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

Percentage Accurate: 71.1% → 94.6%

Time: 14.4s

Alternatives: 14

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: 71.1% 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: 94.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(-x\right) - z\\ t_1 := {\left(e^{-0.5}\right)}^{\left(\mathsf{fma}\left(-1, \log \left(\frac{-1}{y}\right), \log t_0\right)\right)}\\ t_2 := \log \left(\sqrt[3]{\frac{-1}{x}}\right)\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot \frac{1}{\mathsf{fma}\left(0.5, \frac{z}{y} \cdot \frac{x \cdot t_1}{t_0}, t_1\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \left(t_2 + 2 \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- x) z))
        (t_1 (pow (exp -0.5) (fma -1.0 (log (/ -1.0 y)) (log t_0))))
        (t_2 (log (cbrt (/ -1.0 x)))))
   (if (<= y -7.5e+50)
     (* 2.0 (/ 1.0 (fma 0.5 (* (/ z y) (/ (* x t_1) t_0)) t_1)))
     (if (<= y -3.5e-180)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y 6.8e-275)
         (* 2.0 (exp (* 0.5 (- (log (- (- y) z)) (+ t_2 (* 2.0 t_2))))))
         (* 2.0 (* (sqrt z) (sqrt y))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = -x - z;
	double t_1 = pow(exp(-0.5), fma(-1.0, log((-1.0 / y)), log(t_0)));
	double t_2 = log(cbrt((-1.0 / x)));
	double tmp;
	if (y <= -7.5e+50) {
		tmp = 2.0 * (1.0 / fma(0.5, ((z / y) * ((x * t_1) / t_0)), t_1));
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= 6.8e-275) {
		tmp = 2.0 * exp((0.5 * (log((-y - z)) - (t_2 + (2.0 * t_2)))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(-x) - z)
	t_1 = exp(-0.5) ^ fma(-1.0, log(Float64(-1.0 / y)), log(t_0))
	t_2 = log(cbrt(Float64(-1.0 / x)))
	tmp = 0.0
	if (y <= -7.5e+50)
		tmp = Float64(2.0 * Float64(1.0 / fma(0.5, Float64(Float64(z / y) * Float64(Float64(x * t_1) / t_0)), t_1)));
	elseif (y <= -3.5e-180)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= 6.8e-275)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(Float64(-y) - z)) - Float64(t_2 + Float64(2.0 * t_2))))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[((-x) - z), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[-0.5], $MachinePrecision], N[(-1.0 * N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -7.5e+50], N[(2.0 * N[(1.0 / N[(0.5 * N[(N[(z / y), $MachinePrecision] * N[(N[(x * t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.5e-180], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e-275], N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[(t$95$2 + N[(2.0 * t$95$2), $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 4 regimes

2. if y < -7.4999999999999999e50

   1. Initial program 55.9%

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

      1. distribute-lft-out 55.9%

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

   3. Simplified 55.9%

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

      1. flip-+ 18.4%

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

      2. clear-num 18.3%

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

      3. pow2 18.3%

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

      4. pow2 18.3%

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

   5. Applied egg-rr 18.3%

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

      1. inv-pow 18.3%

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

      2. sqrt-pow1 18.3%

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

      3. clear-num 18.4%

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

      4. unpow2 18.4%

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

      5. unpow2 18.4%

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

      6. flip-+ 55.9%

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

      7. fma-udef 56.2%

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

      8. +-commutative 56.2%

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

      9. *-commutative 56.2%

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

      10. metadata-eval 56.2%

          \[\leadsto 2 \cdot {\left(\frac{1}{\mathsf{fma}\left(x, z + y, z \cdot y\right)}\right)}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 56.2%

      \[\leadsto 2 \cdot \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(x, z + y, z \cdot y\right)}\right)}^{-0.5}} \]
   8. Step-by-step derivation

      1. fma-def 55.9%

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

      2. distribute-rgt-in 55.9%

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

      3. +-commutative 55.9%

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

      4. associate-+r+ 55.9%

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

      5. distribute-lft-out 56.0%

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

      6. +-commutative 56.0%

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

      7. *-commutative 56.0%

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

      8. fma-def 56.2%

         \[\leadsto 2 \cdot {\left(\frac{1}{\color{blue}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5} \]

   9. Simplified 56.2%

      \[\leadsto 2 \cdot \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 56.1%

          \[\leadsto 2 \cdot {\color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}}^{-0.5} \]

       2. unpow-prod-down 55.9%

          \[\leadsto 2 \cdot \color{blue}{\left({\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5} \cdot {\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5}\right)} \]

       3. inv-pow 55.9%

          \[\leadsto 2 \cdot \left({\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-1}}}\right)}^{-0.5} \cdot {\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5}\right) \]

       4. sqrt-pow1 55.9%

          \[\leadsto 2 \cdot \left({\color{blue}{\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{-0.5} \cdot {\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5}\right) \]

       5. metadata-eval 55.9%

          \[\leadsto 2 \cdot \left({\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{\color{blue}{-0.5}}\right)}^{-0.5} \cdot {\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5}\right) \]

       6. inv-pow 55.9%

          \[\leadsto 2 \cdot \left({\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-1}}}\right)}^{-0.5}\right) \]

       7. sqrt-pow1 55.9%

          \[\leadsto 2 \cdot \left({\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}\right)}^{-0.5} \cdot {\color{blue}{\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{-0.5}\right) \]

