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

Percentage Accurate: 70.3% → 95.3%

Time: 16.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+73}:\\ \;\;\;\;{\left(\sqrt{2} \cdot e^{0.25 \cdot \left(\log \left(\left(-z\right) - x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x \cdot \left(1 + \frac{y}{z}\right)} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

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+73)
   (pow
    (* (sqrt 2.0) (exp (* 0.25 (- (log (- (- z) x)) (log (/ -1.0 y))))))
    2.0)
   (if (<= y 3.4e-279)
     (* 2.0 (sqrt (* x (+ y z))))
     (* 2.0 (* (sqrt (+ y (* x (+ 1.0 (/ y z))))) (sqrt z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e+73) {
		tmp = pow((sqrt(2.0) * exp((0.25 * (log((-z - x)) - log((-1.0 / y)))))), 2.0);
	} else if (y <= 3.4e-279) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt((y + (x * (1.0 + (y / z))))) * 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+73)) then
        tmp = (sqrt(2.0d0) * exp((0.25d0 * (log((-z - x)) - log(((-1.0d0) / y)))))) ** 2.0d0
    else if (y <= 3.4d-279) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt((y + (x * (1.0d0 + (y / z))))) * 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+73) {
		tmp = Math.pow((Math.sqrt(2.0) * Math.exp((0.25 * (Math.log((-z - x)) - Math.log((-1.0 / y)))))), 2.0);
	} else if (y <= 3.4e-279) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt((y + (x * (1.0 + (y / z))))) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -7.2e+73:
		tmp = math.pow((math.sqrt(2.0) * math.exp((0.25 * (math.log((-z - x)) - math.log((-1.0 / y)))))), 2.0)
	elif y <= 3.4e-279:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt((y + (x * (1.0 + (y / z))))) * 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+73)
		tmp = Float64(sqrt(2.0) * exp(Float64(0.25 * Float64(log(Float64(Float64(-z) - x)) - log(Float64(-1.0 / y)))))) ^ 2.0;
	elseif (y <= 3.4e-279)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + Float64(x * Float64(1.0 + Float64(y / z))))) * 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+73)
		tmp = (sqrt(2.0) * exp((0.25 * (log((-z - x)) - log((-1.0 / y)))))) ^ 2.0;
	elseif (y <= 3.4e-279)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt((y + (x * (1.0 + (y / z))))) * 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+73], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[Exp[N[(0.25 * N[(N[Log[N[((-z) - x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[y, 3.4e-279], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + N[(x * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.1999999999999998e73

   1. Initial program 43.1%

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

      1. associate-+l+ 43.1%

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

      2. +-commutative 43.1%

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

      3. distribute-rgt-in 43.2%

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

   3. Simplified 43.2%

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

      1. add-sqr-sqrt 42.8%

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

      2. sqrt-unprod 43.2%

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

      3. swap-sqr 43.2%

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

      4. add-sqr-sqrt 43.2%

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

      5. distribute-rgt-in 43.1%

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

      6. associate-+r+ 43.1%

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

      7. *-commutative 43.1%

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

      8. distribute-lft-in 43.4%

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

      9. +-commutative 43.4%

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

      10. fma-undefine 43.5%

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

      11. add-sqr-sqrt 43.5%

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

      12. swap-sqr 43.5%

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

      13. sqrt-unprod 43.1%

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

   6. Applied egg-rr 43.1%

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

   7. Taylor expanded in y around -inf 83.3%

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

   if -7.1999999999999998e73 < y < 3.40000000000000015e-279

   1. Initial program 77.7%

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

      1. associate-+l+ 77.7%

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

      2. +-commutative 77.7%

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

      3. distribute-rgt-in 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in x around inf 53.3%

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

   if 3.40000000000000015e-279 < y

   1. Initial program 65.2%

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

      1. associate-+l+ 65.2%

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

      2. +-commutative 65.2%

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

      3. distribute-rgt-in 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in z around inf 55.7%

