	elseif (y <= 5e-293)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * z) + Float64(y * x))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.35e+154)
		tmp = 2.0 * exp(((log((0.0 - y)) - log((-1.0 / x))) * 0.5));
	elseif (y <= -6800000000.0)
		tmp = 2.0 * (sqrt(((y + z) * (y + z))) * sqrt((x / (y + z))));
	elseif (y <= 5e-293)
		tmp = 2.0 * sqrt(((x * z) + (y * x)));
	else
		tmp = (2.0 * sqrt(z)) * sqrt((y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.35e+154], N[(2.0 * N[Exp[N[(N[(N[Log[N[(0.0 - y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -6800000000.0], N[(2.0 * N[(N[Sqrt[N[(N[(y + z), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(x / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-293], N[(2.0 * N[Sqrt[N[(N[(x * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.35000000000000003e154

   1. Initial program 43.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot y}} \]
   4. 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. *-lowering-*.f64 19.0%

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

   5. Simplified 19.0%

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

      1. pow1/2 N/A

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

      2. pow-to-exp N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 17.8%

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

   7. Applied egg-rr 17.8%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\log \left(-1 \cdot y\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}, \frac{1}{2}\right)\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(0 - y\right)\right), \log \left(\frac{-1}{x}\right)\right), \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), \log \left(\frac{-1}{x}\right)\right), \frac{1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), \mathsf{log.f64}\left(\left(\frac{-1}{x}\right)\right)\right), \frac{1}{2}\right)\right)\right) \]

      9. /-lowering-/.f64 48.8%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right)\right), \frac{1}{2}\right)\right)\right) \]

   10. Simplified 48.8%

       \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(0 - y\right) - \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]

   if -1.35000000000000003e154 < y < -6.8e9

   1. Initial program 67.3%

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

      1. flip-+ N/A

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

      2. flip-- N/A

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

      3. associate-/l/ N/A

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

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

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

   4. Applied egg-rr 15.9%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. distribute-lft-in N/A

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

      3. times-frac N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. +-lowering-+.f64 25.2%

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

   7. Simplified 25.2%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. sqrt-prod N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

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

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

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

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 60.1%

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

   9. Applied egg-rr 60.1%

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

   if -6.8e9 < y < 5.0000000000000003e-293

   1. Initial program 80.2%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 80.2%

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

   4. Applied egg-rr 80.2%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 54.7%

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

   7. Simplified 54.7%

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

   if 5.0000000000000003e-293 < y

   1. Initial program 74.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{z \cdot \left(x + y\right)}} \]
   4. 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. +-lowering-+.f64 46.1%

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

   5. Simplified 46.1%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. pow1/2 N/A

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

      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. pow1/2 N/A

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

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

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

      10. +-lowering-+.f64 44.6%

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

   7. Applied egg-rr 44.6%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(0 - y\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -6800000000:\\ \;\;\;\;2 \cdot \left(\sqrt{\left(y + z\right) \cdot \left(y + z\right)} \cdot \sqrt{\frac{x}{y + z}}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -510000000:\\ \;\;\;\;2 \cdot \left(\sqrt{\left(y + z\right) \cdot \left(y + z\right)} \cdot \sqrt{\frac{x}{y + z}}\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -510000000.0)
   (* 2.0 (* (sqrt (* (+ y z) (+ y z))) (sqrt (/ x (+ y z)))))
   (if (<= y 5.5e-293)
     (* 2.0 (sqrt (+ (* x z) (* y x))))
     (* (* 2.0 (sqrt z)) (sqrt (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -510000000.0) {
		tmp = 2.0 * (sqrt(((y + z) * (y + z))) * sqrt((x / (y + z))));
	} else if (y <= 5.5e-293) {
		tmp = 2.0 * sqrt(((x * z) + (y * x)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((y + x));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-510000000.0d0)) then
        tmp = 2.0d0 * (sqrt(((y + z) * (y + z))) * sqrt((x / (y + z))))
    else if (y <= 5.5d-293) then
        tmp = 2.0d0 * sqrt(((x * z) + (y * x)))
    else
        tmp = (2.0d0 * sqrt(z)) * sqrt((y + x))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -510000000.0) {
		tmp = 2.0 * (Math.sqrt(((y + z) * (y + z))) * Math.sqrt((x / (y + z))));
	} else if (y <= 5.5e-293) {
		tmp = 2.0 * Math.sqrt(((x * z) + (y * x)));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.sqrt((y + x));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -510000000.0:
		tmp = 2.0 * (math.sqrt(((y + z) * (y + z))) * math.sqrt((x / (y + z))))
	elif y <= 5.5e-293:
		tmp = 2.0 * math.sqrt(((x * z) + (y * x)))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.sqrt((y + x))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -510000000.0)
		tmp = Float64(2.0 * Float64(sqrt(Float64(Float64(y + z) * Float64(y + z))) * sqrt(Float64(x / Float64(y + z)))));
	elseif (y <= 5.5e-293)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * z) + Float64(y * x))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -510000000.0)
		tmp = 2.0 * (sqrt(((y + z) * (y + z))) * sqrt((x / (y + z))));
	elseif (y <= 5.5e-293)
		tmp = 2.0 * sqrt(((x * z) + (y * x)));
	else
		tmp = (2.0 * sqrt(z)) * sqrt((y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -510000000.0], N[(2.0 * N[(N[Sqrt[N[(N[(y + z), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(x / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-293], N[(2.0 * N[Sqrt[N[(N[(x * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.1e8

