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

Percentage Accurate: 70.3% → 96.7%

Time: 13.2s

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: 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: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\\ \mathbf{if}\;y \leq -6.1 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot t_0}\right)}^{2}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-192}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-306}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(e^{-0.5}\right)}^{t_0}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log (- (- y) z)) (log (/ -1.0 x)))))
   (if (<= y -6.1e+36)
     (* 2.0 (pow (exp (* 0.25 t_0)) 2.0))
     (if (<= y -5.6e-192)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y -5.2e-306)
         (* 2.0 (/ 1.0 (pow (exp -0.5) t_0)))
         (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = log((-y - z)) - log((-1.0 / x));
	double tmp;
	if (y <= -6.1e+36) {
		tmp = 2.0 * pow(exp((0.25 * t_0)), 2.0);
	} else if (y <= -5.6e-192) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -5.2e-306) {
		tmp = 2.0 * (1.0 / pow(exp(-0.5), t_0));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log((-y - z)) - log(((-1.0d0) / x))
    if (y <= (-6.1d+36)) then
        tmp = 2.0d0 * (exp((0.25d0 * t_0)) ** 2.0d0)
    else if (y <= (-5.6d-192)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-5.2d-306)) then
        tmp = 2.0d0 * (1.0d0 / (exp((-0.5d0)) ** t_0))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = Math.log((-y - z)) - Math.log((-1.0 / x));
	double tmp;
	if (y <= -6.1e+36) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * t_0)), 2.0);
	} else if (y <= -5.6e-192) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -5.2e-306) {
		tmp = 2.0 * (1.0 / Math.pow(Math.exp(-0.5), t_0));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = math.log((-y - z)) - math.log((-1.0 / x))
	tmp = 0
	if y <= -6.1e+36:
		tmp = 2.0 * math.pow(math.exp((0.25 * t_0)), 2.0)
	elif y <= -5.6e-192:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -5.2e-306:
		tmp = 2.0 * (1.0 / math.pow(math.exp(-0.5), t_0))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x)))
	tmp = 0.0
	if (y <= -6.1e+36)
		tmp = Float64(2.0 * (exp(Float64(0.25 * t_0)) ^ 2.0));
	elseif (y <= -5.6e-192)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -5.2e-306)
		tmp = Float64(2.0 * Float64(1.0 / (exp(-0.5) ^ t_0)));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = log((-y - z)) - log((-1.0 / x));
	tmp = 0.0;
	if (y <= -6.1e+36)
		tmp = 2.0 * (exp((0.25 * t_0)) ^ 2.0);
	elseif (y <= -5.6e-192)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -5.2e-306)
		tmp = 2.0 * (1.0 / (exp(-0.5) ^ t_0));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.1e+36], N[(2.0 * N[Power[N[Exp[N[(0.25 * t$95$0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.6e-192], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.2e-306], N[(2.0 * N[(1.0 / N[Power[N[Exp[-0.5], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.1e36

   1. Initial program 55.6%

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

      1. distribute-lft-out 55.7%

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

   3. Simplified 55.7%

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

      1. add-sqr-sqrt 55.4%

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

      2. pow2 55.4%

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

      3. pow1/2 55.4%

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

      4. sqrt-pow1 55.4%

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

      5. fma-def 56.1%

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

      6. metadata-eval 56.1%

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

   5. Applied egg-rr 56.1%

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

   6. Taylor expanded in x around -inf 47.5%

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

   if -6.1e36 < y < -5.60000000000000007e-192

   1. Initial program 85.4%

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

      1. distribute-lft-out 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in x around inf 52.8%

