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

Percentage Accurate: 70.3% → 94.8%

Time: 11.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \log \left(\frac{-1}{x}\right)\\ t_1 := -t_0\\ t_2 := \log \left(\left(-z\right) - y\right)\\ t_3 := \sqrt{t_2}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+32}:\\ \;\;\;\;2 \cdot e^{\left(\mathsf{fma}\left(t_3, t_3, t_1\right) + \mathsf{fma}\left(t_1, 1, t_0\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-209}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(t_2 - t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

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 (/ -1.0 x)))
        (t_1 (- t_0))
        (t_2 (log (- (- z) y)))
        (t_3 (sqrt t_2)))
   (if (<= y -2.8e+32)
     (* 2.0 (exp (* (+ (fma t_3 t_3 t_1) (fma t_1 1.0 t_0)) 0.5)))
     (if (<= y -2.8e-209)
       (* 2.0 (sqrt (+ (* z (+ y x)) (* y x))))
       (if (<= y 1.95e-271)
         (* 2.0 (exp (* 0.5 (- t_2 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((-1.0 / x));
	double t_1 = -t_0;
	double t_2 = log((-z - y));
	double t_3 = sqrt(t_2);
	double tmp;
	if (y <= -2.8e+32) {
		tmp = 2.0 * exp(((fma(t_3, t_3, t_1) + fma(t_1, 1.0, t_0)) * 0.5));
	} else if (y <= -2.8e-209) {
		tmp = 2.0 * sqrt(((z * (y + x)) + (y * x)));
	} else if (y <= 1.95e-271) {
		tmp = 2.0 * exp((0.5 * (t_2 - t_0)));
	} 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 = log(Float64(-1.0 / x))
	t_1 = Float64(-t_0)
	t_2 = log(Float64(Float64(-z) - y))
	t_3 = sqrt(t_2)
	tmp = 0.0
	if (y <= -2.8e+32)
		tmp = Float64(2.0 * exp(Float64(Float64(fma(t_3, t_3, t_1) + fma(t_1, 1.0, t_0)) * 0.5)));
	elseif (y <= -2.8e-209)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z * Float64(y + x)) + Float64(y * x))));
	elseif (y <= 1.95e-271)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(t_2 - 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[(-1.0 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[Log[N[((-z) - y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, If[LessEqual[y, -2.8e+32], N[(2.0 * N[Exp[N[(N[(N[(t$95$3 * t$95$3 + t$95$1), $MachinePrecision] + N[(t$95$1 * 1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.8e-209], N[(2.0 * N[Sqrt[N[(N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.95e-271], N[(2.0 * N[Exp[N[(0.5 * N[(t$95$2 - t$95$0), $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 < -2.8e32