       8. metadata-eval 55.9%

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

   11. Applied egg-rr 55.9%

       \[\leadsto 2 \cdot \color{blue}{\left({\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}\right)}^{-0.5} \cdot {\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}\right)}^{-0.5}\right)} \]
   12. Step-by-step derivation

       1. pow-sqr 56.1%

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

       2. metadata-eval 56.1%

          \[\leadsto 2 \cdot {\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}\right)}^{\color{blue}{-1}} \]

       3. unpow-1 56.1%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{{\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}}} \]

       4. *-commutative 56.1%

          \[\leadsto 2 \cdot \frac{1}{{\left(\mathsf{fma}\left(y, x, \color{blue}{z \cdot \left(y + x\right)}\right)\right)}^{-0.5}} \]

       5. +-commutative 56.1%

          \[\leadsto 2 \cdot \frac{1}{{\left(\mathsf{fma}\left(y, x, z \cdot \color{blue}{\left(x + y\right)}\right)\right)}^{-0.5}} \]

   13. Simplified 56.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{1}{{\left(\mathsf{fma}\left(y, x, z \cdot \left(x + y\right)\right)\right)}^{-0.5}}} \]

   14. Taylor expanded in y around -inf 85.0%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{0.5 \cdot \frac{z \cdot \left(e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{y}\right) + \log \left(-1 \cdot z + -1 \cdot x\right)\right)} \cdot x\right)}{y \cdot \left(-1 \cdot z + -1 \cdot x\right)} + e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{y}\right) + \log \left(-1 \cdot z + -1 \cdot x\right)\right)}}} \]
   15. Step-by-step derivation

       1. fma-def 85.0%

          \[\leadsto 2 \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{z \cdot \left(e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{y}\right) + \log \left(-1 \cdot z + -1 \cdot x\right)\right)} \cdot x\right)}{y \cdot \left(-1 \cdot z + -1 \cdot x\right)}, e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{y}\right) + \log \left(-1 \cdot z + -1 \cdot x\right)\right)}\right)}} \]

   16. Simplified 85.0%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{z}{y} \cdot \frac{x \cdot {\left(e^{-0.5}\right)}^{\left(\mathsf{fma}\left(-1, \log \left(\frac{-1}{y}\right), \log \left(-1 \cdot \left(z + x\right)\right)\right)\right)}}{-1 \cdot \left(z + x\right)}, {\left(e^{-0.5}\right)}^{\left(\mathsf{fma}\left(-1, \log \left(\frac{-1}{y}\right), \log \left(-1 \cdot \left(z + x\right)\right)\right)\right)}\right)}} \]

   if -7.4999999999999999e50 < y < -3.5000000000000001e-180

   1. Initial program 86.7%

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

      1. distribute-lft-out 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x around inf 55.8%

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

   if -3.5000000000000001e-180 < y < 6.79999999999999936e-275

   1. Initial program 70.8%

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

      1. distribute-lft-out 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in x around inf 68.5%

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

      1. pow1/2 68.5%

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

      2. metadata-eval 68.5%

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

      3. *-commutative 68.5%

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

      4. pow-to-exp 63.4%

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

      5. +-commutative 63.4%

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

      6. metadata-eval 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in x around -inf 46.1%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot \left(y + z\right)\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. +-commutative 46.1%

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

      2. mul-1-neg 46.1%

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

      3. unsub-neg 46.1%

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

      4. distribute-lft-in 46.1%

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

      5. mul-1-neg 46.1%

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

      6. unsub-neg 46.1%

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

      7. mul-1-neg 46.1%

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

   9. Simplified 46.1%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 46.1%

          \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \log \color{blue}{\left(\left(\sqrt[3]{\frac{-1}{x}} \cdot \sqrt[3]{\frac{-1}{x}}\right) \cdot \sqrt[3]{\frac{-1}{x}}\right)}\right) \cdot 0.5} \]

       2. log-prod 46.1%

          \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \color{blue}{\left(\log \left(\sqrt[3]{\frac{-1}{x}} \cdot \sqrt[3]{\frac{-1}{x}}\right) + \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)}\right) \cdot 0.5} \]

       3. pow2 46.1%

          \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \left(\log \color{blue}{\left({\left(\sqrt[3]{\frac{-1}{x}}\right)}^{2}\right)} + \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)\right) \cdot 0.5} \]

   11. Applied egg-rr 46.1%

       \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \color{blue}{\left(\log \left({\left(\sqrt[3]{\frac{-1}{x}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)}\right) \cdot 0.5} \]
   12. Step-by-step derivation

       1. log-pow 46.1%

          \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{-1}{x}}\right)} + \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)\right) \cdot 0.5} \]

   13. Simplified 46.1%

       \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{\frac{-1}{x}}\right) + \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)}\right) \cdot 0.5} \]

   if 6.79999999999999936e-275 < y

   1. Initial program 69.7%

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

      1. distribute-lft-out 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in x around 0 22.7%