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

      1. associate-/l* 52.0%

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

   7. Simplified 52.0%

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

      1. *-commutative 52.0%

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

      2. sqrt-prod 52.4%

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

      3. +-commutative 52.4%

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

      4. fma-define 52.4%

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

   9. Applied egg-rr 52.4%

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

   10. Taylor expanded in x around 0 52.4%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+73}:\\ \;\;\;\;{\left(\sqrt{2} \cdot e^{0.25 \cdot \left(\log \left(\left(-z\right) - x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x \cdot \left(1 + \frac{y}{z}\right)} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-275}:\\ \;\;\;\;{\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)} \cdot \sqrt{2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x \cdot \left(1 + \frac{y}{z}\right)} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.75e-275)
   (pow
    (* (exp (* 0.25 (- (log (- (- z) y)) (log (/ -1.0 x))))) (sqrt 2.0))
    2.0)
   (* 2.0 (* (sqrt (+ y (* x (+ 1.0 (/ y z))))) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.75e-275) {
		tmp = pow((exp((0.25 * (log((-z - y)) - log((-1.0 / x))))) * sqrt(2.0)), 2.0);
	} else {
		tmp = 2.0 * (sqrt((y + (x * (1.0 + (y / z))))) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.75d-275)) then
        tmp = (exp((0.25d0 * (log((-z - y)) - log(((-1.0d0) / x))))) * sqrt(2.0d0)) ** 2.0d0
    else
        tmp = 2.0d0 * (sqrt((y + (x * (1.0d0 + (y / z))))) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.75e-275) {
		tmp = Math.pow((Math.exp((0.25 * (Math.log((-z - y)) - Math.log((-1.0 / x))))) * Math.sqrt(2.0)), 2.0);
	} else {
		tmp = 2.0 * (Math.sqrt((y + (x * (1.0 + (y / z))))) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.75e-275:
		tmp = math.pow((math.exp((0.25 * (math.log((-z - y)) - math.log((-1.0 / x))))) * math.sqrt(2.0)), 2.0)
	else:
		tmp = 2.0 * (math.sqrt((y + (x * (1.0 + (y / z))))) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.75e-275)
		tmp = Float64(exp(Float64(0.25 * Float64(log(Float64(Float64(-z) - y)) - log(Float64(-1.0 / x))))) * sqrt(2.0)) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + Float64(x * Float64(1.0 + Float64(y / z))))) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.75e-275)
		tmp = (exp((0.25 * (log((-z - y)) - log((-1.0 / x))))) * sqrt(2.0)) ^ 2.0;
	else
		tmp = 2.0 * (sqrt((y + (x * (1.0 + (y / z))))) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.75e-275], N[Power[N[(N[Exp[N[(0.25 * N[(N[Log[N[((-z) - y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + N[(x * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.74999999999999984e-275

   1. Initial program 60.9%

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

      1. associate-+l+ 60.9%

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

      2. +-commutative 60.9%

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

      3. distribute-rgt-in 61.1%

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

   3. Simplified 61.1%

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

      1. add-sqr-sqrt 60.6%

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

      2. sqrt-unprod 61.1%

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

      3. swap-sqr 61.1%

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

      4. add-sqr-sqrt 61.1%

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

      5. distribute-rgt-in 60.9%

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

      6. associate-+r+ 60.9%

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

      7. *-commutative 60.9%

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

      8. distribute-lft-in 61.1%

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

      9. +-commutative 61.1%

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

      10. fma-undefine 61.1%

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

      11. add-sqr-sqrt 61.1%

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

      12. swap-sqr 61.1%

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

      13. sqrt-unprod 60.6%

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

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in x around -inf 41.9%

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

   if -1.74999999999999984e-275 < y

   1. Initial program 66.1%

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

      1. associate-+l+ 66.1%

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

      2. +-commutative 66.1%

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

      3. distribute-rgt-in 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in z around inf 57.0%

