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

Percentage Accurate: 71.2% → 85.3%

Time: 13.7s

Alternatives: 9

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: 71.2% 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: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.16 \cdot 10^{-225}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.16e-225)
   (* 2.0 (sqrt (fma x z (* y (+ x z)))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.16e-225) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (x + z))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.16e-225)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(x + z)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(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, 1.16e-225], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.16000000000000001e-225

   1. Initial program 73.6%

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

      1. associate-+l+ 73.6%

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

      2. *-commutative 73.6%

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

      3. *-commutative 73.6%

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

      4. *-commutative 73.6%

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

      5. +-commutative 73.6%

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

      6. +-commutative 73.6%

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

      7. +-commutative 73.6%

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

      8. *-commutative 73.6%

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

      9. *-commutative 73.6%

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

      10. associate-+l+ 73.6%

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

      11. +-commutative 73.6%

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

      12. fma-def 73.6%

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

      13. distribute-lft-out 73.7%

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

   3. Simplified 73.7%

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

   if 1.16000000000000001e-225 < y

   1. Initial program 63.4%

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

      1. associate-+l+ 63.4%

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

      2. +-commutative 63.4%

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

      3. distribute-rgt-in 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in z around inf 32.1%

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

      1. +-commutative 32.1%

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

   7. Simplified 32.1%

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

      1. *-commutative 32.1%

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

      2. sqrt-prod 43.7%

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

      3. +-commutative 43.7%

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

   9. Applied egg-rr 43.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.16 \cdot 10^{-225}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-292}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e-292)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e-292) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.15d-292) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e-292) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.15e-292:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.15e-292)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.15e-292)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.15e-292], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1499999999999999e-292

   1. Initial program 74.2%

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

      1. associate-+l+ 74.2%

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

      2. +-commutative 74.2%

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

      3. distribute-rgt-in 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in x around inf 50.6%

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

   if 1.1499999999999999e-292 < y

   1. Initial program 63.5%

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

      1. associate-+l+ 63.5%

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

      2. +-commutative 63.5%

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

      3. distribute-rgt-in 63.6%

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

   3. Simplified 63.6%

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

   5. Taylor expanded in z around inf 33.4%

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

      1. +-commutative 33.4%

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

   7. Simplified 33.4%

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

      1. *-commutative 33.4%

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

      2. sqrt-prod 43.3%

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

      3. +-commutative 43.3%

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

   9. Applied egg-rr 43.3%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-292}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.6e-34)
   (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e-34) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.6d-34) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e-34) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.6e-34:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.6e-34)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.6e-34)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.6e-34], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $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 < 1.60000000000000001e-34

   1. Initial program 74.9%

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

      1. associate-+l+ 74.9%

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

      2. +-commutative 74.9%

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

      3. distribute-rgt-in 74.9%

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

   3. Simplified 74.9%

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

   if 1.60000000000000001e-34 < y

   1. Initial program 54.9%

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

      1. associate-+l+ 54.9%

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

      2. +-commutative 54.9%

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

      3. distribute-rgt-in 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in x around 0 18.4%

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

      1. *-commutative 18.4%

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

      2. sqrt-prod 37.0%

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

   7. Applied egg-rr 37.0%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z}}{y + x}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3e-301)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (pow (/ (/ 1.0 z) (+ y x)) -0.5))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e-301) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * pow(((1.0 / z) / (y + x)), -0.5);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3d-301) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (((1.0d0 / z) / (y + x)) ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e-301) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.pow(((1.0 / z) / (y + x)), -0.5);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 3e-301:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.pow(((1.0 / z) / (y + x)), -0.5)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 3e-301)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / z) / Float64(y + x)) ^ -0.5));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3e-301)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (((1.0 / z) / (y + x)) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 3e-301], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / z), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999999e-301

   1. Initial program 73.7%

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

      1. associate-+l+ 73.7%

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

      2. +-commutative 73.7%

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

      3. distribute-rgt-in 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in x around inf 49.1%

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

   if 2.99999999999999999e-301 < y

   1. Initial program 64.5%

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

      1. associate-+l+ 64.5%

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

      2. +-commutative 64.5%

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

      3. distribute-rgt-in 64.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in z around inf 35.8%