   1. Initial program 48.9%

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

      1. distribute-lft-out 48.9%

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

   3. Simplified 48.9%

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

      1. distribute-lft-in 48.9%

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

      2. associate-+l+ 48.9%

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

      3. +-commutative 48.9%

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

      4. distribute-rgt-in 48.9%

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

      5. fma-udef 49.1%

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

      6. pow1/2 49.1%

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

      7. pow-to-exp 45.8%

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

      8. fma-udef 45.6%

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

      9. distribute-rgt-in 45.6%

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

      10. +-commutative 45.6%

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

      11. fma-def 45.6%

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

      12. distribute-rgt-in 45.6%

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

   5. Applied egg-rr 45.6%

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

   6. Taylor expanded in x around -inf 47.9%

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

      1. +-commutative 47.9%

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

      2. mul-1-neg 47.9%

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

      3. unsub-neg 47.9%

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

      4. mul-1-neg 47.9%

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

      5. unsub-neg 47.9%

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

      6. mul-1-neg 47.9%

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

   8. Simplified 47.9%

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

      1. add-sqr-sqrt 47.6%

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

      2. *-un-lft-identity 47.6%

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

      3. prod-diff 47.7%

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

   10. Applied egg-rr 47.7%

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

   if -2.8e32 < y < -2.80000000000000012e-209

   1. Initial program 77.1%

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

      1. associate-+l+ 77.1%

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

      2. +-commutative 77.1%

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

      3. distribute-rgt-out 77.1%

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

      4. fma-def 77.1%

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

   3. Simplified 77.1%

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

      1. fma-udef 77.1%

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

   5. Applied egg-rr 77.1%

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

   if -2.80000000000000012e-209 < y < 1.94999999999999999e-271

   1. Initial program 64.7%

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

      1. distribute-lft-out 64.7%

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

   3. Simplified 64.7%

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

      1. distribute-lft-in 64.7%

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

      2. associate-+l+ 64.7%

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

      3. +-commutative 64.7%

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

      4. distribute-rgt-in 64.7%

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

      5. fma-udef 64.7%

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

      6. pow1/2 64.7%

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

      7. pow-to-exp 60.0%

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

      8. fma-udef 60.0%

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

      9. distribute-rgt-in 60.0%

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

      10. +-commutative 60.0%

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

      11. fma-def 60.0%

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

      12. distribute-rgt-in 60.0%

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

   5. Applied egg-rr 60.0%

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

   6. Taylor expanded in x around -inf 54.5%

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

      1. +-commutative 54.5%

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

      2. mul-1-neg 54.5%

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

      3. unsub-neg 54.5%

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

      4. mul-1-neg 54.5%

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

      5. unsub-neg 54.5%

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

      6. mul-1-neg 54.5%

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

   8. Simplified 54.5%

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

   if 1.94999999999999999e-271 < y

   1. Initial program 76.1%

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

      1. distribute-lft-out 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in x around 0 35.1%

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

      1. pow1/2 35.1%

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

      2. *-commutative 35.1%

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

      3. metadata-eval 35.1%

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

      4. unpow-prod-down 44.7%

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

      5. metadata-eval 44.7%

         \[\leadsto 2 \cdot \left({z}^{\color{blue}{0.5}} \cdot {y}^{\left(\sqrt{0.25}\right)}\right) \]

      6. pow1/2 44.7%

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

      7. metadata-eval 44.7%

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

      8. pow1/2 44.7%

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

   6. Applied egg-rr 44.7%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+32}:\\ \;\;\;\;2 \cdot e^{\left(\mathsf{fma}\left(\sqrt{\log \left(\left(-z\right) - y\right)}, \sqrt{\log \left(\left(-z\right) - y\right)}, -\log \left(\frac{-1}{x}\right)\right) + \mathsf{fma}\left(-\log \left(\frac{-1}{x}\right), 1, \log \left(\frac{-1}{x}\right)\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-209}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2: 94.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot e^{0.5 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-208}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-271}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

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 (- (- z) y)) (log (/ -1.0 x))))))))
   (if (<= y -3.2e+31)
     t_0
     (if (<= y -3.5e-208)
       (* 2.0 (sqrt (+ (* z (+ y x)) (* y x))))
       (if (<= y 4.7e-271) 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((-z - y)) - log((-1.0 / x)))));
	double tmp;
	if (y <= -3.2e+31) {
		tmp = t_0;
	} else if (y <= -3.5e-208) {
		tmp = 2.0 * sqrt(((z * (y + x)) + (y * x)));
	} else if (y <= 4.7e-271) {
		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((-z - y)) - log(((-1.0d0) / x)))))
    if (y <= (-3.2d+31)) then
        tmp = t_0
    else if (y <= (-3.5d-208)) then
        tmp = 2.0d0 * sqrt(((z * (y + x)) + (y * x)))
    else if (y <= 4.7d-271) 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((-z - y)) - Math.log((-1.0 / x)))));
	double tmp;
	if (y <= -3.2e+31) {
		tmp = t_0;
	} else if (y <= -3.5e-208) {
		tmp = 2.0 * Math.sqrt(((z * (y + x)) + (y * x)));
	} else if (y <= 4.7e-271) {
		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((-z - y)) - math.log((-1.0 / x)))))
	tmp = 0
	if y <= -3.2e+31:
		tmp = t_0
	elif y <= -3.5e-208:
		tmp = 2.0 * math.sqrt(((z * (y + x)) + (y * x)))
	elif y <= 4.7e-271:
		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(Float64(-z) - y)) - log(Float64(-1.0 / x))))))
	tmp = 0.0
	if (y <= -3.2e+31)
		tmp = t_0;
	elseif (y <= -3.5e-208)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z * Float64(y + x)) + Float64(y * x))));
	elseif (y <= 4.7e-271)
		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((-z - y)) - log((-1.0 / x)))));
	tmp = 0.0;
	if (y <= -3.2e+31)
		tmp = t_0;
	elseif (y <= -3.5e-208)
		tmp = 2.0 * sqrt(((z * (y + x)) + (y * x)));
	elseif (y <= 4.7e-271)
		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[N[((-z) - y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+31], t$95$0, If[LessEqual[y, -3.5e-208], N[(2.0 * N[Sqrt[N[(N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.7e-271], 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 < -3.2000000000000001e31 or -3.49999999999999991e-208 < y < 4.70000000000000005e-271