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

      1. associate-/l* 53.5%

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

   7. Simplified 53.5%

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

      1. *-commutative 53.5%

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

      2. sqrt-prod 53.8%

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

      3. +-commutative 53.8%

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

      4. fma-define 53.8%

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

   9. Applied egg-rr 53.8%

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

   10. Taylor expanded in x around 0 53.8%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-275}:\\ \;\;\;\;{\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)} \cdot \sqrt{2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x \cdot \left(1 + \frac{y}{z}\right)} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+73}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-284}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x \cdot \left(1 + \frac{y}{z}\right)} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -7.5e+73)
   (* 2.0 (exp (* (- (log (- y)) (log (/ -1.0 x))) 0.5)))
   (if (<= y 3.9e-284)
     (* 2.0 (sqrt (* x (+ y z))))
     (* 2.0 (* (sqrt (+ y (* x (+ 1.0 (/ y z))))) (sqrt z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e+73) {
		tmp = 2.0 * exp(((log(-y) - log((-1.0 / x))) * 0.5));
	} else if (y <= 3.9e-284) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt((y + (x * (1.0 + (y / z))))) * 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 <= (-7.5d+73)) then
        tmp = 2.0d0 * exp(((log(-y) - log(((-1.0d0) / x))) * 0.5d0))
    else if (y <= 3.9d-284) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt((y + (x * (1.0d0 + (y / z))))) * 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 <= -7.5e+73) {
		tmp = 2.0 * Math.exp(((Math.log(-y) - Math.log((-1.0 / x))) * 0.5));
	} else if (y <= 3.9e-284) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt((y + (x * (1.0 + (y / z))))) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -7.5e+73:
		tmp = 2.0 * math.exp(((math.log(-y) - math.log((-1.0 / x))) * 0.5))
	elif y <= 3.9e-284:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt((y + (x * (1.0 + (y / z))))) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -7.5e+73)
		tmp = Float64(2.0 * exp(Float64(Float64(log(Float64(-y)) - log(Float64(-1.0 / x))) * 0.5)));
	elseif (y <= 3.9e-284)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + Float64(x * Float64(1.0 + Float64(y / z))))) * 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 <= -7.5e+73)
		tmp = 2.0 * exp(((log(-y) - log((-1.0 / x))) * 0.5));
	elseif (y <= 3.9e-284)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt((y + (x * (1.0 + (y / z))))) * 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, -7.5e+73], N[(2.0 * N[Exp[N[(N[(N[Log[(-y)], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e-284], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + N[(x * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.5e73

   1. Initial program 43.1%

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

      1. associate-+l+ 43.1%

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

      2. +-commutative 43.1%

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

      3. distribute-rgt-in 43.2%

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

   3. Simplified 43.2%

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

   5. Taylor expanded in z around 0 19.7%

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

      1. *-commutative 19.7%

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

   7. Simplified 19.7%

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

      1. pow1/2 19.9%

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

      2. pow-to-exp 18.5%

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

      3. *-commutative 18.5%

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

   9. Applied egg-rr 18.5%

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

   10. Taylor expanded in x around -inf 40.0%

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

       1. mul-1-neg 40.0%

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

       2. unsub-neg 40.0%

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

       3. neg-mul-1 40.0%

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

   12. Simplified 40.0%

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

   if -7.5e73 < y < 3.8999999999999997e-284

   1. Initial program 77.4%

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

      1. associate-+l+ 77.4%

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

      2. +-commutative 77.4%

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

      3. distribute-rgt-in 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in x around inf 52.6%

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

   if 3.8999999999999997e-284 < y

   1. Initial program 65.5%

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

      1. associate-+l+ 65.5%

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

      2. +-commutative 65.5%

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

      3. distribute-rgt-in 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in z around inf 56.0%

