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

Percentage Accurate: 69.7% → 95.2%

Time: 11.5s

Alternatives: 11

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: 69.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-237}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.12e+52)
   (* 2.0 (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0))
   (if (<= y 8.4e-237)
     (* 2.0 (sqrt (fma x y (* z (+ y x)))))
     (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.12e+52) {
		tmp = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	} else if (y <= 8.4e-237) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.12e+52)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0));
	elseif (y <= 8.4e-237)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.12e+52], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.4e-237], N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.12000000000000002e52

   1. Initial program 51.3%

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

      1. +-commutative 51.3%

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

      2. associate-+r+ 51.3%

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

      3. *-commutative 51.3%

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

      4. +-commutative 51.3%

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

      5. associate-+l+ 51.3%

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

      6. *-commutative 51.3%

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

      7. *-commutative 51.3%

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

      8. *-commutative 51.3%

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

      9. distribute-lft-out 51.3%

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

   3. Simplified 51.3%

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

      1. add-sqr-sqrt 51.1%

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

      2. pow2 51.1%

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

      3. pow1/2 51.1%

         \[\leadsto 2 \cdot {\left(\sqrt{\color{blue}{{\left(x \cdot y + z \cdot \left(y + x\right)\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 51.0%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(x \cdot y + z \cdot \left(y + x\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. +-commutative 51.0%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(z \cdot \left(y + x\right) + x \cdot y\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      6. +-commutative 51.0%

         \[\leadsto 2 \cdot {\left({\left(z \cdot \color{blue}{\left(x + y\right)} + x \cdot y\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      7. fma-def 51.8%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      8. metadata-eval 51.8%

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

   5. Applied egg-rr 51.8%

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

   6. Taylor expanded in x around -inf 47.1%

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

   if -1.12000000000000002e52 < y < 8.4000000000000005e-237

   1. Initial program 80.7%

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

      1. +-commutative 80.7%

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

      2. associate-+r+ 80.7%

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

      3. *-commutative 80.7%

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

      4. +-commutative 80.7%

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

      5. +-commutative 80.7%

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

      6. *-commutative 80.7%

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

      7. *-commutative 80.7%

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

      8. associate-+l+ 80.7%

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

      9. *-commutative 80.7%

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

      10. +-commutative 80.7%

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

      11. associate-+l+ 80.7%

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

      12. *-commutative 80.7%

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

      13. *-commutative 80.7%

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

      14. +-commutative 80.7%

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

   3. Simplified 80.7%

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

   if 8.4000000000000005e-237 < y

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. associate-+r+ 69.6%

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

      3. *-commutative 69.6%

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

      4. +-commutative 69.6%

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

      5. associate-+l+ 69.6%

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

      6. *-commutative 69.6%

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

      7. *-commutative 69.6%

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

      8. *-commutative 69.6%

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

      9. distribute-lft-out 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in z around inf 45.8%

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

      1. +-commutative 45.8%

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

   6. Simplified 45.8%

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

      1. *-commutative 45.8%

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

      2. sqrt-prod 53.1%

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

      3. +-commutative 53.1%

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

   8. Applied egg-rr 53.1%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-237}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 2: 94.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)\right)}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-237}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.12e+52)
   (* 2.0 (pow (exp 0.25) (* 2.0 (- (log (- (- y) z)) (log (/ -1.0 x))))))
   (if (<= y 8.4e-237)
     (* 2.0 (sqrt (fma x y (* z (+ y x)))))
     (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.12e+52) {
		tmp = 2.0 * pow(exp(0.25), (2.0 * (log((-y - z)) - log((-1.0 / x)))));
	} else if (y <= 8.4e-237) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.12e+52)
		tmp = Float64(2.0 * (exp(0.25) ^ Float64(2.0 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))));
	elseif (y <= 8.4e-237)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.12e+52], N[(2.0 * N[Power[N[Exp[0.25], $MachinePrecision], N[(2.0 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.4e-237], N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.12000000000000002e52