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

      1. *-commutative 22.7%

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

      2. sqrt-prod 29.9%

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

   6. Applied egg-rr 29.9%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot \frac{1}{\mathsf{fma}\left(0.5, \frac{z}{y} \cdot \frac{x \cdot {\left(e^{-0.5}\right)}^{\left(\mathsf{fma}\left(-1, \log \left(\frac{-1}{y}\right), \log \left(\left(-x\right) - z\right)\right)\right)}}{\left(-x\right) - z}, {\left(e^{-0.5}\right)}^{\left(\mathsf{fma}\left(-1, \log \left(\frac{-1}{y}\right), \log \left(\left(-x\right) - z\right)\right)\right)}\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \left(\log \left(\sqrt[3]{\frac{-1}{x}}\right) + 2 \cdot \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2: 94.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \log \left(\sqrt[3]{\frac{-1}{x}}\right)\\ t_1 := \log \left(\left(-y\right) - z\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(t_1 - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(t_1 - \left(t_0 + 2 \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (cbrt (/ -1.0 x)))) (t_1 (log (- (- y) z))))
   (if (<= y -2.3e+50)
     (* 2.0 (exp (* 0.5 (- t_1 (log (/ -1.0 x))))))
     (if (<= y -3.5e-180)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y 3.4e-275)
         (* 2.0 (exp (* 0.5 (- t_1 (+ t_0 (* 2.0 t_0))))))
         (* 2.0 (* (sqrt z) (sqrt y))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = log(cbrt((-1.0 / x)));
	double t_1 = log((-y - z));
	double tmp;
	if (y <= -2.3e+50) {
		tmp = 2.0 * exp((0.5 * (t_1 - log((-1.0 / x)))));
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= 3.4e-275) {
		tmp = 2.0 * exp((0.5 * (t_1 - (t_0 + (2.0 * t_0)))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = Math.log(Math.cbrt((-1.0 / x)));
	double t_1 = Math.log((-y - z));
	double tmp;
	if (y <= -2.3e+50) {
		tmp = 2.0 * Math.exp((0.5 * (t_1 - Math.log((-1.0 / x)))));
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= 3.4e-275) {
		tmp = 2.0 * Math.exp((0.5 * (t_1 - (t_0 + (2.0 * t_0)))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = log(cbrt(Float64(-1.0 / x)))
	t_1 = log(Float64(Float64(-y) - z))
	tmp = 0.0
	if (y <= -2.3e+50)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(t_1 - log(Float64(-1.0 / x))))));
	elseif (y <= -3.5e-180)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= 3.4e-275)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(t_1 - Float64(t_0 + Float64(2.0 * t_0))))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -2.3e+50], N[(2.0 * N[Exp[N[(0.5 * N[(t$95$1 - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.5e-180], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e-275], N[(2.0 * N[Exp[N[(0.5 * N[(t$95$1 - N[(t$95$0 + N[(2.0 * t$95$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 4 regimes

2. if y < -2.29999999999999997e50

   1. Initial program 55.9%

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

      1. distribute-lft-out 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in x around inf 33.4%

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

      1. pow1/2 33.4%

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

      2. metadata-eval 33.4%

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

      3. *-commutative 33.4%

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

      4. pow-to-exp 31.1%

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

      5. +-commutative 31.1%

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

      6. metadata-eval 31.1%

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

   6. Applied egg-rr 31.1%

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

   7. Taylor expanded in x around -inf 48.6%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot \left(y + z\right)\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. +-commutative 48.6%

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

      2. mul-1-neg 48.6%

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

      3. unsub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

      5. mul-1-neg 48.6%

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

      6. unsub-neg 48.6%

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

      7. mul-1-neg 48.6%

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

   9. Simplified 48.6%

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

   if -2.29999999999999997e50 < y < -3.5000000000000001e-180

   1. Initial program 86.7%

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

      1. distribute-lft-out 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x around inf 55.8%

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

   if -3.5000000000000001e-180 < y < 3.39999999999999968e-275

   1. Initial program 70.8%

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

      1. distribute-lft-out 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in x around inf 68.5%

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

      1. pow1/2 68.5%

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

      2. metadata-eval 68.5%

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

      3. *-commutative 68.5%

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

      4. pow-to-exp 63.4%

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

      5. +-commutative 63.4%

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

      6. metadata-eval 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in x around -inf 46.1%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot \left(y + z\right)\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. +-commutative 46.1%

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

      2. mul-1-neg 46.1%

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

      3. unsub-neg 46.1%

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

      4. distribute-lft-in 46.1%

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

      5. mul-1-neg 46.1%

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

      6. unsub-neg 46.1%

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

      7. mul-1-neg 46.1%

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

   9. Simplified 46.1%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 46.1%

          \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \log \color{blue}{\left(\left(\sqrt[3]{\frac{-1}{x}} \cdot \sqrt[3]{\frac{-1}{x}}\right) \cdot \sqrt[3]{\frac{-1}{x}}\right)}\right) \cdot 0.5} \]

       2. log-prod 46.1%

          \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \color{blue}{\left(\log \left(\sqrt[3]{\frac{-1}{x}} \cdot \sqrt[3]{\frac{-1}{x}}\right) + \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)}\right) \cdot 0.5} \]

       3. pow2 46.1%

          \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \left(\log \color{blue}{\left({\left(\sqrt[3]{\frac{-1}{x}}\right)}^{2}\right)} + \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)\right) \cdot 0.5} \]

   11. Applied egg-rr 46.1%

       \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \color{blue}{\left(\log \left({\left(\sqrt[3]{\frac{-1}{x}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)}\right) \cdot 0.5} \]
   12. Step-by-step derivation

       1. log-pow 46.1%

          \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{-1}{x}}\right)} + \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)\right) \cdot 0.5} \]

   13. Simplified 46.1%

       \[\leadsto 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{\frac{-1}{x}}\right) + \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)}\right) \cdot 0.5} \]

   if 3.39999999999999968e-275 < y

   1. Initial program 69.7%

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

      1. distribute-lft-out 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in x around 0 22.7%