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

   if -5.60000000000000007e-192 < y < -5.2000000000000001e-306

   1. Initial program 74.3%

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

      1. distribute-lft-out 74.3%

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

   3. Simplified 74.3%

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

      1. flip-+ 58.7%

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

      2. clear-num 58.7%

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

      3. pow2 58.7%

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

      4. pow2 58.7%

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

   5. Applied egg-rr 58.7%

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

      1. inv-pow 58.7%

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

      2. sqrt-pow1 58.7%

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

      3. clear-num 58.7%

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

      4. unpow2 58.7%

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

      5. unpow2 58.7%

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

      6. flip-+ 72.3%

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

      7. fma-udef 72.3%

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

      8. metadata-eval 72.3%

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

   7. Applied egg-rr 72.3%

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

      1. fma-def 72.3%

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

      2. +-commutative 72.3%

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

      3. fma-udef 72.3%

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

   9. Simplified 72.3%

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

       1. add-sqr-sqrt 71.7%

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

       2. unpow-prod-down 71.6%

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

       3. inv-pow 71.6%

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

       4. sqrt-pow1 71.5%

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

       5. metadata-eval 71.5%

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

       6. inv-pow 71.5%

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

       7. sqrt-pow1 73.8%

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

       8. metadata-eval 73.8%

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

   11. Applied egg-rr 73.8%

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

       1. pow-sqr 74.1%

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

       2. metadata-eval 74.1%

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

       3. unpow-1 74.1%

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

       4. fma-udef 74.1%

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

       5. *-commutative 74.1%

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

       6. *-commutative 74.1%

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

       7. fma-def 74.1%

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

       8. *-commutative 74.1%

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

       9. +-commutative 74.1%

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

   13. Simplified 74.1%

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

   14. Taylor expanded in x around -inf 42.4%

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

       1. exp-prod 42.4%

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

       2. +-commutative 42.4%

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

       3. mul-1-neg 42.4%

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

       4. unsub-neg 42.4%

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

       5. distribute-lft-in 42.4%

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

       6. mul-1-neg 42.4%

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

       7. unsub-neg 42.4%

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

       8. mul-1-neg 42.4%

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

   16. Simplified 42.4%

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

   if -5.2000000000000001e-306 < y

   1. Initial program 69.4%

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

      1. distribute-lft-out 69.4%

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

   3. Simplified 69.4%

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

      1. add-sqr-sqrt 69.0%

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

      2. pow2 69.0%

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

      3. pow1/2 69.0%

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

      4. sqrt-pow1 69.1%

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

      5. fma-def 69.5%

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

      6. metadata-eval 69.5%

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

   5. Applied egg-rr 69.5%

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

   6. Taylor expanded in z around inf 42.7%

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

      1. pow-pow 42.9%

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

      2. metadata-eval 42.9%

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

      3. pow1/2 42.9%

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

      4. sqrt-prod 54.7%

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

   8. Applied egg-rr 54.7%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-192}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-306}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(e^{-0.5}\right)}^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 2: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+39}:\\ \;\;\;\;2 \cdot \frac{1}{e^{t_0 \cdot -0.5}}\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-192}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-302}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(e^{-0.5}\right)}^{t_0}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log (- (- y) z)) (log (/ -1.0 x)))))
   (if (<= y -2.6e+39)
     (* 2.0 (/ 1.0 (exp (* t_0 -0.5))))
     (if (<= y -6.5e-192)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y -1.1e-302)
         (* 2.0 (/ 1.0 (pow (exp -0.5) t_0)))
         (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = log((-y - z)) - log((-1.0 / x));
	double tmp;
	if (y <= -2.6e+39) {
		tmp = 2.0 * (1.0 / exp((t_0 * -0.5)));
	} else if (y <= -6.5e-192) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -1.1e-302) {
		tmp = 2.0 * (1.0 / pow(exp(-0.5), t_0));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log((-y - z)) - log(((-1.0d0) / x))
    if (y <= (-2.6d+39)) then
        tmp = 2.0d0 * (1.0d0 / exp((t_0 * (-0.5d0))))
    else if (y <= (-6.5d-192)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-1.1d-302)) then
        tmp = 2.0d0 * (1.0d0 / (exp((-0.5d0)) ** t_0))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = Math.log((-y - z)) - Math.log((-1.0 / x));
	double tmp;
	if (y <= -2.6e+39) {
		tmp = 2.0 * (1.0 / Math.exp((t_0 * -0.5)));
	} else if (y <= -6.5e-192) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -1.1e-302) {
		tmp = 2.0 * (1.0 / Math.pow(Math.exp(-0.5), t_0));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = math.log((-y - z)) - math.log((-1.0 / x))
	tmp = 0
	if y <= -2.6e+39:
		tmp = 2.0 * (1.0 / math.exp((t_0 * -0.5)))
	elif y <= -6.5e-192:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -1.1e-302:
		tmp = 2.0 * (1.0 / math.pow(math.exp(-0.5), t_0))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x)))
	tmp = 0.0
	if (y <= -2.6e+39)
		tmp = Float64(2.0 * Float64(1.0 / exp(Float64(t_0 * -0.5))));
	elseif (y <= -6.5e-192)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -1.1e-302)
		tmp = Float64(2.0 * Float64(1.0 / (exp(-0.5) ^ t_0)));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = log((-y - z)) - log((-1.0 / x));
	tmp = 0.0;
	if (y <= -2.6e+39)
		tmp = 2.0 * (1.0 / exp((t_0 * -0.5)));
	elseif (y <= -6.5e-192)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -1.1e-302)
		tmp = 2.0 * (1.0 / (exp(-0.5) ^ t_0));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.6e+39], N[(2.0 * N[(1.0 / N[Exp[N[(t$95$0 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -6.5e-192], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.1e-302], N[(2.0 * N[(1.0 / N[Power[N[Exp[-0.5], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.6e39