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

      1. associate-/l* 52.4%

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

   7. Simplified 52.4%

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

      1. *-commutative 52.4%

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

      2. sqrt-prod 52.8%

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

      3. +-commutative 52.8%

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

      4. fma-define 52.8%

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

   9. Applied egg-rr 52.8%

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

   10. Taylor expanded in x around 0 52.8%

       \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{y + x \cdot \left(1 + \frac{y}{z}\right)}} \cdot \sqrt{z}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.05 \cdot 10^{-286}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x \cdot \left(1 + \frac{y}{z}\right)} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.05e-286)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt (+ y (* x (+ 1.0 (/ y z))))) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.05e-286) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt((y + (x * (1.0 + (y / z))))) * 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 <= 3.05d-286) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt((y + (x * (1.0d0 + (y / z))))) * 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 <= 3.05e-286) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt((y + (x * (1.0 + (y / z))))) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 3.05e-286:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt((y + (x * (1.0 + (y / z))))) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 3.05e-286)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + Float64(x * Float64(1.0 + Float64(y / z))))) * 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 <= 3.05e-286)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt((y + (x * (1.0 + (y / z))))) * 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, 3.05e-286], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + N[(x * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.05000000000000016e-286

   1. Initial program 61.7%

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

      1. associate-+l+ 61.7%

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

      2. +-commutative 61.7%

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

      3. distribute-rgt-in 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in x around inf 37.7%

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

   if 3.05000000000000016e-286 < y

   1. Initial program 65.5%

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

      1. associate-+l+ 65.5%

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

      2. +-commutative 65.5%

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

      3. distribute-rgt-in 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in z around inf 56.0%

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

      1. associate-/l* 52.4%

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

   7. Simplified 52.4%

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

      1. *-commutative 52.4%

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

      2. sqrt-prod 52.8%

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

      3. +-commutative 52.8%

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

      4. fma-define 52.8%

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

   9. Applied egg-rr 52.8%

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

   10. Taylor expanded in x around 0 52.8%

       \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{y + x \cdot \left(1 + \frac{y}{z}\right)}} \cdot \sqrt{z}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.1 \cdot 10^{-280}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y + x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 8.1e-280)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt z) (sqrt (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.1e-280) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt((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 <= 8.1d-280) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt((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 <= 8.1e-280) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt((y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 8.1e-280:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt((y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 8.1e-280)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(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 <= 8.1e-280)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt((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, 8.1e-280], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.1e-280

   1. Initial program 62.0%

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

      1. associate-+l+ 62.0%

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

      2. +-commutative 62.0%

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

      3. distribute-rgt-in 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in x around inf 38.2%

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

   if 8.1e-280 < y

   1. Initial program 65.2%

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

      1. associate-+l+ 65.2%

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

      2. +-commutative 65.2%

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

      3. distribute-rgt-in 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in z around inf 37.6%

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

      1. +-commutative 37.6%

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

   7. Simplified 37.6%

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

      1. *-commutative 37.6%

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

      2. sqrt-prod 46.1%

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

      3. +-commutative 46.1%

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

   9. Applied egg-rr 46.1%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.1 \cdot 10^{-280}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y + x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5.3e-279)
   (* 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 <= 5.3e-279) {
		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 <= 5.3d-279) 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 <= 5.3e-279) {
		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 <= 5.3e-279:
		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 <= 5.3e-279)
		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 <= 5.3e-279)
		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, 5.3e-279], 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 < 5.3000000000000003e-279

   1. Initial program 62.0%

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

      1. associate-+l+ 62.0%

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

      2. +-commutative 62.0%

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

      3. distribute-rgt-in 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in x around inf 38.2%

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

   if 5.3000000000000003e-279 < y

   1. Initial program 65.2%

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

      1. associate-+l+ 65.2%

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

      2. +-commutative 65.2%

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

      3. distribute-rgt-in 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in x around 0 24.4%