   1. Initial program 51.3%

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

      1. +-commutative 51.3%

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

      2. associate-+r+ 51.3%

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

      3. *-commutative 51.3%

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

      4. +-commutative 51.3%

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

      5. associate-+l+ 51.3%

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

      6. *-commutative 51.3%

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

      7. *-commutative 51.3%

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

      8. *-commutative 51.3%

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

      9. distribute-lft-out 51.3%

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

   3. Simplified 51.3%

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

      1. add-sqr-sqrt 51.1%

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

      2. pow2 51.1%

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

      3. pow1/2 51.1%

         \[\leadsto 2 \cdot {\left(\sqrt{\color{blue}{{\left(x \cdot y + z \cdot \left(y + x\right)\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 51.0%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(x \cdot y + z \cdot \left(y + x\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. +-commutative 51.0%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(z \cdot \left(y + x\right) + x \cdot y\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      6. +-commutative 51.0%

         \[\leadsto 2 \cdot {\left({\left(z \cdot \color{blue}{\left(x + y\right)} + x \cdot y\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      7. fma-def 51.8%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      8. metadata-eval 51.8%

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

   5. Applied egg-rr 51.8%

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

   6. Taylor expanded in x around -inf 47.1%

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

      1. unpow2 47.1%

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

      2. exp-prod 45.5%

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

      3. exp-prod 44.9%

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

      4. pow-sqr 44.9%

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

      5. mul-1-neg 44.9%

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

      6. unsub-neg 44.9%

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

      7. neg-mul-1 44.9%

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

      8. +-commutative 44.9%

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

      9. unsub-neg 44.9%

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

      10. neg-mul-1 44.9%

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

   8. Simplified 44.9%

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

   if -1.12000000000000002e52 < y < 8.4000000000000005e-237

   1. Initial program 80.7%

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

      1. +-commutative 80.7%

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

      2. associate-+r+ 80.7%

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

      3. *-commutative 80.7%

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

      4. +-commutative 80.7%

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

      5. +-commutative 80.7%

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

      6. *-commutative 80.7%

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

      7. *-commutative 80.7%

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

      8. associate-+l+ 80.7%

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

      9. *-commutative 80.7%

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

      10. +-commutative 80.7%

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

      11. associate-+l+ 80.7%

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

      12. *-commutative 80.7%

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

      13. *-commutative 80.7%

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

      14. +-commutative 80.7%

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

   3. Simplified 80.7%

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

   if 8.4000000000000005e-237 < y

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. associate-+r+ 69.6%

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

      3. *-commutative 69.6%

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

      4. +-commutative 69.6%

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

      5. associate-+l+ 69.6%

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

      6. *-commutative 69.6%

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

      7. *-commutative 69.6%

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

      8. *-commutative 69.6%

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

      9. distribute-lft-out 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in z around inf 45.8%

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

      1. +-commutative 45.8%

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

   6. Simplified 45.8%

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

      1. *-commutative 45.8%

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

      2. sqrt-prod 53.1%

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

      3. +-commutative 53.1%

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

   8. Applied egg-rr 53.1%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)\right)}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-237}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 3: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+51}:\\ \;\;\;\;2 \cdot {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)\right)}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-237}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.6e+51)
   (* 2.0 (pow (exp 0.25) (* 2.0 (- (log (- x)) (log (/ -1.0 y))))))
   (if (<= y 8.4e-237)
     (* 2.0 (sqrt (fma x y (* z (+ y x)))))
     (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.6e+51) {
		tmp = 2.0 * pow(exp(0.25), (2.0 * (log(-x) - log((-1.0 / y)))));
	} else if (y <= 8.4e-237) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.6e+51)
		tmp = Float64(2.0 * (exp(0.25) ^ Float64(2.0 * Float64(log(Float64(-x)) - log(Float64(-1.0 / y))))));
	elseif (y <= 8.4e-237)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -2.6e+51], N[(2.0 * N[Power[N[Exp[0.25], $MachinePrecision], N[(2.0 * N[(N[Log[(-x)], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.4e-237], N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6000000000000001e51

   1. Initial program 51.3%

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

      1. +-commutative 51.3%

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

      2. associate-+r+ 51.3%

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

      3. *-commutative 51.3%

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

      4. +-commutative 51.3%

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

      5. associate-+l+ 51.3%

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

      6. *-commutative 51.3%

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

      7. *-commutative 51.3%

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

      8. *-commutative 51.3%

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

      9. distribute-lft-out 51.3%

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

   3. Simplified 51.3%

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

      1. add-sqr-sqrt 51.1%

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

      2. pow2 51.1%

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

      3. pow1/2 51.1%

         \[\leadsto 2 \cdot {\left(\sqrt{\color{blue}{{\left(x \cdot y + z \cdot \left(y + x\right)\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 51.0%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(x \cdot y + z \cdot \left(y + x\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. +-commutative 51.0%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(z \cdot \left(y + x\right) + x \cdot y\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      6. +-commutative 51.0%

         \[\leadsto 2 \cdot {\left({\left(z \cdot \color{blue}{\left(x + y\right)} + x \cdot y\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      7. fma-def 51.8%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      8. metadata-eval 51.8%