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

      1. *-commutative 22.7%

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

      2. sqrt-prod 29.9%

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

   6. Applied egg-rr 29.9%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \left(\log \left(\sqrt[3]{\frac{-1}{x}}\right) + 2 \cdot \log \left(\sqrt[3]{\frac{-1}{x}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3: 94.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.25e+50)
   (* 2.0 (exp (* 0.5 (- (log (- (- y) z)) (log (/ -1.0 x))))))
   (if (<= y -3.5e-180)
     (* 2.0 (sqrt (* x (+ y z))))
     (if (<= y -2e-310)
       (* 2.0 (pow (exp (* 0.25 (- (log (- x)) (log (/ -1.0 y))))) 2.0))
       (* 2.0 (* (sqrt z) (sqrt y)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.25e+50) {
		tmp = 2.0 * exp((0.5 * (log((-y - z)) - log((-1.0 / x)))));
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -2e-310) {
		tmp = 2.0 * pow(exp((0.25 * (log(-x) - log((-1.0 / y))))), 2.0);
	} 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 <= (-2.25d+50)) then
        tmp = 2.0d0 * exp((0.5d0 * (log((-y - z)) - log(((-1.0d0) / x)))))
    else if (y <= (-3.5d-180)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-2d-310)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log(-x) - log(((-1.0d0) / y))))) ** 2.0d0)
    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 <= -2.25e+50) {
		tmp = 2.0 * Math.exp((0.5 * (Math.log((-y - z)) - Math.log((-1.0 / x)))));
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -2e-310) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log(-x) - Math.log((-1.0 / y))))), 2.0);
	} 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 <= -2.25e+50:
		tmp = 2.0 * math.exp((0.5 * (math.log((-y - z)) - math.log((-1.0 / x)))))
	elif y <= -3.5e-180:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -2e-310:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log(-x) - math.log((-1.0 / y))))), 2.0)
	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 <= -2.25e+50)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))));
	elseif (y <= -3.5e-180)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -2e-310)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(-x)) - log(Float64(-1.0 / y))))) ^ 2.0));
	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 <= -2.25e+50)
		tmp = 2.0 * exp((0.5 * (log((-y - z)) - log((-1.0 / x)))));
	elseif (y <= -3.5e-180)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -2e-310)
		tmp = 2.0 * (exp((0.25 * (log(-x) - log((-1.0 / y))))) ^ 2.0);
	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, -2.25e+50], N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.5e-180], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2e-310], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[(-x)], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.25000000000000007e50

   1. Initial program 55.9%

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

      1. distribute-lft-out 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in x around inf 33.4%

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

      1. pow1/2 33.4%

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

      2. metadata-eval 33.4%

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

      3. *-commutative 33.4%

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

      4. pow-to-exp 31.1%

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

      5. +-commutative 31.1%

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

      6. metadata-eval 31.1%

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

   6. Applied egg-rr 31.1%

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

   7. Taylor expanded in x around -inf 48.6%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot \left(y + z\right)\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. +-commutative 48.6%

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

      2. mul-1-neg 48.6%

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

      3. unsub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

      5. mul-1-neg 48.6%

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

      6. unsub-neg 48.6%

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

      7. mul-1-neg 48.6%

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

   9. Simplified 48.6%

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

   if -2.25000000000000007e50 < y < -3.5000000000000001e-180

   1. Initial program 86.7%

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

      1. distribute-lft-out 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x around inf 55.8%

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

   if -3.5000000000000001e-180 < y < -1.999999999999994e-310

   1. Initial program 67.0%

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

      1. distribute-lft-out 67.0%

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

   3. Simplified 67.0%

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

      1. add-sqr-sqrt 66.7%

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

      2. pow2 66.7%

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

      3. pow1/2 66.7%

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

      4. sqrt-pow1 66.7%

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

      5. fma-def 66.7%

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

      6. metadata-eval 66.7%

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

   5. Applied egg-rr 66.7%

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

   6. Taylor expanded in z around 0 8.1%

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

   7. Taylor expanded in y around -inf 7.1%

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

   if -1.999999999999994e-310 < y

   1. Initial program 70.5%

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

      1. distribute-lft-out 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in x around 0 22.2%