   1. Initial program 54.2%

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

      1. distribute-lft-out 54.2%

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

   3. Simplified 54.2%

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

      1. distribute-lft-in 54.2%

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

      2. associate-+l+ 54.2%

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

      3. +-commutative 54.2%

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

      4. distribute-rgt-in 54.2%

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

      5. fma-udef 54.3%

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

      6. pow1/2 54.3%

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

      7. pow-to-exp 50.5%

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

      8. fma-udef 50.4%

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

      9. distribute-rgt-in 50.4%

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

      10. +-commutative 50.4%

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

      11. fma-def 50.4%

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

      12. distribute-rgt-in 50.4%

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

   5. Applied egg-rr 50.4%

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

   6. Taylor expanded in x around -inf 50.1%

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

      1. +-commutative 50.1%

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

      2. mul-1-neg 50.1%

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

      3. unsub-neg 50.1%

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

      4. mul-1-neg 50.1%

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

      5. unsub-neg 50.1%

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

      6. mul-1-neg 50.1%

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

   8. Simplified 50.1%

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

   if -3.2000000000000001e31 < y < -3.49999999999999991e-208

   1. Initial program 77.1%

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

      1. associate-+l+ 77.1%

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

      2. +-commutative 77.1%

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

      3. distribute-rgt-out 77.1%

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

      4. fma-def 77.1%

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

   3. Simplified 77.1%

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

      1. fma-udef 77.1%

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

   5. Applied egg-rr 77.1%

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

   if 4.70000000000000005e-271 < y

   1. Initial program 76.1%

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

      1. distribute-lft-out 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in x around 0 35.1%

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

      1. pow1/2 35.1%

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

      2. *-commutative 35.1%

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

      3. metadata-eval 35.1%

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

      4. unpow-prod-down 44.7%

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

      5. metadata-eval 44.7%

         \[\leadsto 2 \cdot \left({z}^{\color{blue}{0.5}} \cdot {y}^{\left(\sqrt{0.25}\right)}\right) \]

      6. pow1/2 44.7%

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

      7. metadata-eval 44.7%

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

      8. pow1/2 44.7%

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

   6. Applied egg-rr 44.7%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+31}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-208}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3: 94.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot e^{0.5 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-209}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

