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

Percentage Accurate: 70.8% → 95.6%

Time: 8.4s

Alternatives: 9

Speedup: 0.8×

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: 70.8% 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: 95.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{x + z}{y}}\\ \mathbf{if}\;y \leq -6.1 \cdot 10^{+66}:\\ \;\;\;\;\left(\left(t\_0 \cdot -1\right) \cdot 2\right) \cdot y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+18}:\\ \;\;\;\;2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{y}^{-1.5} \cdot z}{\sqrt{x + z}}, x, t\_0 \cdot 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (sqrt (/ (+ x z) y))))
   (if (<= y -6.1e+66)
     (* (* (* t_0 -1.0) 2.0) y)
     (if (<= y 1.3e+18)
       (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z))))
       (* (fma (/ (* (pow y -1.5) z) (sqrt (+ x z))) x (* t_0 2.0)) y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = sqrt(((x + z) / y));
	double tmp;
	if (y <= -6.1e+66) {
		tmp = ((t_0 * -1.0) * 2.0) * y;
	} else if (y <= 1.3e+18) {
		tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	} else {
		tmp = fma(((pow(y, -1.5) * z) / sqrt((x + z))), x, (t_0 * 2.0)) * y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = sqrt(Float64(Float64(x + z) / y))
	tmp = 0.0
	if (y <= -6.1e+66)
		tmp = Float64(Float64(Float64(t_0 * -1.0) * 2.0) * y);
	elseif (y <= 1.3e+18)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))));
	else
		tmp = Float64(fma(Float64(Float64((y ^ -1.5) * z) / sqrt(Float64(x + z))), x, Float64(t_0 * 2.0)) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -6.1e+66], N[(N[(N[(t$95$0 * -1.0), $MachinePrecision] * 2.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.3e+18], N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[y, -1.5], $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(x + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x + N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.10000000000000021e66

   1. Initial program 47.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in y around -inf

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

   10. Applied rewrites 80.2%

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

   if -6.10000000000000021e66 < y < 1.3e18

   1. Initial program 81.9%

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

   if 1.3e18 < y

   1. Initial program 45.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{\frac{1}{{y}^{3}}}{z + x}}, z \cdot x, \sqrt{\frac{z + x}{y}} \cdot 2\right) \cdot y} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{y}^{-1.5} \cdot z}{\sqrt{x + z}}, x, \sqrt{\frac{x + z}{y}} \cdot 2\right) \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(\sqrt{\frac{z}{y}} \cdot 2\right) \cdot y\\ t_1 := 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\\ \mathbf{if}\;t\_1 \leq 10^{-158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (sqrt (/ z y)) 2.0) y))
        (t_1 (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z))))))
   (if (<= t_1 1e-158) t_0 (if (<= t_1 5e+150) t_1 t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (sqrt((z / y)) * 2.0) * y;
	double t_1 = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	double tmp;
	if (t_1 <= 1e-158) {
		tmp = t_0;
	} else if (t_1 <= 5e+150) {
		tmp = t_1;
	} else {
		tmp = t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (sqrt((z / y)) * 2.0d0) * y
    t_1 = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
    if (t_1 <= 1d-158) then
        tmp = t_0
    else if (t_1 <= 5d+150) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = (Math.sqrt((z / y)) * 2.0) * y;
	double t_1 = 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
	double tmp;
	if (t_1 <= 1e-158) {
		tmp = t_0;
	} else if (t_1 <= 5e+150) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (math.sqrt((z / y)) * 2.0) * y
	t_1 = 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))
	tmp = 0
	if t_1 <= 1e-158:
		tmp = t_0
	elif t_1 <= 5e+150:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(sqrt(Float64(z / y)) * 2.0) * y)
	t_1 = Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))))
	tmp = 0.0
	if (t_1 <= 1e-158)
		tmp = t_0;
	elseif (t_1 <= 5e+150)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (sqrt((z / y)) * 2.0) * y;
	t_1 = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	tmp = 0.0;
	if (t_1 <= 1e-158)
		tmp = t_0;
	elseif (t_1 <= 5e+150)
		tmp = t_1;
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(z / y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-158], t$95$0, If[LessEqual[t$95$1, 5e+150], t$95$1, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (sqrt.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)))) < 1.00000000000000006e-158 or 5.00000000000000009e150 < (*.f64 #s(literal 2 binary64) (sqrt.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))))

   1. Initial program 5.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 14.0%

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

   if 1.00000000000000006e-158 < (*.f64 #s(literal 2 binary64) (sqrt.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)))) < 5.00000000000000009e150