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

      1. *-commutative 22.2%

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

      2. sqrt-prod 29.2%

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

   6. Applied egg-rr 29.2%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \frac{1}{e^{-0.5 \cdot \left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5.6e+52)
   (* 2.0 (exp (* 0.5 (- (log (- (- y) z)) (log (/ -1.0 x))))))
   (if (<= y -3.5e-180)
     (* 2.0 (sqrt (* x (+ y z))))
     (if (<= y -2e-310)
       (* 2.0 (/ 1.0 (exp (* -0.5 (- (log (- (- x) z)) (log (/ -1.0 y)))))))
       (* 2.0 (* (sqrt z) (sqrt y)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.6e+52) {
		tmp = 2.0 * exp((0.5 * (log((-y - z)) - log((-1.0 / x)))));
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -2e-310) {
		tmp = 2.0 * (1.0 / exp((-0.5 * (log((-x - z)) - log((-1.0 / y))))));
	} 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 <= (-5.6d+52)) then
        tmp = 2.0d0 * exp((0.5d0 * (log((-y - z)) - log(((-1.0d0) / x)))))
    else if (y <= (-3.5d-180)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-2d-310)) then
        tmp = 2.0d0 * (1.0d0 / exp(((-0.5d0) * (log((-x - z)) - log(((-1.0d0) / y))))))
    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 <= -5.6e+52) {
		tmp = 2.0 * Math.exp((0.5 * (Math.log((-y - z)) - Math.log((-1.0 / x)))));
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -2e-310) {
		tmp = 2.0 * (1.0 / Math.exp((-0.5 * (Math.log((-x - z)) - Math.log((-1.0 / y))))));
	} 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 <= -5.6e+52:
		tmp = 2.0 * math.exp((0.5 * (math.log((-y - z)) - math.log((-1.0 / x)))))
	elif y <= -3.5e-180:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -2e-310:
		tmp = 2.0 * (1.0 / math.exp((-0.5 * (math.log((-x - z)) - math.log((-1.0 / y))))))
	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 <= -5.6e+52)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))));
	elseif (y <= -3.5e-180)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -2e-310)
		tmp = Float64(2.0 * Float64(1.0 / exp(Float64(-0.5 * Float64(log(Float64(Float64(-x) - z)) - log(Float64(-1.0 / y)))))));
	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 <= -5.6e+52)
		tmp = 2.0 * exp((0.5 * (log((-y - z)) - log((-1.0 / x)))));
	elseif (y <= -3.5e-180)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -2e-310)
		tmp = 2.0 * (1.0 / exp((-0.5 * (log((-x - z)) - log((-1.0 / y))))));
	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, -5.6e+52], N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.5e-180], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2e-310], N[(2.0 * N[(1.0 / N[Exp[N[(-0.5 * N[(N[Log[N[((-x) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $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 4 regimes

2. if y < -5.6e52

   1. Initial program 55.9%

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

      1. distribute-lft-out 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in x around inf 33.4%

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

      1. pow1/2 33.4%

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

      2. metadata-eval 33.4%

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

      3. *-commutative 33.4%

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

      4. pow-to-exp 31.1%

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

      5. +-commutative 31.1%

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

      6. metadata-eval 31.1%

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

   6. Applied egg-rr 31.1%

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

   7. Taylor expanded in x around -inf 48.6%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot \left(y + z\right)\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. +-commutative 48.6%

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

      2. mul-1-neg 48.6%

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

      3. unsub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

      5. mul-1-neg 48.6%

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

      6. unsub-neg 48.6%

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

      7. mul-1-neg 48.6%

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

   9. Simplified 48.6%

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

   if -5.6e52 < y < -3.5000000000000001e-180

   1. Initial program 86.7%

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

      1. distribute-lft-out 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x around inf 55.8%

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

   if -3.5000000000000001e-180 < y < -1.999999999999994e-310

   1. Initial program 67.0%

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

      1. distribute-lft-out 67.0%

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

   3. Simplified 67.0%

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

      1. flip-+ 48.6%

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

      2. clear-num 48.4%

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

      3. pow2 48.4%

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

      4. pow2 48.4%

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

   5. Applied egg-rr 48.4%

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

      1. inv-pow 48.4%

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

      2. sqrt-pow1 48.5%

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

      3. clear-num 48.5%

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

      4. unpow2 48.5%

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

      5. unpow2 48.5%

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

      6. flip-+ 67.0%

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

      7. fma-udef 67.0%

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

      8. +-commutative 67.0%

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

      9. *-commutative 67.0%

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

      10. metadata-eval 67.0%

          \[\leadsto 2 \cdot {\left(\frac{1}{\mathsf{fma}\left(x, z + y, z \cdot y\right)}\right)}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 67.0%

      \[\leadsto 2 \cdot \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(x, z + y, z \cdot y\right)}\right)}^{-0.5}} \]
   8. Step-by-step derivation

      1. fma-def 67.0%

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

      2. distribute-rgt-in 67.0%

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

      3. +-commutative 67.0%

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

      4. associate-+r+ 67.0%

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

      5. distribute-lft-out 67.0%

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

      6. +-commutative 67.0%

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

      7. *-commutative 67.0%

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

      8. fma-def 67.0%

         \[\leadsto 2 \cdot {\left(\frac{1}{\color{blue}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5} \]

   9. Simplified 67.0%

      \[\leadsto 2 \cdot \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 66.2%

          \[\leadsto 2 \cdot {\color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}}^{-0.5} \]

       2. unpow-prod-down 66.1%

          \[\leadsto 2 \cdot \color{blue}{\left({\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5} \cdot {\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5}\right)} \]

       3. inv-pow 66.1%

          \[\leadsto 2 \cdot \left({\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-1}}}\right)}^{-0.5} \cdot {\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5}\right) \]

       4. sqrt-pow1 66.2%

          \[\leadsto 2 \cdot \left({\color{blue}{\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{-0.5} \cdot {\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5}\right) \]

       5. metadata-eval 66.2%

          \[\leadsto 2 \cdot \left({\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{\color{blue}{-0.5}}\right)}^{-0.5} \cdot {\left(\sqrt{\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5}\right) \]

       6. inv-pow 66.2%

          \[\leadsto 2 \cdot \left({\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-1}}}\right)}^{-0.5}\right) \]

       7. sqrt-pow1 66.6%

          \[\leadsto 2 \cdot \left({\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}\right)}^{-0.5} \cdot {\color{blue}{\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{-0.5}\right) \]