   1. Initial program 55.6%

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

      1. distribute-lft-out 55.7%

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

   3. Simplified 55.7%

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

      1. flip-+ 21.4%

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

      2. clear-num 21.4%

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

      3. pow2 21.4%

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

      4. pow2 21.4%

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

   5. Applied egg-rr 21.4%

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

      1. inv-pow 21.4%

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

      2. sqrt-pow1 21.4%

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

      3. clear-num 21.4%

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

      4. unpow2 21.4%

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

      5. unpow2 21.4%

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

      6. flip-+ 55.8%

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

      7. fma-udef 56.5%

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

      8. metadata-eval 56.5%

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

   7. Applied egg-rr 56.5%

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

      1. fma-def 55.8%

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

      2. +-commutative 55.8%

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

      3. fma-udef 56.5%

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

   9. Simplified 56.5%

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

       1. add-sqr-sqrt 56.4%

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

       2. unpow-prod-down 56.1%

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

       3. inv-pow 56.1%

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

       4. sqrt-pow1 56.2%

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

       5. metadata-eval 56.2%

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

       6. inv-pow 56.2%

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

       7. sqrt-pow1 56.1%

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

       8. metadata-eval 56.1%

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

   11. Applied egg-rr 56.1%

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

       1. pow-sqr 56.4%

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

       2. metadata-eval 56.4%

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

       3. unpow-1 56.4%

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

       4. fma-udef 55.7%

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

       5. *-commutative 55.7%

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

       6. *-commutative 55.7%

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

       7. fma-def 56.4%

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

       8. *-commutative 56.4%

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

       9. +-commutative 56.4%

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

   13. Simplified 56.4%

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

   14. Taylor expanded in x around -inf 47.5%

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

   if -2.6e39 < y < -6.49999999999999966e-192

   1. Initial program 85.4%

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

      1. distribute-lft-out 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in x around inf 52.8%