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 (- y)) (log (/ -1.0 x))))))))
   (if (<= y -1.4e+34)
     t_0
     (if (<= y -2.8e-209)
       (* 2.0 (sqrt (+ (* z (+ y x)) (* y x))))
       (if (<= y -1e-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(-y) - log((-1.0 / x)))));
	double tmp;
	if (y <= -1.4e+34) {
		tmp = t_0;
	} else if (y <= -2.8e-209) {
		tmp = 2.0 * sqrt(((z * (y + x)) + (y * x)));
	} else if (y <= -1e-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(-y) - log(((-1.0d0) / x)))))
    if (y <= (-1.4d+34)) then
        tmp = t_0
    else if (y <= (-2.8d-209)) then
        tmp = 2.0d0 * sqrt(((z * (y + x)) + (y * x)))
    else if (y <= (-1d-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(-y) - Math.log((-1.0 / x)))));
	double tmp;
	if (y <= -1.4e+34) {
		tmp = t_0;
	} else if (y <= -2.8e-209) {
		tmp = 2.0 * Math.sqrt(((z * (y + x)) + (y * x)));
	} else if (y <= -1e-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(-y) - math.log((-1.0 / x)))))
	tmp = 0
	if y <= -1.4e+34:
		tmp = t_0
	elif y <= -2.8e-209:
		tmp = 2.0 * math.sqrt(((z * (y + x)) + (y * x)))
	elif y <= -1e-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(-y)) - log(Float64(-1.0 / x))))))
	tmp = 0.0
	if (y <= -1.4e+34)
		tmp = t_0;
	elseif (y <= -2.8e-209)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z * Float64(y + x)) + Float64(y * x))));
	elseif (y <= -1e-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(-y) - log((-1.0 / x)))));
	tmp = 0.0;
	if (y <= -1.4e+34)
		tmp = t_0;
	elseif (y <= -2.8e-209)
		tmp = 2.0 * sqrt(((z * (y + x)) + (y * x)));
	elseif (y <= -1e-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[(-y)], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4e+34], t$95$0, If[LessEqual[y, -2.8e-209], N[(2.0 * N[Sqrt[N[(N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1e-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 < -1.40000000000000004e34 or -2.80000000000000012e-209 < y < -9.999999999999969e-311

   1. Initial program 55.1%

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

      1. distribute-lft-out 55.1%

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

   3. Simplified 55.1%

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

      1. distribute-lft-in 55.1%

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

      2. associate-+l+ 55.1%

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

      3. +-commutative 55.1%

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

      4. distribute-rgt-in 55.1%

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

      5. fma-udef 55.2%

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

      6. pow1/2 55.2%

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

      7. pow-to-exp 51.4%

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

      8. fma-udef 51.3%

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

      9. distribute-rgt-in 51.3%

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

      10. +-commutative 51.3%

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

      11. fma-def 51.3%

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

      12. distribute-rgt-in 51.3%

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

   5. Applied egg-rr 51.3%

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

   6. Taylor expanded in x around -inf 51.6%

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

      1. +-commutative 51.6%

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

      2. mul-1-neg 51.6%

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

      3. unsub-neg 51.6%

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

      4. mul-1-neg 51.6%

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

      5. unsub-neg 51.6%

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

      6. mul-1-neg 51.6%

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

   8. Simplified 51.6%

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

   9. Taylor expanded in z around 0 35.7%

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

   if -1.40000000000000004e34 < y < -2.80000000000000012e-209

   1. Initial program 75.8%

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

      1. associate-+l+ 75.8%

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

      2. +-commutative 75.8%

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

      3. distribute-rgt-out 75.8%

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

      4. fma-def 75.7%

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

   3. Simplified 75.7%

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

      1. fma-udef 75.8%

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

   5. Applied egg-rr 75.8%

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

   if -9.999999999999969e-311 < y

   1. Initial program 74.7%

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

      1. distribute-lft-out 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in x around 0 33.3%

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

      1. pow1/2 33.3%

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

      2. *-commutative 33.3%

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

      3. metadata-eval 33.3%

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

      4. unpow-prod-down 42.2%

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

      5. metadata-eval 42.2%

         \[\leadsto 2 \cdot \left({z}^{\color{blue}{0.5}} \cdot {y}^{\left(\sqrt{0.25}\right)}\right) \]

      6. pow1/2 42.2%

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

      7. metadata-eval 42.2%

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

      8. pow1/2 42.2%

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

   6. Applied egg-rr 42.2%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+34}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-209}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y + x, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.7e-271)
   (* 2.0 (sqrt (fma z (+ y x) (* y x))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.7e-271) {
		tmp = 2.0 * sqrt(fma(z, (y + x), (y * x)));
	} 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 <= 4.7e-271)
		tmp = Float64(2.0 * sqrt(fma(z, Float64(y + x), Float64(y * x))));
	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, 4.7e-271], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.70000000000000005e-271

