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

Percentage Accurate: 70.4% → 97.8%

Time: 18.8s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-287}:\\ \;\;\;\;2 \cdot \left({\left(\frac{-1}{y}\right)}^{-0.5} \cdot {\left(\frac{1}{z - x}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -8.8e-287)
   (* 2.0 (* (pow (/ -1.0 y) -0.5) (pow (/ 1.0 (- z x)) -0.5)))
   (* 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 <= -8.8e-287) {
		tmp = 2.0 * (pow((-1.0 / y), -0.5) * pow((1.0 / (z - x)), -0.5));
	} 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 <= (-8.8d-287)) then
        tmp = 2.0d0 * ((((-1.0d0) / y) ** (-0.5d0)) * ((1.0d0 / (z - x)) ** (-0.5d0)))
    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 <= -8.8e-287) {
		tmp = 2.0 * (Math.pow((-1.0 / y), -0.5) * Math.pow((1.0 / (z - x)), -0.5));
	} 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 <= -8.8e-287:
		tmp = 2.0 * (math.pow((-1.0 / y), -0.5) * math.pow((1.0 / (z - x)), -0.5))
	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 <= -8.8e-287)
		tmp = Float64(2.0 * Float64((Float64(-1.0 / y) ^ -0.5) * (Float64(1.0 / Float64(z - x)) ^ -0.5)));
	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 <= -8.8e-287)
		tmp = 2.0 * (((-1.0 / y) ^ -0.5) * ((1.0 / (z - x)) ^ -0.5));
	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, -8.8e-287], N[(2.0 * N[(N[Power[N[(-1.0 / y), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[N[(1.0 / N[(z - x), $MachinePrecision]), $MachinePrecision], -0.5], $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 < -8.8000000000000001e-287

   1. Initial program 69.5%

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

      1. +-commutative 69.5%

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

      2. associate-+r+ 69.5%

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

      3. *-commutative 69.5%

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

      4. +-commutative 69.5%

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

      5. associate-+l+ 69.5%

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

      6. *-commutative 69.5%

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

      7. *-commutative 69.5%

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

      8. *-commutative 69.5%

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

      9. distribute-lft-out 69.5%

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

   3. Simplified 69.5%

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

      1. flip-+ 38.5%

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

      2. clear-num 38.5%

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

      3. +-commutative 38.5%

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

      4. pow2 38.5%

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

      5. pow2 38.5%

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

      6. +-commutative 38.5%

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

   5. Applied egg-rr 38.5%

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

      1. inv-pow 38.5%

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

      2. sqrt-pow1 38.5%

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

      3. clear-num 38.5%

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

      4. unpow2 38.5%

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

      5. unpow2 38.5%

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

      6. flip-+ 69.4%

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

      7. fma-def 69.5%

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

      8. metadata-eval 69.5%

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

   7. Applied egg-rr 69.5%

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

   8. Taylor expanded in y around -inf 42.7%

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

      1. associate-/r* 43.7%

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

      2. neg-mul-1 43.7%

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

      3. +-commutative 43.7%

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

      4. unsub-neg 43.7%

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

      5. mul-1-neg 43.7%

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

   10. Simplified 43.7%

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

       1. div-inv 43.7%

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

       2. unpow-prod-down 62.0%

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

       3. add-sqr-sqrt 48.6%

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

       4. sqrt-unprod 49.9%

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

       5. sqr-neg 49.9%

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

       6. sqrt-unprod 14.3%

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

       7. add-sqr-sqrt 24.1%

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

   12. Applied egg-rr 24.1%

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

   if -8.8000000000000001e-287 < y

   1. Initial program 70.9%

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

      1. +-commutative 70.9%

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

      2. associate-+r+ 70.9%

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

      3. *-commutative 70.9%

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

      4. +-commutative 70.9%

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

      5. associate-+l+ 70.9%

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

      6. *-commutative 70.9%

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

      7. *-commutative 70.9%

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

      8. *-commutative 70.9%

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

      9. distribute-lft-out 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in z around inf 42.9%