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

   if -6.49999999999999966e-192 < y < -1.10000000000000004e-302

   1. Initial program 75.7%

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

      1. distribute-lft-out 75.7%

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

   3. Simplified 75.7%

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

      1. flip-+ 59.3%

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

      2. clear-num 59.3%

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

      3. pow2 59.3%

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

      4. pow2 59.3%

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

   5. Applied egg-rr 59.3%

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

      1. inv-pow 59.3%

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

      2. sqrt-pow1 59.3%

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

      3. clear-num 59.3%

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

      4. unpow2 59.3%

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

      5. unpow2 59.3%

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

      6. flip-+ 73.7%

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

      7. fma-udef 73.7%

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

      8. metadata-eval 73.7%

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

   7. Applied egg-rr 73.7%

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

      1. fma-def 73.7%

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

      2. +-commutative 73.7%

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

      3. fma-udef 73.7%

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

   9. Simplified 73.7%

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

       1. add-sqr-sqrt 73.2%

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

       2. unpow-prod-down 73.1%

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

       3. inv-pow 73.1%

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

       4. sqrt-pow1 73.0%

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

       5. metadata-eval 73.0%

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

       6. inv-pow 73.0%

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

       7. sqrt-pow1 75.2%

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

       8. metadata-eval 75.2%

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

   11. Applied egg-rr 75.2%

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

       1. pow-sqr 75.6%

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

       2. metadata-eval 75.6%

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

       3. unpow-1 75.6%

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

       4. fma-udef 75.6%

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

       5. *-commutative 75.6%

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

       6. *-commutative 75.6%

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

       7. fma-def 75.6%

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

       8. *-commutative 75.6%

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

       9. +-commutative 75.6%

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

   13. Simplified 75.6%

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

   14. Taylor expanded in x around -inf 42.3%

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

       1. exp-prod 42.3%

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

       2. +-commutative 42.3%

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

       3. mul-1-neg 42.3%

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

       4. unsub-neg 42.3%

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

       5. distribute-lft-in 42.3%

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

       6. mul-1-neg 42.3%

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

       7. unsub-neg 42.3%

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

       8. mul-1-neg 42.3%

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

   16. Simplified 42.3%

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

   if -1.10000000000000004e-302 < y

   1. Initial program 69.1%

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

      1. distribute-lft-out 69.1%

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

   3. Simplified 69.1%

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

      1. add-sqr-sqrt 68.8%

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

      2. pow2 68.8%

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

      3. pow1/2 68.8%

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

      4. sqrt-pow1 68.8%

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

      5. fma-def 69.2%

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

      6. metadata-eval 69.2%

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

   5. Applied egg-rr 69.2%

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

   6. Taylor expanded in z around inf 42.8%

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

      1. pow-pow 43.0%

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

      2. metadata-eval 43.0%

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

      3. pow1/2 43.0%

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

      4. sqrt-prod 54.6%

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

   8. Applied egg-rr 54.6%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+39}:\\ \;\;\;\;2 \cdot \frac{1}{e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot -0.5}}\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-192}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-302}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(e^{-0.5}\right)}^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 3: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot \frac{1}{e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot -0.5}}\\ \mathbf{if}\;y \leq -2.85 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-192}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          2.0
          (/ 1.0 (exp (* (- (log (- (- y) z)) (log (/ -1.0 x))) -0.5))))))
   (if (<= y -2.85e+38)
     t_0
     (if (<= y -5.6e-192)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y 4.3e-308) t_0 (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 2.0 * (1.0 / exp(((log((-y - z)) - log((-1.0 / x))) * -0.5)));
	double tmp;
	if (y <= -2.85e+38) {
		tmp = t_0;
	} else if (y <= -5.6e-192) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= 4.3e-308) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (1.0d0 / exp(((log((-y - z)) - log(((-1.0d0) / x))) * (-0.5d0))))
    if (y <= (-2.85d+38)) then
        tmp = t_0
    else if (y <= (-5.6d-192)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= 4.3d-308) then
        tmp = t_0
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 2.0 * (1.0 / Math.exp(((Math.log((-y - z)) - Math.log((-1.0 / x))) * -0.5)));
	double tmp;
	if (y <= -2.85e+38) {
		tmp = t_0;
	} else if (y <= -5.6e-192) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= 4.3e-308) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 2.0 * (1.0 / math.exp(((math.log((-y - z)) - math.log((-1.0 / x))) * -0.5)))
	tmp = 0
	if y <= -2.85e+38:
		tmp = t_0
	elif y <= -5.6e-192:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= 4.3e-308:
		tmp = t_0
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(2.0 * Float64(1.0 / exp(Float64(Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))) * -0.5))))
	tmp = 0.0
	if (y <= -2.85e+38)
		tmp = t_0;
	elseif (y <= -5.6e-192)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= 4.3e-308)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 2.0 * (1.0 / exp(((log((-y - z)) - log((-1.0 / x))) * -0.5)));
	tmp = 0.0;
	if (y <= -2.85e+38)
		tmp = t_0;
	elseif (y <= -5.6e-192)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= 4.3e-308)
		tmp = t_0;
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(2.0 * N[(1.0 / N[Exp[N[(N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.85e+38], t$95$0, If[LessEqual[y, -5.6e-192], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e-308], t$95$0, N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8499999999999999e38 or -5.60000000000000007e-192 < y < 4.3000000000000002e-308

   1. Initial program 62.4%

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

      1. distribute-lft-out 62.5%

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

   3. Simplified 62.5%

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

      1. flip-+ 35.0%

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

      2. clear-num 35.0%

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

      3. pow2 35.0%

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

      4. pow2 35.0%

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

   5. Applied egg-rr 35.0%

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

      1. inv-pow 35.0%

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

      2. sqrt-pow1 35.0%

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

      3. clear-num 35.0%

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

      4. unpow2 35.0%

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

      5. unpow2 35.0%

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

      6. flip-+ 61.8%

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

      7. fma-udef 62.3%

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

      8. metadata-eval 62.3%

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

   7. Applied egg-rr 62.3%

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

      1. fma-def 61.8%

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

      2. +-commutative 61.8%

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

      3. fma-udef 62.3%

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

   9. Simplified 62.3%

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

       1. add-sqr-sqrt 62.0%

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

       2. unpow-prod-down 61.8%

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

       3. inv-pow 61.8%

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

       4. sqrt-pow1 61.8%

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

       5. metadata-eval 61.8%

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

       6. inv-pow 61.8%

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

       7. sqrt-pow1 62.5%

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

       8. metadata-eval 62.5%

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

   11. Applied egg-rr 62.5%

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

       1. pow-sqr 62.9%

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

       2. metadata-eval 62.9%

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

       3. unpow-1 62.9%

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

       4. fma-udef 62.4%

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

       5. *-commutative 62.4%

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

       6. *-commutative 62.4%

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

       7. fma-def 62.9%

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

       8. *-commutative 62.9%

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

       9. +-commutative 62.9%

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

   13. Simplified 62.9%

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

   14. Taylor expanded in x around -inf 45.6%

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

   if -2.8499999999999999e38 < y < -5.60000000000000007e-192

   1. Initial program 85.4%

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

      1. distribute-lft-out 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in x around inf 52.8%