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

      1. *-commutative 24.4%

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

   7. Simplified 24.4%

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

      1. *-commutative 24.4%

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

      2. sqrt-prod 33.7%

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

   9. Applied egg-rr 33.7%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-249}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(\left(y + x\right) \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.85e-249)
   (* 2.0 (sqrt (* x (+ y z))))
   (sqrt (* z (* (+ y x) 4.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.85e-249) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = sqrt((z * ((y + x) * 4.0)));
	}
	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 <= (-1.85d-249)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = sqrt((z * ((y + x) * 4.0d0)))
    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 <= -1.85e-249) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = Math.sqrt((z * ((y + x) * 4.0)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.85e-249:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = math.sqrt((z * ((y + x) * 4.0)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.85e-249)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = sqrt(Float64(z * Float64(Float64(y + x) * 4.0)));
	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 <= -1.85e-249)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = sqrt((z * ((y + x) * 4.0)));
	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, -1.85e-249], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(z * N[(N[(y + x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999988e-249

   1. Initial program 59.4%

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

      1. associate-+l+ 59.4%

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

      2. +-commutative 59.4%

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

      3. distribute-rgt-in 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in x around inf 32.8%

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

   if -1.84999999999999988e-249 < y

   1. Initial program 67.0%

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

      1. associate-+l+ 67.0%

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

      2. +-commutative 67.0%

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

      3. distribute-rgt-in 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in z around inf 41.9%

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

      1. +-commutative 41.9%

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

   7. Simplified 41.9%

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

      1. add-cbrt-cube 29.5%

         \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{z \cdot \left(y + x\right)} \cdot \sqrt{z \cdot \left(y + x\right)}\right) \cdot \sqrt{z \cdot \left(y + x\right)}}} \]

      2. pow1/3 27.7%

         \[\leadsto 2 \cdot \color{blue}{{\left(\left(\sqrt{z \cdot \left(y + x\right)} \cdot \sqrt{z \cdot \left(y + x\right)}\right) \cdot \sqrt{z \cdot \left(y + x\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 27.7%

         \[\leadsto 2 \cdot {\left(\color{blue}{\left(z \cdot \left(y + x\right)\right)} \cdot \sqrt{z \cdot \left(y + x\right)}\right)}^{0.3333333333333333} \]

      4. pow1 27.7%

         \[\leadsto 2 \cdot {\left(\color{blue}{{\left(z \cdot \left(y + x\right)\right)}^{1}} \cdot \sqrt{z \cdot \left(y + x\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 27.7%

         \[\leadsto 2 \cdot {\left({\left(z \cdot \left(y + x\right)\right)}^{1} \cdot \color{blue}{{\left(z \cdot \left(y + x\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 27.7%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(z \cdot \left(y + x\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. +-commutative 27.7%

         \[\leadsto 2 \cdot {\left({\left(z \cdot \color{blue}{\left(x + y\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 27.7%

         \[\leadsto 2 \cdot {\left({\left(z \cdot \left(x + y\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 27.7%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(z \cdot \left(x + y\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 29.5%

          \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(z \cdot \left(x + y\right)\right)}^{1.5}}} \]

       2. +-commutative 29.5%

          \[\leadsto 2 \cdot \sqrt[3]{{\left(z \cdot \color{blue}{\left(y + x\right)}\right)}^{1.5}} \]

   11. Simplified 29.5%

       \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}} \]
   12. Step-by-step derivation

       1. pow1 29.5%

          \[\leadsto \color{blue}{{\left(2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}\right)}^{1}} \]

       2. add-sqr-sqrt 29.4%

          \[\leadsto {\color{blue}{\left(\sqrt{2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}} \cdot \sqrt{2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}}\right)}}^{1} \]

       3. sqrt-unprod 29.5%

          \[\leadsto {\color{blue}{\left(\sqrt{\left(2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}\right) \cdot \left(2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}\right)}\right)}}^{1} \]

       4. *-commutative 29.5%

          \[\leadsto {\left(\sqrt{\color{blue}{\left(\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}} \cdot 2\right)} \cdot \left(2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}\right)}\right)}^{1} \]

       5. *-commutative 29.5%

          \[\leadsto {\left(\sqrt{\left(\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}} \cdot 2\right) \cdot \color{blue}{\left(\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}} \cdot 2\right)}}\right)}^{1} \]