   1. Initial program 56.5%

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

      1. flip-+ N/A

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

      2. flip-- N/A

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

      3. associate-/l/ N/A

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

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

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

   4. Applied egg-rr 10.1%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. distribute-lft-in N/A

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

      3. times-frac N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. +-lowering-+.f64 15.4%

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

   7. Simplified 15.4%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. sqrt-prod N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

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

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

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

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 34.1%

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

   9. Applied egg-rr 34.1%

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

   if -5.1e8 < y < 5.50000000000000028e-293

   1. Initial program 80.2%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 80.2%

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

   4. Applied egg-rr 80.2%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 54.7%

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

   7. Simplified 54.7%

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

   if 5.50000000000000028e-293 < y

   1. Initial program 74.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{z \cdot \left(x + y\right)}} \]
   4. 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. +-lowering-+.f64 46.1%

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

   5. Simplified 46.1%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. pow1/2 N/A

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

      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. pow1/2 N/A

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

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

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

      10. +-lowering-+.f64 44.6%

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

   7. Applied egg-rr 44.6%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -510000000:\\ \;\;\;\;2 \cdot \left(\sqrt{\left(y + z\right) \cdot \left(y + z\right)} \cdot \sqrt{\frac{x}{y + z}}\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot \left(z + x \cdot \left(\frac{z}{y} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.9e-289)
   (* 2.0 (sqrt (* y (+ z (* x (+ (/ z y) 1.0))))))
   (* (* 2.0 (sqrt z)) (sqrt (+ y x)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.9e-289) {
		tmp = 2.0 * sqrt((y * (z + (x * ((z / y) + 1.0)))));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((y + x));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.9d-289)) then
        tmp = 2.0d0 * sqrt((y * (z + (x * ((z / y) + 1.0d0)))))
    else
        tmp = (2.0d0 * sqrt(z)) * sqrt((y + x))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.9e-289) {
		tmp = 2.0 * Math.sqrt((y * (z + (x * ((z / y) + 1.0)))));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.sqrt((y + x));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.9e-289:
		tmp = 2.0 * math.sqrt((y * (z + (x * ((z / y) + 1.0)))))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.sqrt((y + x))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.9e-289)
		tmp = Float64(2.0 * sqrt(Float64(y * Float64(z + Float64(x * Float64(Float64(z / y) + 1.0))))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.9e-289)
		tmp = 2.0 * sqrt((y * (z + (x * ((z / y) + 1.0)))));
	else
		tmp = (2.0 * sqrt(z)) * sqrt((y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.9e-289], N[(2.0 * N[Sqrt[N[(y * N[(z + N[(x * N[(N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.90000000000000005e-289

   1. Initial program 69.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\color{blue}{\left(y \cdot \left(x + \left(z + \frac{x \cdot z}{y}\right)\right)\right)}\right)\right) \]
   4. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

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

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

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

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

      11. /-lowering-/.f64 59.5%

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

   5. Simplified 59.5%

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

   if -1.90000000000000005e-289 < y

   1. Initial program 73.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{z \cdot \left(x + y\right)}} \]
   4. 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. +-lowering-+.f64 47.2%

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

   5. Simplified 47.2%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. pow1/2 N/A

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

      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. pow1/2 N/A

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

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

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

      10. +-lowering-+.f64 46.2%

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

   7. Applied egg-rr 46.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot \left(z + x \cdot \left(\frac{z}{y} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot \left(z + x \cdot \left(\frac{z}{y} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e-293)
   (* 2.0 (sqrt (* y (+ z (* x (+ (/ z y) 1.0))))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e-293) {
		tmp = 2.0 * sqrt((y * (z + (x * ((z / y) + 1.0)))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.4d-293) then
        tmp = 2.0d0 * sqrt((y * (z + (x * ((z / y) + 1.0d0)))))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e-293) {
		tmp = 2.0 * Math.sqrt((y * (z + (x * ((z / y) + 1.0)))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 4.4e-293:
		tmp = 2.0 * math.sqrt((y * (z + (x * ((z / y) + 1.0)))))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.4e-293)
		tmp = Float64(2.0 * sqrt(Float64(y * Float64(z + Float64(x * Float64(Float64(z / y) + 1.0))))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.4e-293)
		tmp = 2.0 * sqrt((y * (z + (x * ((z / y) + 1.0)))));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 4.4e-293], N[(2.0 * N[Sqrt[N[(y * N[(z + N[(x * N[(N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.4e-293

