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

Percentage Accurate: 71.3% → 96.3%

Time: 5.9s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-2\right) \cdot \left(\sqrt{{\left(-y\right)}^{-1}} \cdot \sqrt{-\left(z + x\right)}\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \end{array} \]

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 (pow (- y) -1.0)) (sqrt (- (+ z x))))) y)
   (* (/ (* (sqrt y) 2.0) (sqrt z)) 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(pow(-y, -1.0)) * sqrt(-(z + x)))) * y;
	} else {
		tmp = ((sqrt(y) * 2.0) / sqrt(z)) * 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 ** (-1.0d0))) * sqrt(-(z + x)))) * y
    else
        tmp = ((sqrt(y) * 2.0d0) / sqrt(z)) * 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(Math.pow(-y, -1.0)) * Math.sqrt(-(z + x)))) * y;
	} else {
		tmp = ((Math.sqrt(y) * 2.0) / Math.sqrt(z)) * 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(math.pow(-y, -1.0)) * math.sqrt(-(z + x)))) * y
	else:
		tmp = ((math.sqrt(y) * 2.0) / math.sqrt(z)) * 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(Float64(Float64(-2.0) * Float64(sqrt((Float64(-y) ^ -1.0)) * sqrt(Float64(-Float64(z + x))))) * y);
	else
		tmp = Float64(Float64(Float64(sqrt(y) * 2.0) / sqrt(z)) * 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 ^ -1.0)) * sqrt(-(z + x)))) * y;
	else
		tmp = ((sqrt(y) * 2.0) / sqrt(z)) * 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[(N[((-2.0) * N[(N[Sqrt[N[Power[(-y), -1.0], $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-N[(z + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[Sqrt[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 65.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 56.6%

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

   10. Applied rewrites 62.8%

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

   if -4.999999999999985e-310 < y

   1. Initial program 66.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.8%

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

   10. Applied rewrites 41.0%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-2\right) \cdot \left(\sqrt{{\left(-y\right)}^{-1}} \cdot \sqrt{-\left(z + x\right)}\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{y} \cdot 2}{\sqrt{z}} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.6× speedup

Math

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

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 (- (+ z x))) (sqrt (- y)))) y)
   (* (/ (* (sqrt y) 2.0) (sqrt z)) 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(-(z + x)) / sqrt(-y))) * y;
	} else {
		tmp = ((sqrt(y) * 2.0) / sqrt(z)) * 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(-(z + x)) / sqrt(-y))) * y
    else
        tmp = ((sqrt(y) * 2.0d0) / sqrt(z)) * 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(-(z + x)) / Math.sqrt(-y))) * y;
	} else {
		tmp = ((Math.sqrt(y) * 2.0) / Math.sqrt(z)) * 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(-(z + x)) / math.sqrt(-y))) * y
	else:
		tmp = ((math.sqrt(y) * 2.0) / math.sqrt(z)) * 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(Float64(Float64(-2.0) * Float64(sqrt(Float64(-Float64(z + x))) / sqrt(Float64(-y)))) * y);
	else
		tmp = Float64(Float64(Float64(sqrt(y) * 2.0) / sqrt(z)) * 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(-(z + x)) / sqrt(-y))) * y;
	else
		tmp = ((sqrt(y) * 2.0) / sqrt(z)) * 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[(N[((-2.0) * N[(N[Sqrt[(-N[(z + x), $MachinePrecision])], $MachinePrecision] / N[Sqrt[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[Sqrt[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 65.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 56.6%