       6. swap-sqr 29.5%

          \[\leadsto {\left(\sqrt{\color{blue}{\left(\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}\right) \cdot \left(2 \cdot 2\right)}}\right)}^{1} \]

       7. cbrt-unprod 17.7%

          \[\leadsto {\left(\sqrt{\color{blue}{\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5} \cdot {\left(z \cdot \left(y + x\right)\right)}^{1.5}}} \cdot \left(2 \cdot 2\right)}\right)}^{1} \]

       8. pow-prod-up 17.8%

          \[\leadsto {\left(\sqrt{\sqrt[3]{\color{blue}{{\left(z \cdot \left(y + x\right)\right)}^{\left(1.5 + 1.5\right)}}} \cdot \left(2 \cdot 2\right)}\right)}^{1} \]

       9. metadata-eval 17.8%

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

       10. pow3 17.8%

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

       11. add-cbrt-cube 41.9%

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

       12. +-commutative 41.9%

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

       13. metadata-eval 41.9%

           \[\leadsto {\left(\sqrt{\left(z \cdot \left(x + y\right)\right) \cdot \color{blue}{4}}\right)}^{1} \]

   13. Applied egg-rr 41.9%

       \[\leadsto \color{blue}{{\left(\sqrt{\left(z \cdot \left(x + y\right)\right) \cdot 4}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 41.9%

          \[\leadsto \color{blue}{\sqrt{\left(z \cdot \left(x + y\right)\right) \cdot 4}} \]

       2. associate-*l* 41.9%

          \[\leadsto \sqrt{\color{blue}{z \cdot \left(\left(x + y\right) \cdot 4\right)}} \]

       3. +-commutative 41.9%

          \[\leadsto \sqrt{z \cdot \left(\color{blue}{\left(y + x\right)} \cdot 4\right)} \]

   15. Simplified 41.9%

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

Alternative 8: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-261}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(\left(y + x\right) \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4.5e-261) (* 2.0 (sqrt (* y x))) (sqrt (* z (* (+ y x) 4.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.5e-261) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = sqrt((z * ((y + x) * 4.0)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.5d-261)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = sqrt((z * ((y + x) * 4.0d0)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.5e-261) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = Math.sqrt((z * ((y + x) * 4.0)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -4.5e-261:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = math.sqrt((z * ((y + x) * 4.0)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -4.5e-261)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = sqrt(Float64(z * Float64(Float64(y + x) * 4.0)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.5e-261)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = sqrt((z * ((y + x) * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -4.5e-261], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(z * N[(N[(y + x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5000000000000001e-261

   1. Initial program 59.6%

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

      1. associate-+l+ 59.6%

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

      2. +-commutative 59.6%

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

      3. distribute-rgt-in 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in z around 0 22.4%

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

      1. *-commutative 22.4%

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

   7. Simplified 22.4%

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

   if -4.5000000000000001e-261 < y

   1. Initial program 67.0%

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

      1. associate-+l+ 67.0%

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

      2. +-commutative 67.0%

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

      3. distribute-rgt-in 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in z around inf 41.4%

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

      1. +-commutative 41.4%

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

   7. Simplified 41.4%

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

      1. add-cbrt-cube 28.9%

         \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{z \cdot \left(y + x\right)} \cdot \sqrt{z \cdot \left(y + x\right)}\right) \cdot \sqrt{z \cdot \left(y + x\right)}}} \]

      2. pow1/3 27.1%

         \[\leadsto 2 \cdot \color{blue}{{\left(\left(\sqrt{z \cdot \left(y + x\right)} \cdot \sqrt{z \cdot \left(y + x\right)}\right) \cdot \sqrt{z \cdot \left(y + x\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 27.1%

         \[\leadsto 2 \cdot {\left(\color{blue}{\left(z \cdot \left(y + x\right)\right)} \cdot \sqrt{z \cdot \left(y + x\right)}\right)}^{0.3333333333333333} \]