   1. Initial program 99.8%

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

Alternative 3: 95.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+66}:\\ \;\;\;\;\left(\left(\sqrt{\frac{x + z}{y}} \cdot -1\right) \cdot 2\right) \cdot y\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-42}:\\ \;\;\;\;2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{z}{y}} \cdot 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6.1e+66)
   (* (* (* (sqrt (/ (+ x z) y)) -1.0) 2.0) y)
   (if (<= y 8.5e-42)
     (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z))))
     (* (* (sqrt (/ z y)) 2.0) y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.1e+66) {
		tmp = ((sqrt(((x + z) / y)) * -1.0) * 2.0) * y;
	} else if (y <= 8.5e-42) {
		tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	} else {
		tmp = (sqrt((z / y)) * 2.0) * 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 <= (-6.1d+66)) then
        tmp = ((sqrt(((x + z) / y)) * (-1.0d0)) * 2.0d0) * y
    else if (y <= 8.5d-42) then
        tmp = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
    else
        tmp = (sqrt((z / y)) * 2.0d0) * 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 <= -6.1e+66) {
		tmp = ((Math.sqrt(((x + z) / y)) * -1.0) * 2.0) * y;
	} else if (y <= 8.5e-42) {
		tmp = 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
	} else {
		tmp = (Math.sqrt((z / y)) * 2.0) * y;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -6.1e+66:
		tmp = ((math.sqrt(((x + z) / y)) * -1.0) * 2.0) * y
	elif y <= 8.5e-42:
		tmp = 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))
	else:
		tmp = (math.sqrt((z / y)) * 2.0) * y
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -6.1e+66)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(x + z) / y)) * -1.0) * 2.0) * y);
	elseif (y <= 8.5e-42)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))));
	else
		tmp = Float64(Float64(sqrt(Float64(z / y)) * 2.0) * 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 <= -6.1e+66)
		tmp = ((sqrt(((x + z) / y)) * -1.0) * 2.0) * y;
	elseif (y <= 8.5e-42)
		tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
	else
		tmp = (sqrt((z / y)) * 2.0) * 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, -6.1e+66], N[(N[(N[(N[Sqrt[N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision] * 2.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 8.5e-42], N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(z / y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.10000000000000021e66

   1. Initial program 47.0%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in y around -inf

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

   10. Applied rewrites 80.2%

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

   if -6.10000000000000021e66 < y < 8.4999999999999996e-42

   1. Initial program 81.3%

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

   if 8.4999999999999996e-42 < y

   1. Initial program 53.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 77.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 33.1%

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

Alternative 4: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-42}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{z}{y}} \cdot 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.95e-275)
   (* 2.0 (sqrt (* (+ z y) x)))
   (if (<= y 8.5e-42)
     (* 2.0 (sqrt (* (+ y x) z)))
     (* (* (sqrt (/ z y)) 2.0) y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.95e-275) {
		tmp = 2.0 * sqrt(((z + y) * x));
	} else if (y <= 8.5e-42) {
		tmp = 2.0 * sqrt(((y + x) * z));
	} else {
		tmp = (sqrt((z / y)) * 2.0) * 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 <= (-1.95d-275)) then
        tmp = 2.0d0 * sqrt(((z + y) * x))
    else if (y <= 8.5d-42) then
        tmp = 2.0d0 * sqrt(((y + x) * z))
    else
        tmp = (sqrt((z / y)) * 2.0d0) * 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 <= -1.95e-275) {
		tmp = 2.0 * Math.sqrt(((z + y) * x));
	} else if (y <= 8.5e-42) {
		tmp = 2.0 * Math.sqrt(((y + x) * z));
	} else {
		tmp = (Math.sqrt((z / y)) * 2.0) * y;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.95e-275:
		tmp = 2.0 * math.sqrt(((z + y) * x))
	elif y <= 8.5e-42:
		tmp = 2.0 * math.sqrt(((y + x) * z))
	else:
		tmp = (math.sqrt((z / y)) * 2.0) * y
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.95e-275)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + y) * x)));
	elseif (y <= 8.5e-42)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + x) * z)));
	else
		tmp = Float64(Float64(sqrt(Float64(z / y)) * 2.0) * 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 <= -1.95e-275)
		tmp = 2.0 * sqrt(((z + y) * x));
	elseif (y <= 8.5e-42)
		tmp = 2.0 * sqrt(((y + x) * z));
	else
		tmp = (sqrt((z / y)) * 2.0) * 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, -1.95e-275], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e-42], N[(2.0 * N[Sqrt[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(z / y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.94999999999999986e-275