       8. metadata-eval 66.6%

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

   11. Applied egg-rr 66.6%

       \[\leadsto 2 \cdot \color{blue}{\left({\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}\right)}^{-0.5} \cdot {\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}\right)}^{-0.5}\right)} \]
   12. Step-by-step derivation

       1. pow-sqr 66.8%

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

       2. metadata-eval 66.8%

          \[\leadsto 2 \cdot {\left({\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}\right)}^{\color{blue}{-1}} \]

       3. unpow-1 66.8%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{{\left(\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)\right)}^{-0.5}}} \]

       4. *-commutative 66.8%

          \[\leadsto 2 \cdot \frac{1}{{\left(\mathsf{fma}\left(y, x, \color{blue}{z \cdot \left(y + x\right)}\right)\right)}^{-0.5}} \]

       5. +-commutative 66.8%

          \[\leadsto 2 \cdot \frac{1}{{\left(\mathsf{fma}\left(y, x, z \cdot \color{blue}{\left(x + y\right)}\right)\right)}^{-0.5}} \]

   13. Simplified 66.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{1}{{\left(\mathsf{fma}\left(y, x, z \cdot \left(x + y\right)\right)\right)}^{-0.5}}} \]

   14. Taylor expanded in y around -inf 18.4%

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

   if -1.999999999999994e-310 < y

   1. Initial program 70.5%

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

      1. distribute-lft-out 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in x around 0 22.2%

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

      1. *-commutative 22.2%

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

      2. sqrt-prod 29.2%

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

   6. Applied egg-rr 29.2%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \frac{1}{e^{-0.5 \cdot \left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 5: 94.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot e^{0.5 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{if}\;y \leq -7 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 2.0 (exp (* 0.5 (- (log (- x)) (log (/ -1.0 y))))))))
   (if (<= y -7e+50)
     t_0
     (if (<= y -3.5e-180)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y -2e-310) t_0 (* 2.0 (* (sqrt z) (sqrt y))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 2.0 * exp((0.5 * (log(-x) - log((-1.0 / y)))));
	double tmp;
	if (y <= -7e+50) {
		tmp = t_0;
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -2e-310) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * exp((0.5d0 * (log(-x) - log(((-1.0d0) / y)))))
    if (y <= (-7d+50)) then
        tmp = t_0
    else if (y <= (-3.5d-180)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-2d-310)) then
        tmp = t_0
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 2.0 * Math.exp((0.5 * (Math.log(-x) - Math.log((-1.0 / y)))));
	double tmp;
	if (y <= -7e+50) {
		tmp = t_0;
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -2e-310) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 2.0 * math.exp((0.5 * (math.log(-x) - math.log((-1.0 / y)))))
	tmp = 0
	if y <= -7e+50:
		tmp = t_0
	elif y <= -3.5e-180:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -2e-310:
		tmp = t_0
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(-x)) - log(Float64(-1.0 / y))))))
	tmp = 0.0
	if (y <= -7e+50)
		tmp = t_0;
	elseif (y <= -3.5e-180)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -2e-310)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 2.0 * exp((0.5 * (log(-x) - log((-1.0 / y)))));
	tmp = 0.0;
	if (y <= -7e+50)
		tmp = t_0;
	elseif (y <= -3.5e-180)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -2e-310)
		tmp = t_0;
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[(-x)], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e+50], t$95$0, If[LessEqual[y, -3.5e-180], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2e-310], t$95$0, N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.00000000000000012e50 or -3.5000000000000001e-180 < y < -1.999999999999994e-310

   1. Initial program 58.7%

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

      1. distribute-lft-out 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in x around inf 41.1%

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

      1. pow1/2 41.2%

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

      2. metadata-eval 41.2%

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

      3. *-commutative 41.2%

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

      4. pow-to-exp 38.3%

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

      5. +-commutative 38.3%

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

      6. metadata-eval 38.3%

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

   6. Applied egg-rr 38.3%

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

   7. Taylor expanded in y around -inf 35.5%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{y}\right) + \log \left(-1 \cdot x\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. +-commutative 35.5%

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

      2. mul-1-neg 35.5%

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

      3. unsub-neg 35.5%

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

      4. mul-1-neg 35.5%

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

   9. Simplified 35.5%

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

   if -7.00000000000000012e50 < y < -3.5000000000000001e-180

   1. Initial program 86.7%

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

      1. distribute-lft-out 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x around inf 55.8%

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

   if -1.999999999999994e-310 < y

   1. Initial program 70.5%

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

      1. distribute-lft-out 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in x around 0 22.2%