      4. pow1 27.1%

         \[\leadsto 2 \cdot {\left(\color{blue}{{\left(z \cdot \left(y + x\right)\right)}^{1}} \cdot \sqrt{z \cdot \left(y + x\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 27.2%

         \[\leadsto 2 \cdot {\left({\left(z \cdot \left(y + x\right)\right)}^{1} \cdot \color{blue}{{\left(z \cdot \left(y + x\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 27.2%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(z \cdot \left(y + x\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. +-commutative 27.2%

         \[\leadsto 2 \cdot {\left({\left(z \cdot \color{blue}{\left(x + y\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 27.2%

         \[\leadsto 2 \cdot {\left({\left(z \cdot \left(x + y\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 27.2%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(z \cdot \left(x + y\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 28.9%

          \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(z \cdot \left(x + y\right)\right)}^{1.5}}} \]

       2. +-commutative 28.9%

          \[\leadsto 2 \cdot \sqrt[3]{{\left(z \cdot \color{blue}{\left(y + x\right)}\right)}^{1.5}} \]

   11. Simplified 28.9%

       \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}} \]
   12. Step-by-step derivation

       1. pow1 28.9%

          \[\leadsto \color{blue}{{\left(2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}\right)}^{1}} \]

       2. add-sqr-sqrt 28.8%

          \[\leadsto {\color{blue}{\left(\sqrt{2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}} \cdot \sqrt{2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}}\right)}}^{1} \]

       3. sqrt-unprod 28.9%

          \[\leadsto {\color{blue}{\left(\sqrt{\left(2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}\right) \cdot \left(2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}\right)}\right)}}^{1} \]

       4. *-commutative 28.9%

          \[\leadsto {\left(\sqrt{\color{blue}{\left(\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}} \cdot 2\right)} \cdot \left(2 \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}\right)}\right)}^{1} \]

       5. *-commutative 28.9%

          \[\leadsto {\left(\sqrt{\left(\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}} \cdot 2\right) \cdot \color{blue}{\left(\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}} \cdot 2\right)}}\right)}^{1} \]

       6. swap-sqr 28.9%

          \[\leadsto {\left(\sqrt{\color{blue}{\left(\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}} \cdot \sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5}}\right) \cdot \left(2 \cdot 2\right)}}\right)}^{1} \]

       7. cbrt-unprod 18.0%

          \[\leadsto {\left(\sqrt{\color{blue}{\sqrt[3]{{\left(z \cdot \left(y + x\right)\right)}^{1.5} \cdot {\left(z \cdot \left(y + x\right)\right)}^{1.5}}} \cdot \left(2 \cdot 2\right)}\right)}^{1} \]

       8. pow-prod-up 18.1%

          \[\leadsto {\left(\sqrt{\sqrt[3]{\color{blue}{{\left(z \cdot \left(y + x\right)\right)}^{\left(1.5 + 1.5\right)}}} \cdot \left(2 \cdot 2\right)}\right)}^{1} \]

       9. metadata-eval 18.1%

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

       10. pow3 18.1%

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

       11. add-cbrt-cube 41.4%

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

       12. +-commutative 41.4%

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

       13. metadata-eval 41.4%

           \[\leadsto {\left(\sqrt{\left(z \cdot \left(x + y\right)\right) \cdot \color{blue}{4}}\right)}^{1} \]

   13. Applied egg-rr 41.4%

       \[\leadsto \color{blue}{{\left(\sqrt{\left(z \cdot \left(x + y\right)\right) \cdot 4}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 41.4%

          \[\leadsto \color{blue}{\sqrt{\left(z \cdot \left(x + y\right)\right) \cdot 4}} \]

       2. associate-*l* 41.4%

          \[\leadsto \sqrt{\color{blue}{z \cdot \left(\left(x + y\right) \cdot 4\right)}} \]

       3. +-commutative 41.4%

          \[\leadsto \sqrt{z \cdot \left(\color{blue}{\left(y + x\right)} \cdot 4\right)} \]