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 42.2

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

   5. Applied rewrites 42.2%

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

   if -1.94999999999999986e-275 < y < 8.4999999999999996e-42

   1. Initial program 72.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 56.0

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

   5. Applied rewrites 56.0%

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

   if 8.4999999999999996e-42 < y

   1. Initial program 53.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 77.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 33.1%

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

Alternative 5: 70.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.95e-275)
   (* 2.0 (sqrt (* (+ z y) x)))
   (* 2.0 (sqrt (* (+ y x) z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.95e-275) {
		tmp = 2.0 * sqrt(((z + y) * x));
	} else {
		tmp = 2.0 * sqrt(((y + x) * 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 <= (-1.95d-275)) then
        tmp = 2.0d0 * sqrt(((z + y) * x))
    else
        tmp = 2.0d0 * sqrt(((y + x) * 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 <= -1.95e-275) {
		tmp = 2.0 * Math.sqrt(((z + y) * x));
	} else {
		tmp = 2.0 * Math.sqrt(((y + x) * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.95e-275:
		tmp = 2.0 * math.sqrt(((z + y) * x))
	else:
		tmp = 2.0 * math.sqrt(((y + x) * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.95e-275)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + y) * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + x) * 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 <= -1.95e-275)
		tmp = 2.0 * sqrt(((z + y) * x));
	else
		tmp = 2.0 * sqrt(((y + x) * 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, -1.95e-275], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.94999999999999986e-275

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 42.2

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

   5. Applied rewrites 42.2%

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

   if -1.94999999999999986e-275 < y

   1. Initial program 63.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 40.6

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

   5. Applied rewrites 40.6%

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

Alternative 6: 69.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.95e-275) (* 2.0 (sqrt (* (+ z y) x))) (* 2.0 (sqrt (* z y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.95e-275) {
		tmp = 2.0 * sqrt(((z + y) * x));
	} else {
		tmp = 2.0 * sqrt((z * 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 <= (-1.95d-275)) then
        tmp = 2.0d0 * sqrt(((z + y) * x))
    else
        tmp = 2.0d0 * sqrt((z * 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 <= -1.95e-275) {
		tmp = 2.0 * Math.sqrt(((z + y) * x));
	} else {
		tmp = 2.0 * Math.sqrt((z * y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.95e-275:
		tmp = 2.0 * math.sqrt(((z + y) * x))
	else:
		tmp = 2.0 * math.sqrt((z * y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.95e-275)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z + y) * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * 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 <= -1.95e-275)
		tmp = 2.0 * sqrt(((z + y) * x));
	else
		tmp = 2.0 * sqrt((z * 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, -1.95e-275], N[(2.0 * N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.94999999999999986e-275

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 42.2

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

   5. Applied rewrites 42.2%

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

   if -1.94999999999999986e-275 < y

   1. Initial program 63.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 20.6

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

   5. Applied rewrites 20.6%

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

Alternative 7: 68.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.95e-275) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* z y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.95e-275) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((z * 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 <= (-1.95d-275)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((z * 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 <= -1.95e-275) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((z * y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -1.95e-275:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((z * y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.95e-275)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * 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 <= -1.95e-275)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((z * 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, -1.95e-275], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.94999999999999986e-275

   1. Initial program 70.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 23.3

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

   5. Applied rewrites 23.3%

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

   if -1.94999999999999986e-275 < y

   1. Initial program 63.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 20.6

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

   5. Applied rewrites 20.6%

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

Alternative 8: 36.0% accurate, 1.8× 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 67.1%

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

3. Taylor expanded in z around 0

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

   1. lower-sqrt.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 24.3

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

5. Applied rewrites 24.3%

   \[\leadsto 2 \cdot \color{blue}{\sqrt{y \cdot x}} \]
6. Add Preprocessing

Alternative 9: 0.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \frac{0}{0} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ 0.0 0.0))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 0.0 / 0.0;
}

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 = 0.0d0 / 0.0d0
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return 0.0 / 0.0;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 0.0 / 0.0

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(0.0 / 0.0)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 0.0 / 0.0;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(0.0 / 0.0), $MachinePrecision]

Derivation

1. Initial program 67.1%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. flip-+ N/A

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

4. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\frac{0}{0}} \]
5. Add Preprocessing

Developer Target 1: 83.2% accurate, 0.0× 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 2024339 
(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)))))