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

   if 4.3000000000000002e-308 < y

   1. Initial program 69.4%

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

      1. distribute-lft-out 69.4%

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

   3. Simplified 69.4%

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

      1. add-sqr-sqrt 69.0%

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

      2. pow2 69.0%

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

      3. pow1/2 69.0%

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

      4. sqrt-pow1 69.1%

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

      5. fma-def 69.5%

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

      6. metadata-eval 69.5%

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

   5. Applied egg-rr 69.5%

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

   6. Taylor expanded in z around inf 42.7%

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

      1. pow-pow 42.9%

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

      2. metadata-eval 42.9%

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

      3. pow1/2 42.9%

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

      4. sqrt-prod 54.7%

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

   8. Applied egg-rr 54.7%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+38}:\\ \;\;\;\;2 \cdot \frac{1}{e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot -0.5}}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-192}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-308}:\\ \;\;\;\;2 \cdot \frac{1}{e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot -0.5}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 4: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot e^{-0.5 \cdot \left(\log \left(\frac{-1}{x}\right) + \log \left(\frac{-1}{y + z}\right)\right)}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-192}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-301}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (* 2.0 (exp (* -0.5 (+ (log (/ -1.0 x)) (log (/ -1.0 (+ y z)))))))))
   (if (<= y -7.5e+36)
     t_0
     (if (<= y -5.6e-192)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y -3.4e-301) t_0 (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 2.0 * exp((-0.5 * (log((-1.0 / x)) + log((-1.0 / (y + z))))));
	double tmp;
	if (y <= -7.5e+36) {
		tmp = t_0;
	} else if (y <= -5.6e-192) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -3.4e-301) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * exp(((-0.5d0) * (log(((-1.0d0) / x)) + log(((-1.0d0) / (y + z))))))
    if (y <= (-7.5d+36)) then
        tmp = t_0
    else if (y <= (-5.6d-192)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-3.4d-301)) then
        tmp = t_0
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 2.0 * Math.exp((-0.5 * (Math.log((-1.0 / x)) + Math.log((-1.0 / (y + z))))));
	double tmp;
	if (y <= -7.5e+36) {
		tmp = t_0;
	} else if (y <= -5.6e-192) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -3.4e-301) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 2.0 * math.exp((-0.5 * (math.log((-1.0 / x)) + math.log((-1.0 / (y + z))))))
	tmp = 0
	if y <= -7.5e+36:
		tmp = t_0
	elif y <= -5.6e-192:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -3.4e-301:
		tmp = t_0
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(2.0 * exp(Float64(-0.5 * Float64(log(Float64(-1.0 / x)) + log(Float64(-1.0 / Float64(y + z)))))))
	tmp = 0.0
	if (y <= -7.5e+36)
		tmp = t_0;
	elseif (y <= -5.6e-192)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -3.4e-301)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 2.0 * exp((-0.5 * (log((-1.0 / x)) + log((-1.0 / (y + z))))));
	tmp = 0.0;
	if (y <= -7.5e+36)
		tmp = t_0;
	elseif (y <= -5.6e-192)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -3.4e-301)
		tmp = t_0;
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(2.0 * N[Exp[N[(-0.5 * N[(N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Log[N[(-1.0 / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+36], t$95$0, If[LessEqual[y, -5.6e-192], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.4e-301], t$95$0, N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.50000000000000054e36 or -5.60000000000000007e-192 < y < -3.4000000000000002e-301

   1. Initial program 63.3%

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

      1. distribute-lft-out 63.4%

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

   3. Simplified 63.4%

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

      1. flip-+ 35.0%

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

      2. clear-num 35.1%

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

      3. pow2 35.1%

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

      4. pow2 35.1%

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

   5. Applied egg-rr 35.1%

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

      1. inv-pow 35.1%

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

      2. sqrt-pow1 35.1%

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

      3. clear-num 35.1%

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

      4. unpow2 35.1%

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

      5. unpow2 35.1%

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

      6. flip-+ 62.7%

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

      7. fma-udef 63.2%

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

      8. metadata-eval 63.2%

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

   7. Applied egg-rr 63.2%

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

      1. fma-def 62.7%

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

      2. +-commutative 62.7%

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

      3. fma-udef 63.2%

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

   9. Simplified 63.2%

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

   10. Taylor expanded in x around -inf 46.2%

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

   if -7.50000000000000054e36 < y < -5.60000000000000007e-192

   1. Initial program 85.4%

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

      1. distribute-lft-out 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in x around inf 52.8%