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

      1. +-commutative 42.9%

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

   6. Simplified 42.9%

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

      1. +-commutative 42.9%

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

      2. *-commutative 42.9%

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

      3. sqrt-prod 43.1%

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

   8. Applied egg-rr 43.1%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-287}:\\ \;\;\;\;2 \cdot \left({\left(\frac{-1}{y}\right)}^{-0.5} \cdot {\left(\frac{1}{z - x}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 2: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{-1}{x}}{\left(-y\right) - z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 7.2e-304)
   (* 2.0 (pow (/ (/ -1.0 x) (- (- y) z)) -0.5))
   (* 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 <= 7.2e-304) {
		tmp = 2.0 * pow(((-1.0 / x) / (-y - z)), -0.5);
	} 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 <= 7.2d-304) then
        tmp = 2.0d0 * ((((-1.0d0) / x) / (-y - z)) ** (-0.5d0))
    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 <= 7.2e-304) {
		tmp = 2.0 * Math.pow(((-1.0 / x) / (-y - z)), -0.5);
	} 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 <= 7.2e-304:
		tmp = 2.0 * math.pow(((-1.0 / x) / (-y - z)), -0.5)
	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 <= 7.2e-304)
		tmp = Float64(2.0 * (Float64(Float64(-1.0 / x) / Float64(Float64(-y) - z)) ^ -0.5));
	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 <= 7.2e-304)
		tmp = 2.0 * (((-1.0 / x) / (-y - z)) ^ -0.5);
	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, 7.2e-304], N[(2.0 * N[Power[N[(N[(-1.0 / x), $MachinePrecision] / N[((-y) - z), $MachinePrecision]), $MachinePrecision], -0.5], $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 < 7.2000000000000003e-304

   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. +-commutative 70.5%

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

      2. associate-+r+ 70.5%

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

      3. *-commutative 70.5%

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

      4. +-commutative 70.5%

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

      5. associate-+l+ 70.5%

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

      6. *-commutative 70.5%

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

      7. *-commutative 70.5%

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

      8. *-commutative 70.5%

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

      9. distribute-lft-out 70.6%

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

   3. Simplified 70.6%

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

      1. flip-+ 38.2%

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

      2. clear-num 38.1%

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

      3. +-commutative 38.1%

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

      4. pow2 38.1%

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

      5. pow2 38.1%

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

      6. +-commutative 38.1%

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

   5. Applied egg-rr 38.1%

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

      1. inv-pow 38.1%

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

      2. sqrt-pow1 38.1%

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

      3. clear-num 38.1%

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

      4. unpow2 38.1%

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

      5. unpow2 38.1%

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

      6. flip-+ 70.5%

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

      7. fma-def 70.5%

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

      8. metadata-eval 70.5%

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

   7. Applied egg-rr 70.5%

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

   8. Taylor expanded in x around -inf 48.0%

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

      1. associate-/r* 49.1%

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

      2. mul-1-neg 49.1%

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

      3. unsub-neg 49.1%

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

      4. mul-1-neg 49.1%

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

   10. Simplified 49.1%

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

   if 7.2000000000000003e-304 < y

   1. Initial program 70.1%

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

      1. +-commutative 70.1%

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

      2. associate-+r+ 70.1%

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

      3. *-commutative 70.1%

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

      4. +-commutative 70.1%

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

      5. associate-+l+ 70.1%

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

      6. *-commutative 70.1%

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

      7. *-commutative 70.1%

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

      8. *-commutative 70.1%

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

      9. distribute-lft-out 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in z around inf 41.3%

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

      1. +-commutative 41.3%

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

   6. Simplified 41.3%

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

      1. +-commutative 41.3%

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

      2. *-commutative 41.3%

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

      3. sqrt-prod 42.9%

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

   8. Applied egg-rr 42.9%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{-1}{x}}{\left(-y\right) - z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 3: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-303}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{-1}{x}}{\left(-y\right) - z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.25e-303)
   (* 2.0 (pow (/ (/ -1.0 x) (- (- y) z)) -0.5))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e-303) {
		tmp = 2.0 * pow(((-1.0 / x) / (-y - z)), -0.5);
	} 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 <= 1.25d-303) then
        tmp = 2.0d0 * ((((-1.0d0) / x) / (-y - z)) ** (-0.5d0))
    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 <= 1.25e-303) {
		tmp = 2.0 * Math.pow(((-1.0 / x) / (-y - z)), -0.5);
	} 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 <= 1.25e-303:
		tmp = 2.0 * math.pow(((-1.0 / x) / (-y - z)), -0.5)
	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 <= 1.25e-303)
		tmp = Float64(2.0 * (Float64(Float64(-1.0 / x) / Float64(Float64(-y) - z)) ^ -0.5));
	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 <= 1.25e-303)
		tmp = 2.0 * (((-1.0 / x) / (-y - z)) ^ -0.5);
	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, 1.25e-303], N[(2.0 * N[Power[N[(N[(-1.0 / x), $MachinePrecision] / N[((-y) - z), $MachinePrecision]), $MachinePrecision], -0.5], $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 < 1.25e-303

