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

Percentage Accurate: 70.0% → 83.9%

Time: 34.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.0% 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: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.65 \cdot 10^{-270}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.65e-270)
   (* 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 <= 3.65e-270) {
		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 <= 3.65d-270) 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 <= 3.65e-270) {
		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 <= 3.65e-270:
		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 <= 3.65e-270)
		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 <= 3.65e-270)
		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, 3.65e-270], 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 < 3.64999999999999996e-270

   1. Initial program 74.8%

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

      1. associate-+l+ 74.8%

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

      2. +-commutative 74.8%

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

      3. distribute-rgt-in 74.8%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in x around inf 51.5%

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

      1. +-commutative 51.5%

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

   7. Simplified 51.5%

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

   if 3.64999999999999996e-270 < y

   1. Initial program 73.5%

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

      1. associate-+l+ 73.5%

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

      2. +-commutative 73.5%

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

      3. distribute-rgt-in 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in z around inf 44.3%

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

      1. *-commutative 44.3%

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

      2. sqrt-prod 49.8%

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

      3. +-commutative 49.8%

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

   7. Applied egg-rr 49.8%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.65 \cdot 10^{-270}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(\left(y + x\right) \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e-298)
   (* 2.0 (sqrt (* x (+ y z))))
   (sqrt (* z (* (+ y x) 4.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e-298) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = sqrt((z * ((y + x) * 4.0)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.5d-298) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = sqrt((z * ((y + x) * 4.0d0)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e-298) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = Math.sqrt((z * ((y + x) * 4.0)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 3.5e-298:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = math.sqrt((z * ((y + x) * 4.0)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 3.5e-298)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = sqrt(Float64(z * Float64(Float64(y + x) * 4.0)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.5e-298)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = sqrt((z * ((y + x) * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 3.5e-298], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(z * N[(N[(y + x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4999999999999998e-298

   1. Initial program 73.8%

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

      1. associate-+l+ 73.8%

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

      2. +-commutative 73.8%

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

      3. distribute-rgt-in 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in x around inf 49.0%

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

      1. +-commutative 49.0%

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

   7. Simplified 49.0%

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

   if 3.4999999999999998e-298 < y

   1. Initial program 74.7%

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

      1. associate-+l+ 74.7%

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

      2. +-commutative 74.7%

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

      3. distribute-rgt-in 74.7%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in z around inf 47.7%

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

      1. pow1 47.7%

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

      2. add-sqr-sqrt 47.3%

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

      3. sqrt-unprod 47.7%

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

      4. *-commutative 47.7%

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

      5. *-commutative 47.7%

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

      6. swap-sqr 47.7%

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

      7. add-sqr-sqrt 47.7%

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

      8. +-commutative 47.7%

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

      9. metadata-eval 47.7%

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

   7. Applied egg-rr 47.7%

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

      1. unpow1 47.7%

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

      2. associate-*l* 47.7%

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

      3. +-commutative 47.7%

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

   9. Simplified 47.7%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(\left(y + x\right) \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-263}:\\ \;\;\;\;\sqrt{y \cdot \left(x \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(\left(y + x\right) \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -9e-263) (sqrt (* y (* x 4.0))) (sqrt (* z (* (+ y x) 4.0)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -9e-263) {
		tmp = sqrt((y * (x * 4.0)));
	} else {
		tmp = sqrt((z * ((y + x) * 4.0)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-9d-263)) then
        tmp = sqrt((y * (x * 4.0d0)))
    else
        tmp = sqrt((z * ((y + x) * 4.0d0)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -9e-263) {
		tmp = Math.sqrt((y * (x * 4.0)));
	} else {
		tmp = Math.sqrt((z * ((y + x) * 4.0)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -9e-263:
		tmp = math.sqrt((y * (x * 4.0)))
	else:
		tmp = math.sqrt((z * ((y + x) * 4.0)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -9e-263)
		tmp = sqrt(Float64(y * Float64(x * 4.0)));
	else
		tmp = sqrt(Float64(z * Float64(Float64(y + x) * 4.0)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -9e-263)
		tmp = sqrt((y * (x * 4.0)));
	else
		tmp = sqrt((z * ((y + x) * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -9e-263], N[Sqrt[N[(y * N[(x * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(z * N[(N[(y + x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.9999999999999994e-263

   1. Initial program 72.5%

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

      1. associate-+l+ 72.5%

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

      2. +-commutative 72.5%

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

      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 z around 0 27.9%

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

      1. add-cbrt-cube 21.4%

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

      2. add-sqr-sqrt 21.4%

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

      3. pow1 21.4%

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

      4. pow1/2 21.4%

         \[\leadsto 2 \cdot \sqrt[3]{{\left(x \cdot y\right)}^{1} \cdot \color{blue}{{\left(x \cdot y\right)}^{0.5}}} \]

      5. pow-prod-up 21.4%

         \[\leadsto 2 \cdot \sqrt[3]{\color{blue}{{\left(x \cdot y\right)}^{\left(1 + 0.5\right)}}} \]

      6. *-commutative 21.4%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(y \cdot x\right)}}^{\left(1 + 0.5\right)}} \]