   1. Initial program 63.8%

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

      1. associate-+l+ 63.8%

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

      2. +-commutative 63.8%

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

      3. distribute-rgt-out 63.8%

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

      4. fma-def 63.8%

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

   3. Simplified 63.8%

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

   if 4.70000000000000005e-271 < y

   1. Initial program 76.1%

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

      1. distribute-lft-out 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in x around 0 35.1%

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

      1. pow1/2 35.1%

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

      2. *-commutative 35.1%

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

      3. metadata-eval 35.1%

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

      4. unpow-prod-down 44.7%

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

      5. metadata-eval 44.7%

         \[\leadsto 2 \cdot \left({z}^{\color{blue}{0.5}} \cdot {y}^{\left(\sqrt{0.25}\right)}\right) \]

      6. pow1/2 44.7%

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

      7. metadata-eval 44.7%

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

      8. pow1/2 44.7%

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

   6. Applied egg-rr 44.7%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y + x, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 5: 82.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.7e-271)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.7e-271) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.7d-271) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.7e-271) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 4.7e-271:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.7e-271)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.7e-271)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 4.7e-271], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.70000000000000005e-271

   1. Initial program 63.8%

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

      1. distribute-lft-out 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in x around inf 44.7%

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

   if 4.70000000000000005e-271 < y

   1. Initial program 76.1%

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

      1. distribute-lft-out 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in x around 0 35.1%

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

      1. pow1/2 35.1%

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

      2. *-commutative 35.1%

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

      3. metadata-eval 35.1%

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

      4. unpow-prod-down 44.7%

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

      5. metadata-eval 44.7%

         \[\leadsto 2 \cdot \left({z}^{\color{blue}{0.5}} \cdot {y}^{\left(\sqrt{0.25}\right)}\right) \]

      6. pow1/2 44.7%

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

      7. metadata-eval 44.7%

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

      8. pow1/2 44.7%

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

   6. Applied egg-rr 44.7%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 6: 70.4% 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 69.9%

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

   1. distribute-lft-out 69.9%

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

3. Simplified 69.9%

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

4. Final simplification 69.9%

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

Alternative 7: 70.3% accurate, 1.0× speedup

Math

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

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. associate-+l+ 69.9%

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

   2. +-commutative 69.9%

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

   3. distribute-rgt-out 69.9%

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

   4. fma-def 69.9%

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

3. Simplified 69.9%

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

   1. fma-udef 69.9%

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

5. Applied egg-rr 69.9%

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

6. Final simplification 69.9%

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

Alternative 8: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-295}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e-295) (* 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 <= -2.3e-295) {
		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 <= (-2.3d-295)) 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 <= -2.3e-295) {
		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 <= -2.3e-295:
		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 <= -2.3e-295)
		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 <= -2.3e-295)
		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, -2.3e-295], 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 < -2.3e-295

   1. Initial program 64.5%

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

      1. distribute-lft-out 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in z around 0 24.0%

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

   if -2.3e-295 < y

   1. Initial program 74.7%

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

      1. distribute-lft-out 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in z around inf 54.8%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-295}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 9: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-266) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-266) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1d-266)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-266) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1e-266:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-266)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-266)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1e-266], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999998e-267

   1. Initial program 63.8%

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

      1. distribute-lft-out 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in x around inf 42.6%

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

   if -9.9999999999999998e-267 < y

   1. Initial program 75.0%

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

      1. distribute-lft-out 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in z around inf 55.8%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 10: 68.3% accurate, 1.1× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-310) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-310) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1d-310)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-310) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1e-310:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-310)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-310)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1e-310], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.999999999999969e-311

   1. Initial program 64.5%

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

      1. distribute-lft-out 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in z around 0 24.0%

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

   if -9.999999999999969e-311 < y

   1. Initial program 74.7%

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

      1. distribute-lft-out 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in x around 0 33.3%

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

4. Final simplification 28.9%

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

Alternative 11: 35.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. distribute-lft-out 69.9%

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

3. Simplified 69.9%

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

4. Taylor expanded in z around 0 22.7%

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

5. Final simplification 22.7%

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

Developer target: 82.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023230 
(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)))))