   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. +-commutative 70.5%

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

      2. associate-+r+ 70.5%

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

      3. *-commutative 70.5%

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

      4. +-commutative 70.5%

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

      5. associate-+l+ 70.5%

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

      6. *-commutative 70.5%

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

      7. *-commutative 70.5%

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

      8. *-commutative 70.5%

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

      9. distribute-lft-out 70.6%

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

   3. Simplified 70.6%

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

      1. flip-+ 38.2%

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

      2. clear-num 38.1%

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

      3. +-commutative 38.1%

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

      4. pow2 38.1%

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

      5. pow2 38.1%

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

      6. +-commutative 38.1%

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

   5. Applied egg-rr 38.1%

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

      1. inv-pow 38.1%

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

      2. sqrt-pow1 38.1%

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

      3. clear-num 38.1%

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

      4. unpow2 38.1%

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

      5. unpow2 38.1%

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

      6. flip-+ 70.5%

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

      7. fma-def 70.5%

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

      8. metadata-eval 70.5%

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

   7. Applied egg-rr 70.5%

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

   8. Taylor expanded in x around -inf 48.0%

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

      1. associate-/r* 49.1%

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

      2. mul-1-neg 49.1%

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

      3. unsub-neg 49.1%

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

      4. mul-1-neg 49.1%

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

   10. Simplified 49.1%

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

   if 1.25e-303 < y

   1. Initial program 70.1%

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

      1. +-commutative 70.1%

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

      2. associate-+r+ 70.1%

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

      3. *-commutative 70.1%

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

      4. +-commutative 70.1%

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

      5. associate-+l+ 70.1%

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

      6. *-commutative 70.1%

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

      7. *-commutative 70.1%

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

      8. *-commutative 70.1%

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

      9. distribute-lft-out 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 20.9%

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

      1. sqrt-prod 29.4%

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

   6. Applied egg-rr 29.4%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-303}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{-1}{x}}{\left(-y\right) - z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-147}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{y}}{z + x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z}}{y + x}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -7.5e-147)
   (* 2.0 (pow (/ (/ 1.0 y) (+ z x)) -0.5))
   (* 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 (x <= -7.5e-147) {
		tmp = 2.0 * pow(((1.0 / y) / (z + x)), -0.5);
	} 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 (x <= (-7.5d-147)) then
        tmp = 2.0d0 * (((1.0d0 / y) / (z + x)) ** (-0.5d0))
    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 (x <= -7.5e-147) {
		tmp = 2.0 * Math.pow(((1.0 / y) / (z + x)), -0.5);
	} 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 x <= -7.5e-147:
		tmp = 2.0 * math.pow(((1.0 / y) / (z + x)), -0.5)
	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 (x <= -7.5e-147)
		tmp = Float64(2.0 * (Float64(Float64(1.0 / y) / Float64(z + x)) ^ -0.5));
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / 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 (x <= -7.5e-147)
		tmp = 2.0 * (((1.0 / y) / (z + x)) ^ -0.5);
	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[x, -7.5e-147], N[(2.0 * N[Power[N[(N[(1.0 / y), $MachinePrecision] / N[(z + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / z), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.50000000000000047e-147

