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

Percentage Accurate: 71.1% → 83.1%

Time: 8.3s

Alternatives: 6

Speedup: 1.2×

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: 71.1% 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: 83.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{{\left(y + x\right)}^{-1}}{{z}^{3}}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e+22)
   (* (sqrt (fma (+ y x) z (* y x))) 2.0)
   (*
    (fma
     (sqrt (/ (pow (+ y x) -1.0) (pow z 3.0)))
     (* y x)
     (* (sqrt (/ (+ y x) z)) 2.0))
    z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e+22) {
		tmp = sqrt(fma((y + x), z, (y * x))) * 2.0;
	} else {
		tmp = fma(sqrt((pow((y + x), -1.0) / pow(z, 3.0))), (y * x), (sqrt(((y + x) / z)) * 2.0)) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 8.2e+22)
		tmp = Float64(sqrt(fma(Float64(y + x), z, Float64(y * x))) * 2.0);
	else
		tmp = Float64(fma(sqrt(Float64((Float64(y + x) ^ -1.0) / (z ^ 3.0))), Float64(y * x), Float64(sqrt(Float64(Float64(y + x) / z)) * 2.0)) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 8.2e+22], N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Power[N[(y + x), $MachinePrecision], -1.0], $MachinePrecision] / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y * x), $MachinePrecision] + N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.19999999999999958e22

   1. Initial program 73.9%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. *-commutative N/A

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

      3. lower-*.f64 73.9

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-out N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 74.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 74.0

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

   4. Applied rewrites 74.0%

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

   if 8.19999999999999958e22 < y

   1. Initial program 46.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 29.9%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y + x, z, y \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{{\left(y + x\right)}^{-1}}{{z}^{3}}}, y \cdot x, \sqrt{\frac{y + x}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4.9e-278)
   (* 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 <= -4.9e-278) {
		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 <= (-4.9d-278)) 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 <= -4.9e-278) {
		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 <= -4.9e-278:
		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 <= -4.9e-278)
		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 <= -4.9e-278)
		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, -4.9e-278], 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 < -4.9000000000000002e-278

   1. Initial program 72.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 53.0

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

   5. Applied rewrites 53.0%

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

   if -4.9000000000000002e-278 < y

   1. Initial program 64.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

Alternative 3: 70.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{\left(z + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4.9e-278) (* 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 <= -4.9e-278) {
		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 <= (-4.9d-278)) 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 <= -4.9e-278) {
		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 <= -4.9e-278:
		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 <= -4.9e-278)
		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 <= -4.9e-278)
		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, -4.9e-278], 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 < -4.9000000000000002e-278

   1. Initial program 72.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 53.0

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

   5. Applied rewrites 53.0%

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

   if -4.9000000000000002e-278 < y

   1. Initial program 64.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 19.7

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

   5. Applied rewrites 19.7%

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

Alternative 4: 71.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 68.2

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-rgt-out N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 68.3

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 68.3

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

4. Applied rewrites 68.3%

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

Alternative 5: 69.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4.9e-278) (* 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 <= -4.9e-278) {
		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 <= (-4.9d-278)) 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 <= -4.9e-278) {
		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 <= -4.9e-278:
		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 <= -4.9e-278)
		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 <= -4.9e-278)
		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, -4.9e-278], 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 < -4.9000000000000002e-278

   1. Initial program 72.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 25.9

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

   5. Applied rewrites 25.9%

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

   if -4.9000000000000002e-278 < y

   1. Initial program 64.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 19.7

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

   5. Applied rewrites 19.7%

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

Alternative 6: 35.9% 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 68.2%

   \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
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 24.1

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

5. Applied rewrites 24.1%

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

Developer Target 1: 83.6% 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 2024324 
(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)))))