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

   10. Applied rewrites 62.8%

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

   if -4.999999999999985e-310 < y

   1. Initial program 66.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.8%

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

   10. Applied rewrites 41.0%

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

4. Final simplification 53.3%

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

Alternative 3: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1500000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{x}{y}}\right) \cdot y\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x + y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1500000000000.0)
   (* (* -2.0 (sqrt (/ x y))) y)
   (if (<= y 4.3e-55)
     (* (sqrt (fma y (+ z x) (* z x))) 2.0)
     (* (* (sqrt (/ (+ x y) z)) 2.0) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1500000000000.0) {
		tmp = (-2.0 * sqrt((x / y))) * y;
	} else if (y <= 4.3e-55) {
		tmp = sqrt(fma(y, (z + x), (z * x))) * 2.0;
	} else {
		tmp = (sqrt(((x + y) / z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1500000000000.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(x / y))) * y);
	elseif (y <= 4.3e-55)
		tmp = Float64(sqrt(fma(y, Float64(z + x), Float64(z * x))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(x + y) / z)) * 2.0) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1500000000000.0], N[(N[(-2.0 * N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 4.3e-55], N[(N[Sqrt[N[(y * N[(z + x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e12

   1. Initial program 44.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 85.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.1%

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

   if -1.5e12 < y < 4.3000000000000001e-55

   1. Initial program 80.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 80.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 80.2

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

   4. Applied rewrites 80.2%

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

   if 4.3000000000000001e-55 < y

   1. Initial program 55.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 55.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1500000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{x}{y}}\right) \cdot y\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{x + y}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1500000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{x}{y}}\right) \cdot y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{z}{y}} \cdot 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1500000000000.0)
   (* (* -2.0 (sqrt (/ x y))) y)
   (if (<= y 1.7e+15)
     (* (sqrt (fma y (+ z x) (* z x))) 2.0)
     (* (* (sqrt (/ z y)) 2.0) y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1500000000000.0) {
		tmp = (-2.0 * sqrt((x / y))) * y;
	} else if (y <= 1.7e+15) {
		tmp = sqrt(fma(y, (z + x), (z * x))) * 2.0;
	} else {
		tmp = (sqrt((z / y)) * 2.0) * y;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1500000000000.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(x / y))) * y);
	elseif (y <= 1.7e+15)
		tmp = Float64(sqrt(fma(y, Float64(z + x), Float64(z * x))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(z / y)) * 2.0) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1500000000000.0], N[(N[(-2.0 * N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.7e+15], N[(N[Sqrt[N[(y * N[(z + x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(z / y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e12

   1. Initial program 44.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 85.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.1%

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

   if -1.5e12 < y < 1.7e15

   1. Initial program 79.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 79.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

   if 1.7e15 < y

   1. Initial program 50.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.8%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1500000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{x}{y}}\right) \cdot y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{z}{y}} \cdot 2\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1500000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{x}{y}}\right) \cdot y\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1500000000000.0)
   (* (* -2.0 (sqrt (/ x y))) y)
   (if (<= y 9.5e+16)
     (* (sqrt (fma y (+ z x) (* z x))) 2.0)
     (* (* (sqrt (/ y z)) 2.0) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1500000000000.0) {
		tmp = (-2.0 * sqrt((x / y))) * y;
	} else if (y <= 9.5e+16) {
		tmp = sqrt(fma(y, (z + x), (z * x))) * 2.0;
	} else {
		tmp = (sqrt((y / z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1500000000000.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(x / y))) * y);
	elseif (y <= 9.5e+16)
		tmp = Float64(sqrt(fma(y, Float64(z + x), Float64(z * x))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(y / z)) * 2.0) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1500000000000.0], N[(N[(-2.0 * N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 9.5e+16], N[(N[Sqrt[N[(y * N[(z + x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e12

   1. Initial program 44.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 85.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.1%

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

   if -1.5e12 < y < 9.5e16

   1. Initial program 79.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 79.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

   if 9.5e16 < y

   1. Initial program 50.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.5%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1500000000000:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{x}{y}}\right) \cdot y\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 9.5e+16)
   (* (sqrt (fma y (+ z x) (* z x))) 2.0)
   (* (* (sqrt (/ y z)) 2.0) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.5e+16) {
		tmp = sqrt(fma(y, (z + x), (z * x))) * 2.0;
	} else {
		tmp = (sqrt((y / z)) * 2.0) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 9.5e+16)
		tmp = Float64(sqrt(fma(y, Float64(z + x), Float64(z * x))) * 2.0);
	else
		tmp = Float64(Float64(sqrt(Float64(y / z)) * 2.0) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 9.5e+16], N[(N[Sqrt[N[(y * N[(z + x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5e16