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

      1. *-commutative 22.2%

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

      2. sqrt-prod 29.2%

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

   6. Applied egg-rr 29.2%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 6: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+51}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1e+51)
   (* 2.0 (exp (* 0.5 (- (log (- (- y) z)) (log (/ -1.0 x))))))
   (if (<= y -3.5e-180)
     (* 2.0 (sqrt (* x (+ y z))))
     (if (<= y -2e-310)
       (* 2.0 (exp (* 0.5 (- (log (- x)) (log (/ -1.0 y))))))
       (* 2.0 (* (sqrt z) (sqrt y)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+51) {
		tmp = 2.0 * exp((0.5 * (log((-y - z)) - log((-1.0 / x)))));
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -2e-310) {
		tmp = 2.0 * exp((0.5 * (log(-x) - log((-1.0 / y)))));
	} 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 <= (-1d+51)) then
        tmp = 2.0d0 * exp((0.5d0 * (log((-y - z)) - log(((-1.0d0) / x)))))
    else if (y <= (-3.5d-180)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-2d-310)) then
        tmp = 2.0d0 * exp((0.5d0 * (log(-x) - log(((-1.0d0) / y)))))
    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 <= -1e+51) {
		tmp = 2.0 * Math.exp((0.5 * (Math.log((-y - z)) - Math.log((-1.0 / x)))));
	} else if (y <= -3.5e-180) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -2e-310) {
		tmp = 2.0 * Math.exp((0.5 * (Math.log(-x) - Math.log((-1.0 / y)))));
	} 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 <= -1e+51:
		tmp = 2.0 * math.exp((0.5 * (math.log((-y - z)) - math.log((-1.0 / x)))))
	elif y <= -3.5e-180:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -2e-310:
		tmp = 2.0 * math.exp((0.5 * (math.log(-x) - math.log((-1.0 / y)))))
	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 <= -1e+51)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))));
	elseif (y <= -3.5e-180)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -2e-310)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(-x)) - log(Float64(-1.0 / y))))));
	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 <= -1e+51)
		tmp = 2.0 * exp((0.5 * (log((-y - z)) - log((-1.0 / x)))));
	elseif (y <= -3.5e-180)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -2e-310)
		tmp = 2.0 * exp((0.5 * (log(-x) - log((-1.0 / y)))));
	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, -1e+51], N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.5e-180], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2e-310], N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[(-x)], $MachinePrecision] - N[Log[N[(-1.0 / y), $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 4 regimes

2. if y < -1e51

   1. Initial program 55.9%

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

      1. distribute-lft-out 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in x around inf 33.4%

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

      1. pow1/2 33.4%

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

      2. metadata-eval 33.4%

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

      3. *-commutative 33.4%

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

      4. pow-to-exp 31.1%

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

      5. +-commutative 31.1%

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

      6. metadata-eval 31.1%

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

   6. Applied egg-rr 31.1%

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

   7. Taylor expanded in x around -inf 48.6%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot \left(y + z\right)\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. +-commutative 48.6%

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

      2. mul-1-neg 48.6%

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

      3. unsub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

      5. mul-1-neg 48.6%

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

      6. unsub-neg 48.6%

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

      7. mul-1-neg 48.6%

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

   9. Simplified 48.6%

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

   if -1e51 < y < -3.5000000000000001e-180

   1. Initial program 86.7%

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

      1. distribute-lft-out 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x around inf 55.8%

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

   if -3.5000000000000001e-180 < y < -1.999999999999994e-310

   1. Initial program 67.0%

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

      1. distribute-lft-out 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in x around inf 64.4%

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

      1. pow1/2 64.4%

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

      2. metadata-eval 64.4%

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

      3. *-commutative 64.4%

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

      4. pow-to-exp 59.7%

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

      5. +-commutative 59.7%

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

      6. metadata-eval 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in y around -inf 7.1%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{y}\right) + \log \left(-1 \cdot x\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. +-commutative 7.1%

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

      2. mul-1-neg 7.1%

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

      3. unsub-neg 7.1%

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

      4. mul-1-neg 7.1%

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

   9. Simplified 7.1%

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

   if -1.999999999999994e-310 < y

   1. Initial program 70.5%

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

      1. distribute-lft-out 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in x around 0 22.2%

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

      1. *-commutative 22.2%

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

      2. sqrt-prod 29.2%

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

   6. Applied egg-rr 29.2%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+51}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 7: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 210000000000:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y, x \cdot \left(y + z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 210000000000.0)
   (* 2.0 (sqrt (fma z y (* x (+ y z)))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 210000000000.0) {
		tmp = 2.0 * sqrt(fma(z, y, (x * (y + z))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 210000000000.0)
		tmp = Float64(2.0 * sqrt(fma(z, y, Float64(x * Float64(y + z)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 210000000000.0], N[(2.0 * N[Sqrt[N[(z * y + N[(x * N[(y + z), $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 < 2.1e11

   1. Initial program 74.2%

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

      1. distribute-lft-out 74.2%

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

   3. Simplified 74.2%

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

      1. +-commutative 74.2%

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

      2. *-commutative 74.2%

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

      3. fma-def 74.3%

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

   5. Applied egg-rr 74.3%

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

   if 2.1e11 < y

   1. Initial program 51.1%

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

      1. distribute-lft-out 51.2%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in x around 0 25.0%

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

      1. *-commutative 25.0%

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

      2. sqrt-prod 40.1%

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

   6. Applied egg-rr 40.1%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 210000000000:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y, x \cdot \left(y + z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 8: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 530000000000:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 530000000000.0)
   (* 2.0 (sqrt (+ (* x (+ y z)) (* y z))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 530000000000.0) {
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 530000000000.0d0) then
        tmp = 2.0d0 * sqrt(((x * (y + z)) + (y * z)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 530000000000.0) {
		tmp = 2.0 * Math.sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 530000000000.0:
		tmp = 2.0 * math.sqrt(((x * (y + z)) + (y * z)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 530000000000.0)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * Float64(y + z)) + Float64(y * z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 530000000000.0)
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 530000000000.0], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.3e11

   1. Initial program 74.2%

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

      1. distribute-lft-out 74.2%

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

   3. Simplified 74.2%

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

   if 5.3e11 < y

   1. Initial program 51.1%

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

      1. distribute-lft-out 51.2%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in x around 0 25.0%