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

   5. Applied egg-rr 51.8%

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

   6. Taylor expanded in z around 0 28.1%

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

   7. Taylor expanded in y around -inf 45.1%

      \[\leadsto 2 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log \left(-1 \cdot x\right) + -1 \cdot \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 45.1%

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

      2. exp-prod 43.6%

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

      3. exp-prod 43.0%

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

      4. pow-sqr 43.0%

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

      5. mul-1-neg 43.0%

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

      6. unsub-neg 43.0%

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

      7. mul-1-neg 43.0%

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

   9. Simplified 43.0%

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

   if -2.6000000000000001e51 < y < 8.4000000000000005e-237

   1. Initial program 80.7%

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

      1. +-commutative 80.7%

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

      2. associate-+r+ 80.7%

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

      3. *-commutative 80.7%

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

      4. +-commutative 80.7%

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

      5. +-commutative 80.7%

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

      6. *-commutative 80.7%

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

      7. *-commutative 80.7%

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

      8. associate-+l+ 80.7%

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

      9. *-commutative 80.7%

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

      10. +-commutative 80.7%

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

      11. associate-+l+ 80.7%

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

      12. *-commutative 80.7%

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

      13. *-commutative 80.7%

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

      14. +-commutative 80.7%

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

   3. Simplified 80.7%

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

   if 8.4000000000000005e-237 < y

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. associate-+r+ 69.6%

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

      3. *-commutative 69.6%

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

      4. +-commutative 69.6%

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

      5. associate-+l+ 69.6%

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

      6. *-commutative 69.6%

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

      7. *-commutative 69.6%

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

      8. *-commutative 69.6%

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

      9. distribute-lft-out 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in z around inf 45.8%

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

      1. +-commutative 45.8%

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

   6. Simplified 45.8%

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

      1. *-commutative 45.8%

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

      2. sqrt-prod 53.1%

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

      3. +-commutative 53.1%

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

   8. Applied egg-rr 53.1%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+51}:\\ \;\;\;\;2 \cdot {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)\right)}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-237}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 4: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-237}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e+53)
   (* 2.0 (pow (exp (* 0.25 (- (log (- y)) (log (/ -1.0 x))))) 2.0))
   (if (<= y 8.4e-237)
     (* 2.0 (sqrt (fma x y (* z (+ y x)))))
     (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+53) {
		tmp = 2.0 * pow(exp((0.25 * (log(-y) - log((-1.0 / x))))), 2.0);
	} else if (y <= 8.4e-237) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e+53)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(-y)) - log(Float64(-1.0 / x))))) ^ 2.0));
	elseif (y <= 8.4e-237)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -2.4e+53], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[(-y)], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.4e-237], N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e53

   1. Initial program 51.3%

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

      1. +-commutative 51.3%

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

      2. associate-+r+ 51.3%

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

      3. *-commutative 51.3%

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

      4. +-commutative 51.3%

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

      5. associate-+l+ 51.3%

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

      6. *-commutative 51.3%

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

      7. *-commutative 51.3%

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

      8. *-commutative 51.3%

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

      9. distribute-lft-out 51.3%

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

   3. Simplified 51.3%

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

      1. add-sqr-sqrt 51.1%

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

      2. pow2 51.1%

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

      3. pow1/2 51.1%

         \[\leadsto 2 \cdot {\left(\sqrt{\color{blue}{{\left(x \cdot y + z \cdot \left(y + x\right)\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 51.0%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(x \cdot y + z \cdot \left(y + x\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. +-commutative 51.0%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(z \cdot \left(y + x\right) + x \cdot y\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      6. +-commutative 51.0%

         \[\leadsto 2 \cdot {\left({\left(z \cdot \color{blue}{\left(x + y\right)} + x \cdot y\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      7. fma-def 51.8%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      8. metadata-eval 51.8%