   15. Simplified 41.4%

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

Alternative 9: 70.3% 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 63.6%

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

   1. associate-+l+ 63.6%

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

   2. +-commutative 63.6%

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

   3. distribute-rgt-in 63.7%

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

3. Simplified 63.7%

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

5. Final simplification 63.7%

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

Alternative 10: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-249}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot \left(z \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.85e-249) (* 2.0 (sqrt (* y x))) (sqrt (* y (* z 4.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.85e-249) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = sqrt((y * (z * 4.0)));
	}
	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 <= (-1.85d-249)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = sqrt((y * (z * 4.0d0)))
    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 <= -1.85e-249) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = Math.sqrt((y * (z * 4.0)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.85e-249:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = math.sqrt((y * (z * 4.0)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.85e-249)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = sqrt(Float64(y * Float64(z * 4.0)));
	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 <= -1.85e-249)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = sqrt((y * (z * 4.0)));
	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, -1.85e-249], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(y * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999988e-249

   1. Initial program 59.4%

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

      1. associate-+l+ 59.4%

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

      2. +-commutative 59.4%

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

      3. distribute-rgt-in 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in z around 0 22.8%

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

      1. *-commutative 22.8%

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

   7. Simplified 22.8%

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

   if -1.84999999999999988e-249 < y

   1. Initial program 67.0%

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

      1. associate-+l+ 67.0%

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

      2. +-commutative 67.0%

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

      3. distribute-rgt-in 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in x around 0 22.6%

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

      1. *-commutative 22.6%

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

   7. Simplified 22.6%

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

      1. add-sqr-sqrt 22.4%

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

      2. sqrt-unprod 22.6%

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

      3. swap-sqr 22.6%

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

      4. add-sqr-sqrt 22.6%

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

      5. metadata-eval 22.6%

         \[\leadsto \sqrt{\left(y \cdot z\right) \cdot \color{blue}{4}} \]

   9. Applied egg-rr 22.6%

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

       1. associate-*l* 22.6%

          \[\leadsto \sqrt{\color{blue}{y \cdot \left(z \cdot 4\right)}} \]

   11. Simplified 22.6%

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

Alternative 11: 35.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \sqrt{y \cdot \left(z \cdot 4\right)} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (sqrt (* y (* z 4.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return sqrt((y * (z * 4.0)));
}

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 = sqrt((y * (z * 4.0d0)))
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return Math.sqrt((y * (z * 4.0)));
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return math.sqrt((y * (z * 4.0)))

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return sqrt(Float64(y * Float64(z * 4.0)))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = sqrt((y * (z * 4.0)));
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[Sqrt[N[(y * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 63.6%

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

   1. associate-+l+ 63.6%

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

   2. +-commutative 63.6%

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

   3. distribute-rgt-in 63.7%

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

3. Simplified 63.7%

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

5. Taylor expanded in x around 0 25.7%

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

   1. *-commutative 25.7%

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

7. Simplified 25.7%

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

   1. add-sqr-sqrt 25.5%

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

   2. sqrt-unprod 25.7%

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

   3. swap-sqr 25.7%

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

   4. add-sqr-sqrt 25.7%

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

   5. metadata-eval 25.7%

      \[\leadsto \sqrt{\left(y \cdot z\right) \cdot \color{blue}{4}} \]

9. Applied egg-rr 25.7%

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

    1. associate-*l* 25.7%

       \[\leadsto \sqrt{\color{blue}{y \cdot \left(z \cdot 4\right)}} \]

11. Simplified 25.7%

    \[\leadsto \color{blue}{\sqrt{y \cdot \left(z \cdot 4\right)}} \]
12. Add Preprocessing

Developer Target 1: 82.9% 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 2024117 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 763695009057367500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 2 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 1/4 (* (* (pow y -3/4) (* (pow z -3/4) x)) (+ y z))) (* (pow z 1/4) (pow y 1/4))) (+ (* 1/4 (* (* (pow y -3/4) (* (pow z -3/4) x)) (+ y z))) (* (pow z 1/4) (pow y 1/4)))) 2)))

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