      7. metadata-eval 21.4%

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

   7. Applied egg-rr 21.4%

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

      1. pow1 21.4%

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

      2. add-sqr-sqrt 21.4%

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

      3. sqrt-unprod 21.4%

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

      4. *-commutative 21.4%

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

      5. *-commutative 21.4%

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

      6. swap-sqr 21.4%

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

      7. cbrt-unprod 13.8%

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

      8. pow-prod-up 14.1%

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

      9. metadata-eval 14.1%

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

      10. pow3 14.1%

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

      11. add-cbrt-cube 27.9%

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

      12. metadata-eval 27.9%

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

   9. Applied egg-rr 27.9%

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

       1. unpow1 27.9%

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

       2. associate-*l* 27.9%

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

   11. Simplified 27.9%

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

   if -8.9999999999999994e-263 < y

   1. Initial program 75.7%

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

      1. associate-+l+ 75.7%

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

      2. +-commutative 75.7%

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

      3. distribute-rgt-in 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in z around inf 51.1%

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

      1. pow1 51.1%

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

      2. add-sqr-sqrt 50.7%

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

      3. sqrt-unprod 51.1%

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

      4. *-commutative 51.1%

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

      5. *-commutative 51.1%

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

      6. swap-sqr 51.1%

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

      7. add-sqr-sqrt 51.1%

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

      8. +-commutative 51.1%

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

      9. metadata-eval 51.1%

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

   7. Applied egg-rr 51.1%

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

      1. unpow1 51.1%

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

      2. associate-*l* 51.1%

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

      3. +-commutative 51.1%

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

   9. Simplified 51.1%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-263}:\\ \;\;\;\;\sqrt{y \cdot \left(x \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(\left(y + x\right) \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.1% 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 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. Final simplification 74.2%

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

Alternative 5: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{y \cdot \left(x \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-310) (sqrt (* y (* x 4.0))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = sqrt((y * (x * 4.0)));
	} 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 <= (-4d-310)) then
        tmp = sqrt((y * (x * 4.0d0)))
    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 <= -4e-310) {
		tmp = Math.sqrt((y * (x * 4.0)));
	} 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 <= -4e-310:
		tmp = math.sqrt((y * (x * 4.0)))
	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 <= -4e-310)
		tmp = sqrt(Float64(y * Float64(x * 4.0)));
	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 <= -4e-310)
		tmp = sqrt((y * (x * 4.0)));
	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, -4e-310], N[Sqrt[N[(y * N[(x * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.999999999999988e-310

   1. Initial program 74.1%

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

      1. associate-+l+ 74.1%

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

      2. +-commutative 74.1%

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

      3. distribute-rgt-in 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in z around 0 25.8%

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

      1. add-cbrt-cube 19.8%

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

      2. add-sqr-sqrt 19.8%

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

      3. pow1 19.8%

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

      4. pow1/2 19.8%

         \[\leadsto 2 \cdot \sqrt[3]{{\left(x \cdot y\right)}^{1} \cdot \color{blue}{{\left(x \cdot y\right)}^{0.5}}} \]

      5. pow-prod-up 19.9%

         \[\leadsto 2 \cdot \sqrt[3]{\color{blue}{{\left(x \cdot y\right)}^{\left(1 + 0.5\right)}}} \]

      6. *-commutative 19.9%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(y \cdot x\right)}}^{\left(1 + 0.5\right)}} \]

      7. metadata-eval 19.9%

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

   7. Applied egg-rr 19.9%

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

      1. pow1 19.9%

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

      2. add-sqr-sqrt 19.8%

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

      3. sqrt-unprod 19.9%

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

      4. *-commutative 19.9%

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

      5. *-commutative 19.9%

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

      6. swap-sqr 19.9%

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

      7. cbrt-unprod 12.9%

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

      8. pow-prod-up 13.2%

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

      9. metadata-eval 13.2%

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

      10. pow3 13.2%

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

      11. add-cbrt-cube 25.8%

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

      12. metadata-eval 25.8%

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

   9. Applied egg-rr 25.8%

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

       1. unpow1 25.8%

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

       2. associate-*l* 25.8%

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

   11. Simplified 25.8%

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

   if -3.999999999999988e-310 < y

   1. Initial program 74.3%

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

      1. associate-+l+ 74.3%

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

      2. +-commutative 74.3%

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

      3. distribute-rgt-in 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in x around 0 25.9%

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

      1. *-commutative 25.9%

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

   7. Simplified 25.9%

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

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{y \cdot \left(x \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt((y * (x * 4.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 z around 0 26.6%

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

   1. add-cbrt-cube 19.8%

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

   2. add-sqr-sqrt 19.8%

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

   3. pow1 19.8%

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

   4. pow1/2 19.8%

      \[\leadsto 2 \cdot \sqrt[3]{{\left(x \cdot y\right)}^{1} \cdot \color{blue}{{\left(x \cdot y\right)}^{0.5}}} \]

   5. pow-prod-up 19.9%

      \[\leadsto 2 \cdot \sqrt[3]{\color{blue}{{\left(x \cdot y\right)}^{\left(1 + 0.5\right)}}} \]

   6. *-commutative 19.9%

      \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(y \cdot x\right)}}^{\left(1 + 0.5\right)}} \]

   7. metadata-eval 19.9%

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

7. Applied egg-rr 19.9%

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

   1. pow1 19.9%

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

   2. add-sqr-sqrt 19.8%

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

   3. sqrt-unprod 19.9%

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

   4. *-commutative 19.9%

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

   5. *-commutative 19.9%

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

   6. swap-sqr 19.9%

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

   7. cbrt-unprod 12.1%

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

   8. pow-prod-up 12.4%

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

   9. metadata-eval 12.4%

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

   10. pow3 12.4%

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

   11. add-cbrt-cube 26.6%

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

   12. metadata-eval 26.6%

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

9. Applied egg-rr 26.6%

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

    1. unpow1 26.6%

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

    2. associate-*l* 26.6%

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

11. Simplified 26.6%

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

Developer Target 1: 82.4% 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 2024143 
(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)))))