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

   5. Applied egg-rr 51.8%

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

   6. Taylor expanded in z around 0 28.1%

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

   7. Taylor expanded in x around -inf 45.1%

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

   if -2.4e53 < y < 8.4000000000000005e-237

   1. Initial program 80.7%

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

      1. +-commutative 80.7%

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

      2. associate-+r+ 80.7%

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

      3. *-commutative 80.7%

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

      4. +-commutative 80.7%

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

      5. +-commutative 80.7%

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

      6. *-commutative 80.7%

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

      7. *-commutative 80.7%

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

      8. associate-+l+ 80.7%

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

      9. *-commutative 80.7%

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

      10. +-commutative 80.7%

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

      11. associate-+l+ 80.7%

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

      12. *-commutative 80.7%

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

      13. *-commutative 80.7%

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

      14. +-commutative 80.7%

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

   3. Simplified 80.7%

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

   if 8.4000000000000005e-237 < y

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. associate-+r+ 69.6%

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

      3. *-commutative 69.6%

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

      4. +-commutative 69.6%

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

      5. associate-+l+ 69.6%

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

      6. *-commutative 69.6%

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

      7. *-commutative 69.6%

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

      8. *-commutative 69.6%

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

      9. distribute-lft-out 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in z around inf 45.8%

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

      1. +-commutative 45.8%

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

   6. Simplified 45.8%

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

      1. *-commutative 45.8%

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

      2. sqrt-prod 53.1%

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

      3. +-commutative 53.1%

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

   8. Applied egg-rr 53.1%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-237}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 5: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.18e-275)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.18e-275) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(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 <= 1.18d-275) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(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 <= 1.18e-275) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.18e-275:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.18e-275)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(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 <= 1.18e-275)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(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, 1.18e-275], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.17999999999999995e-275

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. associate-+r+ 69.6%

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

      3. *-commutative 69.6%

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

      4. +-commutative 69.6%

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

      5. associate-+l+ 69.6%

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

      6. *-commutative 69.6%

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

      7. *-commutative 69.6%

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

      8. *-commutative 69.6%

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

      9. distribute-lft-out 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in x around inf 51.6%

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

   if 1.17999999999999995e-275 < y

   1. Initial program 70.0%

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

      1. +-commutative 70.0%

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

      2. associate-+r+ 70.0%

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

      3. *-commutative 70.0%

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

      4. +-commutative 70.0%

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

      5. associate-+l+ 70.0%

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

      6. *-commutative 70.0%

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

      7. *-commutative 70.0%

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

      8. *-commutative 70.0%

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

      9. distribute-lft-out 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in z around inf 47.6%

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

      1. +-commutative 47.6%

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

   6. Simplified 47.6%

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

      1. *-commutative 47.6%

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

      2. sqrt-prod 51.9%

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

      3. +-commutative 51.9%

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

   8. Applied egg-rr 51.9%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 6: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00058:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 0.00058)
   (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00058) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.00058d0) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00058) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 0.00058:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 0.00058)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.00058)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 0.00058], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $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 < 5.8e-4

   1. Initial program 74.9%

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

      1. +-commutative 74.9%

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

      2. associate-+r+ 74.9%

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

      3. *-commutative 74.9%

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

      4. +-commutative 74.9%

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

      5. associate-+l+ 74.9%

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

      6. *-commutative 74.9%

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

      7. *-commutative 74.9%

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

      8. *-commutative 74.9%

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

      9. distribute-lft-out 74.9%

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

   3. Simplified 74.9%

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

   if 5.8e-4 < y

   1. Initial program 57.4%

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

      1. +-commutative 57.4%

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

      2. associate-+r+ 57.4%

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

      3. *-commutative 57.4%

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

      4. +-commutative 57.4%

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

      5. associate-+l+ 57.4%

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

      6. *-commutative 57.4%

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

      7. *-commutative 57.4%

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

      8. *-commutative 57.4%

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

      9. distribute-lft-out 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in x around 0 30.9%

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

      1. *-commutative 30.9%

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

      2. sqrt-prod 52.4%

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

   6. Applied egg-rr 52.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00058:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 7: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Initial program 69.8%