   1. Initial program 68.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\color{blue}{\left(y \cdot \left(x + \left(z + \frac{x \cdot z}{y}\right)\right)\right)}\right)\right) \]
   4. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

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

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

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

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

      11. /-lowering-/.f64 56.8%

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

   5. Simplified 56.8%

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

   if 4.4e-293 < y

   1. Initial program 74.6%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 74.6%

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

   4. Applied egg-rr 74.6%

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

   5. Taylor expanded in x around 0

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

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

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

      2. *-lowering-*.f64 23.6%

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

   7. Simplified 23.6%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

      5. pow1/2 N/A

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

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

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

      7. pow1/2 N/A

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

      8. sqrt-lowering-sqrt.f64 30.2%

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

   9. Applied egg-rr 30.2%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot \left(z + x \cdot \left(\frac{z}{y} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-308}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\sqrt{\frac{\frac{1}{z}}{y + x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-308)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (/ 1.0 (sqrt (/ (/ 1.0 z) (+ y x)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-308) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (1.0 / sqrt(((1.0 / 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 <= 5d-308) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (1.0d0 / sqrt(((1.0d0 / 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 <= 5e-308) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (1.0 / Math.sqrt(((1.0 / z) / (y + x))));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 5e-308:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (1.0 / math.sqrt(((1.0 / z) / (y + x))))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-308)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(1.0 / sqrt(Float64(Float64(1.0 / 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 <= 5e-308)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (1.0 / sqrt(((1.0 / 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, 5e-308], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[Sqrt[N[(N[(1.0 / z), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999955e-308

   1. Initial program 68.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right)}} \]
   4. 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 42.7%

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

   5. Simplified 42.7%

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

   if 4.99999999999999955e-308 < y

   1. Initial program 74.4%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}\right)}\right)\right) \]

   4. Applied egg-rr 74.0%

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

   5. Taylor expanded in z around inf

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

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

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

      2. associate-/r* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 47.6%

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

   7. Simplified 47.6%

      \[\leadsto 2 \cdot \frac{1}{\color{blue}{\sqrt{\frac{\frac{1}{z}}{y + x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-302}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-302) (* 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 <= -1e-302) {
		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 <= (-1d-302)) 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 <= -1e-302) {
		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 <= -1e-302:
		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 <= -1e-302)
		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 <= -1e-302)
		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, -1e-302], 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 < -9.9999999999999996e-303

   1. Initial program 68.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right)}} \]
   4. 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 42.6%

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

   5. Simplified 42.6%

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

   if -9.9999999999999996e-303 < y

   1. Initial program 74.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{z \cdot \left(x + y\right)}} \]
   4. 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. +-lowering-+.f64 46.8%

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

   5. Simplified 46.8%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-302}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5.5e-293) (* 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 <= 5.5e-293) {
		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 <= 5.5d-293) 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 <= 5.5e-293) {
		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 <= 5.5e-293:
		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 <= 5.5e-293)
		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 <= 5.5e-293)
		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, 5.5e-293], 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 < 5.50000000000000028e-293

   1. Initial program 68.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right)}} \]
   4. 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 43.3%

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

   5. Simplified 43.3%

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

   if 5.50000000000000028e-293 < y

   1. Initial program 74.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \sqrt{y \cdot z}} \]
   4. 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 23.6%

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

   5. Simplified 23.6%

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

Alternative 8: 69.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. *-commutative N/A

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

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

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

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

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. distribute-lft-out N/A

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

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

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

   11. +-lowering-+.f64 71.2%

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

4. Applied egg-rr 71.2%

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

5. Final simplification 71.2%

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

Alternative 9: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310) (* 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 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 68.4%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot y}} \]
   4. 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. *-lowering-*.f64 20.9%

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

   5. Simplified 20.9%

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

   if -4.999999999999985e-310 < y

   1. Initial program 74.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \sqrt{y \cdot z}} \]
   4. 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 23.1%

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

   5. Simplified 23.1%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.5% 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 71.2%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot y}} \]
4. 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. *-lowering-*.f64 25.4%

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

5. Simplified 25.4%

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

6. Final simplification 25.4%

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

Developer Target 1: 82.3% 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 2024192 
(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)))))