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

      1. *-commutative 25.0%

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

      2. sqrt-prod 40.1%

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

   6. Applied egg-rr 40.1%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 530000000000:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 9: 71.2% accurate, 1.0× speedup

Math

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

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 (+ y z)) (* y z)))))

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 69.0%

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

   1. distribute-lft-out 69.0%

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

3. Simplified 69.0%

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

4. Final simplification 69.0%

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

Alternative 10: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ 2 \cdot {\left(\frac{\frac{1}{y}}{z + x}\right)}^{-0.5} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* 2.0 (pow (/ (/ 1.0 y) (+ z x)) -0.5)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 2.0 * pow(((1.0 / y) / (z + x)), -0.5);
}

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 * (((1.0d0 / y) / (z + x)) ** (-0.5d0))
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return 2.0 * Math.pow(((1.0 / y) / (z + x)), -0.5);
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 2.0 * math.pow(((1.0 / y) / (z + x)), -0.5)

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(2.0 * (Float64(Float64(1.0 / y) / Float64(z + x)) ^ -0.5))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 2.0 * (((1.0 / y) / (z + x)) ^ -0.5);
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[Power[N[(N[(1.0 / y), $MachinePrecision] / N[(z + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

   1. distribute-lft-out 69.0%

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

3. Simplified 69.0%

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

   1. flip-+ 34.7%

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

   2. clear-num 34.7%

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

   3. pow2 34.7%

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

   4. pow2 34.7%

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

5. Applied egg-rr 34.7%

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

   1. inv-pow 34.7%

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

   2. sqrt-pow1 34.7%

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

   3. clear-num 34.6%

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

   4. unpow2 34.6%

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

   5. unpow2 34.6%

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

   6. flip-+ 68.8%

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

   7. fma-udef 69.0%

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

   8. +-commutative 69.0%

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

   9. *-commutative 69.0%

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

   10. metadata-eval 69.0%

       \[\leadsto 2 \cdot {\left(\frac{1}{\mathsf{fma}\left(x, z + y, z \cdot y\right)}\right)}^{\color{blue}{-0.5}} \]

7. Applied egg-rr 69.0%

   \[\leadsto 2 \cdot \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(x, z + y, z \cdot y\right)}\right)}^{-0.5}} \]
8. Step-by-step derivation

   1. fma-def 68.8%

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

   2. distribute-rgt-in 68.8%

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

   3. +-commutative 68.8%

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

   4. associate-+r+ 68.8%

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

   5. distribute-lft-out 68.8%

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

   6. +-commutative 68.8%

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

   7. *-commutative 68.8%

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

   8. fma-def 69.0%

      \[\leadsto 2 \cdot {\left(\frac{1}{\color{blue}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}}\right)}^{-0.5} \]

9. Simplified 69.0%

   \[\leadsto 2 \cdot \color{blue}{{\left(\frac{1}{\mathsf{fma}\left(y, x, \left(y + x\right) \cdot z\right)}\right)}^{-0.5}} \]

10. Taylor expanded in y around inf 44.9%

    \[\leadsto 2 \cdot {\color{blue}{\left(\frac{1}{y \cdot \left(z + x\right)}\right)}}^{-0.5} \]
11. Step-by-step derivation

    1. associate-/r* 45.6%

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

12. Simplified 45.6%

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

13. Final simplification 45.6%

    \[\leadsto 2 \cdot {\left(\frac{\frac{1}{y}}{z + x}\right)}^{-0.5} \]

Alternative 11: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-264}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e-264) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e-264) {
		tmp = 2.0 * sqrt((y * x));
	} 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 <= (-4.8d-264)) then
        tmp = 2.0d0 * sqrt((y * x))
    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 <= -4.8e-264) {
		tmp = 2.0 * Math.sqrt((y * x));
	} 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 <= -4.8e-264:
		tmp = 2.0 * math.sqrt((y * x))
	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 <= -4.8e-264)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	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 <= -4.8e-264)
		tmp = 2.0 * sqrt((y * x));
	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, -4.8e-264], N[(2.0 * N[Sqrt[N[(y * x), $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 < -4.7999999999999997e-264

   1. Initial program 68.3%

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

      1. distribute-lft-out 68.3%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in z around 0 28.6%

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

   if -4.7999999999999997e-264 < 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. distribute-lft-out 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in z around inf 52.1%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-264}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 12: 71.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-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 <= -2e-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 <= (-2d-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 <= -2e-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 <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.99999999999999986e-303

   1. Initial program 67.6%

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

      1. distribute-lft-out 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in x around inf 45.8%

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

   if -1.99999999999999986e-303 < y

   1. Initial program 70.5%

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

      1. distribute-lft-out 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in z around inf 51.6%

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

4. Final simplification 48.6%

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

Alternative 13: 68.9% accurate, 1.1× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-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 <= -2e-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 <= (-2d-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 <= -2e-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 <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.999999999999994e-310

   1. Initial program 67.6%

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

      1. distribute-lft-out 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in z around 0 26.9%

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

   if -1.999999999999994e-310 < y

   1. Initial program 70.5%

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

      1. distribute-lft-out 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in x around 0 22.2%

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

4. Final simplification 24.7%

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

Alternative 14: 36.0% 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 69.0%

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

   1. distribute-lft-out 69.0%

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

3. Simplified 69.0%

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

4. Taylor expanded in z around 0 24.5%

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

5. Final simplification 24.5%

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

Developer target: 83.3% 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 2023229 
(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)))))