   1. Initial program 64.5%

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

      1. +-commutative 64.5%

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

      2. associate-+r+ 64.5%

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

      3. *-commutative 64.5%

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

      4. +-commutative 64.5%

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

      5. associate-+l+ 64.5%

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

      6. *-commutative 64.5%

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

      7. *-commutative 64.5%

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

      8. *-commutative 64.5%

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

      9. distribute-lft-out 64.5%

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

   3. Simplified 64.5%

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

      1. flip-+ 29.6%

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

      2. clear-num 29.6%

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

      3. +-commutative 29.6%

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

      4. pow2 29.6%

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

      5. pow2 29.6%

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

      6. +-commutative 29.6%

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

   5. Applied egg-rr 29.6%

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

      1. inv-pow 29.6%

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

      2. sqrt-pow1 29.6%

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

      3. clear-num 29.6%

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

      4. unpow2 29.6%

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

      5. unpow2 29.6%

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

      6. flip-+ 64.4%

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

      7. fma-def 64.6%

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

      8. metadata-eval 64.6%

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

   7. Applied egg-rr 64.6%

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

   8. Taylor expanded in y around -inf 34.4%

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

      1. associate-/r* 34.4%

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

      2. neg-mul-1 34.4%

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

      3. +-commutative 34.4%

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

      4. unsub-neg 34.4%

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

      5. mul-1-neg 34.4%

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

   10. Simplified 34.4%

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

   11. Taylor expanded in y around 0 34.4%

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

       1. associate-/r* 34.4%

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

   13. Simplified 34.4%

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

   if -7.50000000000000047e-147 < x

   1. Initial program 73.4%

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

      1. +-commutative 73.4%

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

      2. associate-+r+ 73.4%

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

      3. *-commutative 73.4%

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

      4. +-commutative 73.4%

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

      5. associate-+l+ 73.4%

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

      6. *-commutative 73.4%

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

      7. *-commutative 73.4%

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

      8. *-commutative 73.4%

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

      9. distribute-lft-out 73.4%

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

   3. Simplified 73.4%

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

      1. flip-+ 39.3%

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

      2. clear-num 39.2%

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

      3. +-commutative 39.2%

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

      4. pow2 39.2%

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

      5. pow2 39.2%

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

      6. +-commutative 39.2%

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

   5. Applied egg-rr 39.2%

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

      1. inv-pow 39.2%

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

      2. sqrt-pow1 39.2%

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

      3. clear-num 39.2%

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

      4. unpow2 39.2%

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

      5. unpow2 39.2%

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

      6. flip-+ 73.4%

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

      7. fma-def 73.5%

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

      8. metadata-eval 73.5%

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

   7. Applied egg-rr 73.5%

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

   8. Taylor expanded in z around inf 49.2%

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

      1. associate-/r* 50.4%

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

      2. +-commutative 50.4%

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

   10. Simplified 50.4%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-147}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{y}}{z + x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z}}{y + x}\right)}^{-0.5}\\ \end{array} \]

Alternative 5: 70.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. +-commutative 70.3%

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

   2. associate-+r+ 70.3%

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

   3. *-commutative 70.3%

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

   4. +-commutative 70.3%

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

   5. associate-+l+ 70.3%

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

   6. *-commutative 70.3%

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

   7. *-commutative 70.3%

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

   8. *-commutative 70.3%

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

   9. distribute-lft-out 70.3%

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

3. Simplified 70.3%

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

4. Final simplification 70.3%

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

Alternative 6: 69.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * (((1.0d0 / y) / (z + x)) ** (-0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. +-commutative 70.3%

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

   2. associate-+r+ 70.3%

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

   3. *-commutative 70.3%

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

   4. +-commutative 70.3%

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

   5. associate-+l+ 70.3%

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

   6. *-commutative 70.3%

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

   7. *-commutative 70.3%

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

   8. *-commutative 70.3%

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

   9. distribute-lft-out 70.3%

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

3. Simplified 70.3%

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

   1. flip-+ 35.9%

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

   2. clear-num 35.9%

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

   3. +-commutative 35.9%

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

   4. pow2 35.9%

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

   5. pow2 35.9%

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

   6. +-commutative 35.9%

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

5. Applied egg-rr 35.9%

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

   1. inv-pow 35.9%

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

   2. sqrt-pow1 35.9%

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

   3. clear-num 35.9%

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

   4. unpow2 35.9%

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

   5. unpow2 35.9%

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

   6. flip-+ 70.3%

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

   7. fma-def 70.5%

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

   8. metadata-eval 70.5%

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

7. Applied egg-rr 70.5%

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

8. Taylor expanded in y around -inf 45.3%

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

   1. associate-/r* 45.7%

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

   2. neg-mul-1 45.7%

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

   3. +-commutative 45.7%

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

   4. unsub-neg 45.7%

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

   5. mul-1-neg 45.7%

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

10. Simplified 45.7%

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

11. Taylor expanded in y around 0 45.3%

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

    1. associate-/r* 45.7%

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

13. Simplified 45.7%

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

14. Final simplification 45.7%

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

Alternative 7: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.65e-265) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.65e-265) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.65d-265)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.65e-265) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.65e-265:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.65e-265)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.65e-265)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.65e-265], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65000000000000001e-265