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

   1. +-commutative 69.8%

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

   2. associate-+r+ 69.8%

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

   3. *-commutative 69.8%

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

   4. +-commutative 69.8%

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

   5. associate-+l+ 69.8%

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

   6. *-commutative 69.8%

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

   7. *-commutative 69.8%

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

   8. *-commutative 69.8%

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

   9. distribute-lft-out 69.9%

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

3. Simplified 69.9%

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

4. Final simplification 69.9%

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

Alternative 8: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{-242}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.52e-242) (* 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 <= 1.52e-242) {
		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 <= 1.52d-242) 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 <= 1.52e-242) {
		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 <= 1.52e-242:
		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 <= 1.52e-242)
		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 <= 1.52e-242)
		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, 1.52e-242], 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 < 1.5199999999999999e-242

   1. Initial program 70.0%

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

      1. +-commutative 70.0%

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

      2. associate-+r+ 70.0%

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

      3. *-commutative 70.0%

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

      4. +-commutative 70.0%

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

      5. associate-+l+ 70.0%

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

      6. *-commutative 70.0%

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

      7. *-commutative 70.0%

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

      8. *-commutative 70.0%

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

      9. distribute-lft-out 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in x around inf 53.1%

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

   if 1.5199999999999999e-242 < y

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. associate-+r+ 69.6%

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

      3. *-commutative 69.6%

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

      4. +-commutative 69.6%

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

      5. associate-+l+ 69.6%

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

      6. *-commutative 69.6%

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

      7. *-commutative 69.6%

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

      8. *-commutative 69.6%

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

      9. distribute-lft-out 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in x around 0 26.8%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{-242}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 9: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-303}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

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-303) (* 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-303) {
		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-303) 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-303) {
		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-303:
		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-303)
		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-303)
		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-303], 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 < 9.99999999999999931e-304

   1. Initial program 69.4%

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

      1. +-commutative 69.4%

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

      2. associate-+r+ 69.4%

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

      3. *-commutative 69.4%

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

      4. +-commutative 69.4%

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

      5. associate-+l+ 69.4%

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

      6. *-commutative 69.4%

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

      7. *-commutative 69.4%

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

      8. *-commutative 69.4%

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

      9. distribute-lft-out 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in x around inf 50.7%

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

   if 9.99999999999999931e-304 < y

   1. Initial program 70.2%

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

      1. +-commutative 70.2%

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

      2. associate-+r+ 70.2%

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

      3. *-commutative 70.2%

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

      4. +-commutative 70.2%

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

      5. associate-+l+ 70.2%

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

      6. *-commutative 70.2%

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

      7. *-commutative 70.2%

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

      8. *-commutative 70.2%

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

      9. distribute-lft-out 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in z around inf 48.4%

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

      1. +-commutative 48.4%

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

   6. Simplified 48.4%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-303}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 10: 67.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-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 <= -1e-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 <= (-1d-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 <= -1e-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 <= -1e-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 <= -1e-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 <= -1e-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, -1e-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 < -9.999999999999969e-311

   1. Initial program 69.1%

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

      1. +-commutative 69.1%

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

      2. associate-+r+ 69.1%

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

      3. *-commutative 69.1%

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

      4. +-commutative 69.1%

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

      5. associate-+l+ 69.1%

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

      6. *-commutative 69.1%

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

      7. *-commutative 69.1%

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

      8. *-commutative 69.1%

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

      9. distribute-lft-out 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in z around 0 28.9%

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

      1. *-commutative 28.9%

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

   6. Simplified 28.9%

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

   if -9.999999999999969e-311 < y

   1. Initial program 70.4%

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

      1. +-commutative 70.4%

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

      2. associate-+r+ 70.4%

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

      3. *-commutative 70.4%

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

      4. +-commutative 70.4%

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

      5. associate-+l+ 70.4%

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

      6. *-commutative 70.4%

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

      7. *-commutative 70.4%

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

      8. *-commutative 70.4%

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

      9. distribute-lft-out 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in x around 0 24.5%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 11: 35.1% 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 69.8%

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

   1. +-commutative 69.8%

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

   2. associate-+r+ 69.8%

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

   3. *-commutative 69.8%

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

   4. +-commutative 69.8%

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

   5. associate-+l+ 69.8%

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

   6. *-commutative 69.8%

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

   7. *-commutative 69.8%

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

   8. *-commutative 69.8%

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

   9. distribute-lft-out 69.9%

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

3. Simplified 69.9%

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

4. Taylor expanded in z around 0 25.9%

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

   1. *-commutative 25.9%

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

6. Simplified 25.9%

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

7. Final simplification 25.9%

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

Developer target: 82.5% 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 2023290 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< z 7.636950090573675e+176) (* 2.0 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25))) (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25)))) 2.0))

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