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

   if -3.4000000000000002e-301 < y

   1. Initial program 68.6%

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

      1. distribute-lft-out 68.6%

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

   3. Simplified 68.6%

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

      1. add-sqr-sqrt 68.3%

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

      2. pow2 68.3%

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

      3. pow1/2 68.3%

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

      4. sqrt-pow1 68.3%

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

      5. fma-def 68.7%

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

      6. metadata-eval 68.7%

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

   5. Applied egg-rr 68.7%

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

   6. Taylor expanded in z around inf 42.5%

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

      1. pow-pow 42.7%

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

      2. metadata-eval 42.7%

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

      3. pow1/2 42.7%

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

      4. sqrt-prod 55.0%

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

   8. Applied egg-rr 55.0%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot e^{-0.5 \cdot \left(\log \left(\frac{-1}{x}\right) + \log \left(\frac{-1}{y + z}\right)\right)}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-192}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot e^{-0.5 \cdot \left(\log \left(\frac{-1}{x}\right) + \log \left(\frac{-1}{y + z}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 5: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+37}:\\ \;\;\;\;2 \cdot e^{-0.5 \cdot \left(\log \left(\frac{-1}{x}\right) + \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.4e+37)
   (* 2.0 (exp (* -0.5 (+ (log (/ -1.0 x)) (log (/ -1.0 y))))))
   (if (<= y 9.2e-301)
     (* 2.0 (sqrt (* x (+ y z))))
     (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e+37) {
		tmp = 2.0 * exp((-0.5 * (log((-1.0 / x)) + log((-1.0 / y)))));
	} else if (y <= 9.2e-301) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.4d+37)) then
        tmp = 2.0d0 * exp(((-0.5d0) * (log(((-1.0d0) / x)) + log(((-1.0d0) / y)))))
    else if (y <= 9.2d-301) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e+37) {
		tmp = 2.0 * Math.exp((-0.5 * (Math.log((-1.0 / x)) + Math.log((-1.0 / y)))));
	} else if (y <= 9.2e-301) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -3.4e+37:
		tmp = 2.0 * math.exp((-0.5 * (math.log((-1.0 / x)) + math.log((-1.0 / y)))))
	elif y <= 9.2e-301:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -3.4e+37)
		tmp = Float64(2.0 * exp(Float64(-0.5 * Float64(log(Float64(-1.0 / x)) + log(Float64(-1.0 / y))))));
	elseif (y <= 9.2e-301)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.4e+37)
		tmp = 2.0 * exp((-0.5 * (log((-1.0 / x)) + log((-1.0 / y)))));
	elseif (y <= 9.2e-301)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -3.4e+37], N[(2.0 * N[Exp[N[(-0.5 * N[(N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e-301], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.40000000000000006e37

   1. Initial program 55.6%

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

      1. distribute-lft-out 55.7%

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

   3. Simplified 55.7%

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

      1. flip-+ 21.4%

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

      2. clear-num 21.4%

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

      3. pow2 21.4%

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

      4. pow2 21.4%

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

   5. Applied egg-rr 21.4%

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

      1. inv-pow 21.4%

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

      2. sqrt-pow1 21.4%

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

      3. clear-num 21.4%

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

      4. unpow2 21.4%

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

      5. unpow2 21.4%

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

      6. flip-+ 55.8%

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

      7. fma-udef 56.5%

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

      8. metadata-eval 56.5%

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

   7. Applied egg-rr 56.5%

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

      1. fma-def 55.8%

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

      2. +-commutative 55.8%

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

      3. fma-udef 56.5%

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

   9. Simplified 56.5%

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

   10. Taylor expanded in x around -inf 47.5%

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

   11. Taylor expanded in y around inf 0.0%

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

       1. +-commutative 0.0%

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

       2. log-rec 0.0%

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

       3. sub-neg 0.0%

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

       4. log-div 45.3%

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

   13. Simplified 45.3%

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

   if -3.40000000000000006e37 < y < 9.2000000000000007e-301

   1. Initial program 80.9%

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

      1. distribute-lft-out 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in x around inf 59.1%