   1. Initial program 70.1%

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

      1. +-commutative 70.1%

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

      2. associate-+r+ 70.1%

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

      3. *-commutative 70.1%

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

      4. +-commutative 70.1%

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

      5. associate-+l+ 70.1%

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

      6. *-commutative 70.1%

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

      7. *-commutative 70.1%

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

      8. *-commutative 70.1%

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

      9. distribute-lft-out 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in x around inf 45.6%

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

   if -1.65000000000000001e-265 < y

   1. Initial program 70.4%

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

      1. +-commutative 70.4%

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

      2. associate-+r+ 70.4%

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

      3. *-commutative 70.4%

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

      4. +-commutative 70.4%

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

      5. associate-+l+ 70.4%

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

      6. *-commutative 70.4%

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

      7. *-commutative 70.4%

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

      8. *-commutative 70.4%

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

      9. distribute-lft-out 70.4%

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

   3. Simplified 70.4%

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

   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 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 8: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.65e-265)
   (* 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 <= -1.65e-265) {
		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 <= (-1.65d-265)) 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 <= -1.65e-265) {
		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 <= -1.65e-265:
		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 <= -1.65e-265)
		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 <= -1.65e-265)
		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, -1.65e-265], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65000000000000001e-265

   1. Initial program 70.1%

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

      1. +-commutative 70.1%

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

      2. associate-+r+ 70.1%

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

      3. *-commutative 70.1%

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

      4. +-commutative 70.1%

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

      5. associate-+l+ 70.1%

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

      6. *-commutative 70.1%

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

      7. *-commutative 70.1%

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

      8. *-commutative 70.1%

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

      9. distribute-lft-out 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in x around inf 45.6%

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

   if -1.65000000000000001e-265 < y

   1. Initial program 70.4%

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

      1. +-commutative 70.4%

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

      2. associate-+r+ 70.4%

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

      3. *-commutative 70.4%

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

      4. +-commutative 70.4%

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

      5. associate-+l+ 70.4%

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

      6. *-commutative 70.4%

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

      7. *-commutative 70.4%

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

      8. *-commutative 70.4%

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

      9. distribute-lft-out 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in z around inf 43.7%

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

      1. +-commutative 43.7%

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

   6. Simplified 43.7%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 9: 68.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.65e-265) (* 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 <= -1.65e-265) {
		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 <= (-1.65d-265)) 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 <= -1.65e-265) {
		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 <= -1.65e-265:
		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 <= -1.65e-265)
		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 <= -1.65e-265)
		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, -1.65e-265], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65000000000000001e-265

   1. Initial program 70.1%

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

      1. +-commutative 70.1%

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

      2. associate-+r+ 70.1%

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

      3. *-commutative 70.1%

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

      4. +-commutative 70.1%

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

      5. associate-+l+ 70.1%

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

      6. *-commutative 70.1%

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

      7. *-commutative 70.1%

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

      8. *-commutative 70.1%

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

      9. distribute-lft-out 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in z around 0 20.5%

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

      1. *-commutative 20.5%

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

   6. Simplified 20.5%

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

   if -1.65000000000000001e-265 < y

   1. Initial program 70.4%

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

      1. +-commutative 70.4%

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

      2. associate-+r+ 70.4%

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

      3. *-commutative 70.4%

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

      4. +-commutative 70.4%

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

      5. associate-+l+ 70.4%

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

      6. *-commutative 70.4%

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

      7. *-commutative 70.4%

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

      8. *-commutative 70.4%

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

      9. distribute-lft-out 70.4%

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

   3. Simplified 70.4%

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

   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 20.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 10: 36.4% 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 70.3%

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

   1. +-commutative 70.3%

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

   2. associate-+r+ 70.3%

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

   3. *-commutative 70.3%

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

   4. +-commutative 70.3%

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

   5. associate-+l+ 70.3%

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

   6. *-commutative 70.3%

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

   7. *-commutative 70.3%

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

   8. *-commutative 70.3%

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

   9. distribute-lft-out 70.3%

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

3. Simplified 70.3%

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

4. Taylor expanded in z around 0 24.6%

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

   1. *-commutative 24.6%

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

6. Simplified 24.6%

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

7. Final simplification 24.6%

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

Developer target: 83.2% 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 2023279 
(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)))))