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

      1. +-commutative 35.8%

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

   7. Simplified 35.8%

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

      1. *-commutative 35.8%

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

      2. sqrt-prod 41.3%

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

      3. +-commutative 41.3%

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

   9. Applied egg-rr 41.3%

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

       1. sqrt-unprod 35.8%

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

       2. +-commutative 35.8%

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

       3. /-rgt-identity 35.8%

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

       4. clear-num 35.8%

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

       5. div-inv 35.8%

          \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{y + x}{\frac{1}{z}}}} \]

       6. clear-num 35.8%

          \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{1}{\frac{\frac{1}{z}}{y + x}}}} \]

       7. metadata-eval 35.8%

          \[\leadsto 2 \cdot \sqrt{\frac{\color{blue}{1 \cdot 1}}{\frac{\frac{1}{z}}{y + x}}} \]

       8. add-sqr-sqrt 35.8%

          \[\leadsto 2 \cdot \sqrt{\frac{1 \cdot 1}{\color{blue}{\sqrt{\frac{\frac{1}{z}}{y + x}} \cdot \sqrt{\frac{\frac{1}{z}}{y + x}}}}} \]

       9. frac-times 35.8%

          \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{1}{\sqrt{\frac{\frac{1}{z}}{y + x}}} \cdot \frac{1}{\sqrt{\frac{\frac{1}{z}}{y + x}}}}} \]

       10. sqrt-unprod 36.5%

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

       11. add-sqr-sqrt 36.7%

           \[\leadsto 2 \cdot \color{blue}{\frac{1}{\sqrt{\frac{\frac{1}{z}}{y + x}}}} \]

       12. inv-pow 36.7%

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

       13. sqrt-pow2 36.9%

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

       14. associate-/r* 35.9%

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

       15. +-commutative 35.9%

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

       16. metadata-eval 35.9%

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

   11. Applied egg-rr 35.9%

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

       1. associate-/r* 36.9%

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

       2. +-commutative 36.9%

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

   13. Simplified 36.9%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z}}{y + x}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-292}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e-292) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e-292) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.15d-292) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e-292) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 1.15e-292:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.15e-292)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.15e-292)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.15e-292], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1499999999999999e-292

   1. Initial program 74.2%

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

      1. associate-+l+ 74.2%

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

      2. +-commutative 74.2%

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

      3. distribute-rgt-in 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in x around inf 50.6%

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

   if 1.1499999999999999e-292 < y

   1. Initial program 63.5%

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

      1. associate-+l+ 63.5%

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

      2. +-commutative 63.5%

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

      3. distribute-rgt-in 63.6%

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

   3. Simplified 63.6%

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

   5. Taylor expanded in x around 0 20.4%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-292}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-285) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-285) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-285)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-285) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -5e-285:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-285)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-285)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -5e-285], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000018e-285

   1. Initial program 72.6%

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

      1. associate-+l+ 72.6%

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

      2. +-commutative 72.6%

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

      3. distribute-rgt-in 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in x around inf 47.0%

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

   if -5.00000000000000018e-285 < y

   1. Initial program 65.8%

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

      1. associate-+l+ 65.8%

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

      2. +-commutative 65.8%

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

      3. distribute-rgt-in 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in z around inf 37.8%

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

      1. +-commutative 37.8%

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

   7. Simplified 37.8%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% 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 69.2%

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

   1. associate-+l+ 69.2%

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

   2. +-commutative 69.2%

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

   3. distribute-rgt-in 69.2%

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

3. Simplified 69.2%

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

5. Final simplification 69.2%

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

Alternative 8: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 73.3%

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

      1. associate-+l+ 73.3%

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

      2. +-commutative 73.3%

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

      3. distribute-rgt-in 73.3%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in z around 0 25.4%

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

      1. *-commutative 25.4%

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

   7. Simplified 25.4%

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

   if -4.999999999999985e-310 < y

   1. Initial program 65.0%

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

      1. associate-+l+ 65.0%

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

      2. +-commutative 65.0%

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

      3. distribute-rgt-in 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in x around 0 19.3%

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

4. Final simplification 22.3%

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

Alternative 9: 36.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

   1. associate-+l+ 69.2%

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

   2. +-commutative 69.2%

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

   3. distribute-rgt-in 69.2%

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

3. Simplified 69.2%

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

5. Taylor expanded in z around 0 27.7%

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

   1. *-commutative 27.7%

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

7. Simplified 27.7%

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

8. Final simplification 27.7%

   \[\leadsto 2 \cdot \sqrt{y \cdot x} \]
9. Add Preprocessing

Developer target: 83.9% 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 2024024 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< z 7.636950090573675e+176) (* 2.0 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25))) (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25)))) 2.0))

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