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

   if 9.2000000000000007e-301 < y

   1. Initial program 69.4%

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

      1. distribute-lft-out 69.4%

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

   3. Simplified 69.4%

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

      1. add-sqr-sqrt 69.0%

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

      2. pow2 69.0%

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

      3. pow1/2 69.0%

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

      4. sqrt-pow1 69.1%

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

      5. fma-def 69.5%

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

      6. metadata-eval 69.5%

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

   5. Applied egg-rr 69.5%

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

   6. Taylor expanded in z around inf 42.7%

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

      1. pow-pow 42.9%

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

      2. metadata-eval 42.9%

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

      3. pow1/2 42.9%

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

      4. sqrt-prod 54.7%

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

   8. Applied egg-rr 54.7%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+37}:\\ \;\;\;\;2 \cdot e^{-0.5 \cdot \left(\log \left(\frac{-1}{x}\right) + \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 6: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e-301)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e-301) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9.2d-301) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e-301) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 9.2e-301:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 9.2e-301)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.2e-301)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 9.2e-301], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.2000000000000007e-301

   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 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in x around inf 47.1%

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

   if 9.2000000000000007e-301 < y

   1. Initial program 69.4%

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

      1. distribute-lft-out 69.4%

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

   3. Simplified 69.4%

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

      1. add-sqr-sqrt 69.0%

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

      2. pow2 69.0%

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

      3. pow1/2 69.0%

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

      4. sqrt-pow1 69.1%

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

      5. fma-def 69.5%

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

      6. metadata-eval 69.5%

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

   5. Applied egg-rr 69.5%

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

   6. Taylor expanded in z around inf 42.7%

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

      1. pow-pow 42.9%

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

      2. metadata-eval 42.9%

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

      3. pow1/2 42.9%

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

      4. sqrt-prod 54.7%

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

   8. Applied egg-rr 54.7%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 7: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e-301)
   (* 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 <= 9.2e-301) {
		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 <= 9.2d-301) 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 <= 9.2e-301) {
		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 <= 9.2e-301:
		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 <= 9.2e-301)
		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 <= 9.2e-301)
		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, 9.2e-301], 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 < 9.2000000000000007e-301

   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 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in x around inf 47.1%

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

   if 9.2000000000000007e-301 < y

   1. Initial program 69.4%

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

      1. distribute-lft-out 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in x around 0 19.7%

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

      1. *-commutative 19.7%

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

      2. sqrt-prod 34.7%

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

   6. Applied egg-rr 34.7%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 8: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x + x \cdot z\right) + y \cdot z \leq 5 \cdot 10^{+264}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{-1}{y}}{\left(-z\right) - x}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ (+ (* y x) (* x z)) (* y z)) 5e+264)
   (* 2.0 (sqrt (+ (* y z) (* x (+ y z)))))
   (* 2.0 (pow (/ (/ -1.0 y) (- (- z) x)) -0.5))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((((y * x) + (x * z)) + (y * z)) <= 5e+264) {
		tmp = 2.0 * sqrt(((y * z) + (x * (y + z))));
	} else {
		tmp = 2.0 * pow(((-1.0 / y) / (-z - x)), -0.5);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((((y * x) + (x * z)) + (y * z)) <= 5d+264) then
        tmp = 2.0d0 * sqrt(((y * z) + (x * (y + z))))
    else
        tmp = 2.0d0 * ((((-1.0d0) / y) / (-z - x)) ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((((y * x) + (x * z)) + (y * z)) <= 5e+264) {
		tmp = 2.0 * Math.sqrt(((y * z) + (x * (y + z))));
	} else {
		tmp = 2.0 * Math.pow(((-1.0 / y) / (-z - x)), -0.5);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (((y * x) + (x * z)) + (y * z)) <= 5e+264:
		tmp = 2.0 * math.sqrt(((y * z) + (x * (y + z))))
	else:
		tmp = 2.0 * math.pow(((-1.0 / y) / (-z - x)), -0.5)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(y * x) + Float64(x * z)) + Float64(y * z)) <= 5e+264)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * z) + Float64(x * Float64(y + z)))));
	else
		tmp = Float64(2.0 * (Float64(Float64(-1.0 / y) / Float64(Float64(-z) - x)) ^ -0.5));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((((y * x) + (x * z)) + (y * z)) <= 5e+264)
		tmp = 2.0 * sqrt(((y * z) + (x * (y + z))));
	else
		tmp = 2.0 * (((-1.0 / y) / (-z - x)) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(N[(N[(y * x), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision], 5e+264], N[(2.0 * N[Sqrt[N[(N[(y * z), $MachinePrecision] + N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(-1.0 / y), $MachinePrecision] / N[((-z) - x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)) < 5.00000000000000033e264

   1. Initial program 96.2%

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

      1. distribute-lft-out 96.2%

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

   3. Simplified 96.2%

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

   if 5.00000000000000033e264 < (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))

   1. Initial program 19.0%

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

      1. distribute-lft-out 19.1%

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

   3. Simplified 19.1%

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

      1. flip-+ 0.5%

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

      2. clear-num 0.5%

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

      3. pow2 0.5%

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

      4. pow2 0.5%

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

   5. Applied egg-rr 0.5%

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

      1. inv-pow 0.5%

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

      2. sqrt-pow1 0.5%

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

      3. clear-num 0.5%

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

      4. unpow2 0.5%

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

      5. unpow2 0.5%

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

      6. flip-+ 19.1%

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

      7. fma-udef 20.1%

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

      8. metadata-eval 20.1%

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

   7. Applied egg-rr 20.1%

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

      1. fma-def 19.1%

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

      2. +-commutative 19.1%

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

      3. fma-udef 20.1%

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

   9. Simplified 20.1%

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

   10. Taylor expanded in y around -inf 13.9%

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

       1. associate-/r* 17.2%

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

       2. mul-1-neg 17.2%

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

       3. unsub-neg 17.2%

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

       4. mul-1-neg 17.2%

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

   12. Simplified 17.2%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\frac{-1}{y}}{\left(-z\right) - x}\right)}}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x + x \cdot z\right) + y \cdot z \leq 5 \cdot 10^{+264}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{-1}{y}}{\left(-z\right) - x}\right)}^{-0.5}\\ \end{array} \]

Alternative 9: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{z \cdot \left(y + x\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1e-307)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (pow (/ 1.0 (* z (+ y x))) -0.5))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-307) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * pow((1.0 / (z * (y + x))), -0.5);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1d-307) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * ((1.0d0 / (z * (y + x))) ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-307) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.pow((1.0 / (z * (y + x))), -0.5);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1e-307:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.pow((1.0 / (z * (y + x))), -0.5)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1e-307)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * (Float64(1.0 / Float64(z * Float64(y + x))) ^ -0.5));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e-307)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * ((1.0 / (z * (y + x))) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1e-307], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(1.0 / N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999909e-308

   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 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in x around inf 47.1%

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

   if 9.99999999999999909e-308 < y

   1. Initial program 69.4%

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

      1. distribute-lft-out 69.4%

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

   3. Simplified 69.4%

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

      1. flip-+ 37.9%

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

      2. clear-num 37.9%

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

      3. pow2 37.9%

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

      4. pow2 37.9%

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

   5. Applied egg-rr 37.9%

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

      1. inv-pow 37.9%

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

      2. sqrt-pow1 37.9%

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

      3. clear-num 37.9%

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

      4. unpow2 37.9%

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

      5. unpow2 37.9%

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

      6. flip-+ 69.4%

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

      7. fma-udef 69.8%

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

      8. metadata-eval 69.8%

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

   7. Applied egg-rr 69.8%

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

      1. fma-def 69.4%

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

      2. +-commutative 69.4%

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

      3. fma-udef 69.8%

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

   9. Simplified 69.8%

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

   10. Taylor expanded in z around inf 42.9%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{1}{\left(y + x\right) \cdot z}\right)}}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{1}{z \cdot \left(y + x\right)}\right)}^{-0.5}\\ \end{array} \]

Alternative 10: 70.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 70.0%

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

   1. distribute-lft-out 70.0%

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

3. Simplified 70.0%

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

4. Final simplification 70.0%

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

Alternative 11: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-288}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

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-288) (* 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-288) {
		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-288)) 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-288) {
		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-288:
		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-288)
		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-288)
		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-288], 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-288

   1. Initial program 70.6%

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

      1. distribute-lft-out 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in z around 0 29.1%

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

   if -2.3e-288 < y

   1. Initial program 69.4%

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

      1. distribute-lft-out 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in z around inf 44.0%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-288}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 12: 70.5% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-302) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-302) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1d-302)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-302) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1e-302:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-302)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-302)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1e-302], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999996e-303

   1. Initial program 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 47.0%

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

   if -9.9999999999999996e-303 < y

   1. Initial program 69.1%

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

      1. distribute-lft-out 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in z around inf 43.0%

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

4. Final simplification 45.0%

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

Alternative 13: 68.4% accurate, 1.1× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-310)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 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 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in z around 0 27.9%

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

   if -4.999999999999985e-310 < y

   1. Initial program 69.4%

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

      1. distribute-lft-out 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in x around 0 19.7%

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

4. Final simplification 23.9%

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

Alternative 14: 35.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. distribute-lft-out 70.0%

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

3. Simplified 70.0%

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

4. Taylor expanded in z around 0 28.8%

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

5. Final simplification 28.8%

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

Developer target: 83.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023221 
(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)))))