   1. Initial program 69.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 69.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 69.8

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

   4. Applied rewrites 69.8%

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

   if 9.5e16 < y

   1. Initial program 50.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y, z + x, z \cdot x\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{y}{z}} \cdot 2\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, x, z \cdot y\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e-291)
   (* (sqrt (* (+ z y) x)) 2.0)
   (* (sqrt (fma z x (* z y))) 2.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e-291) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = sqrt(fma(z, x, (z * y))) * 2.0;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.5e-291)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	else
		tmp = Float64(sqrt(fma(z, x, Float64(z * y))) * 2.0);
	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, -2.5e-291], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Sqrt[N[(z * x + N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5000000000000002e-291

   1. Initial program 65.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 46.6

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

   5. Applied rewrites 46.6%

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

   if -2.5000000000000002e-291 < y

   1. Initial program 65.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 46.8

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

   5. Applied rewrites 46.8%

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

   7. Applied rewrites 46.8%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, x, z \cdot y\right)} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e-291)
   (* (sqrt (* (+ z y) x)) 2.0)
   (* (sqrt (* (+ x y) z)) 2.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e-291) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = sqrt(((x + y) * z)) * 2.0;
	}
	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 <= (-2.5d-291)) then
        tmp = sqrt(((z + y) * x)) * 2.0d0
    else
        tmp = sqrt(((x + y) * z)) * 2.0d0
    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 <= -2.5e-291) {
		tmp = Math.sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = Math.sqrt(((x + y) * z)) * 2.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -2.5e-291:
		tmp = math.sqrt(((z + y) * x)) * 2.0
	else:
		tmp = math.sqrt(((x + y) * z)) * 2.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.5e-291)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	else
		tmp = Float64(sqrt(Float64(Float64(x + y) * z)) * 2.0);
	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 <= -2.5e-291)
		tmp = sqrt(((z + y) * x)) * 2.0;
	else
		tmp = sqrt(((x + y) * z)) * 2.0;
	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, -2.5e-291], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Sqrt[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5000000000000002e-291

   1. Initial program 65.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 46.6

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

   5. Applied rewrites 46.6%

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

   if -2.5000000000000002e-291 < y

   1. Initial program 65.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + y\right) \cdot z} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot y} \cdot 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.2e-291) (* (sqrt (* (+ z y) x)) 2.0) (* (sqrt (* z y)) 2.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e-291) {
		tmp = sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = sqrt((z * y)) * 2.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.2d-291) then
        tmp = sqrt(((z + y) * x)) * 2.0d0
    else
        tmp = sqrt((z * y)) * 2.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e-291) {
		tmp = Math.sqrt(((z + y) * x)) * 2.0;
	} else {
		tmp = Math.sqrt((z * y)) * 2.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 4.2e-291:
		tmp = math.sqrt(((z + y) * x)) * 2.0
	else:
		tmp = math.sqrt((z * y)) * 2.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.2e-291)
		tmp = Float64(sqrt(Float64(Float64(z + y) * x)) * 2.0);
	else
		tmp = Float64(sqrt(Float64(z * y)) * 2.0);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.2e-291)
		tmp = sqrt(((z + y) * x)) * 2.0;
	else
		tmp = sqrt((z * y)) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 4.2e-291], N[(N[Sqrt[N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Sqrt[N[(z * y), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.1999999999999999e-291

   1. Initial program 65.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 47.2

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

   5. Applied rewrites 47.2%

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

   if 4.1999999999999999e-291 < y

   1. Initial program 65.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 27.0

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

   5. Applied rewrites 27.0%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{\left(z + y\right) \cdot x} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot y} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-out N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 65.6

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 65.6

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

4. Applied rewrites 65.6%

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

5. Final simplification 65.6%

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

Alternative 11: 69.2% accurate, 1.4× speedup

Math

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

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

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = sqrt((x * y)) * 2.0;
	} else {
		tmp = sqrt((z * y)) * 2.0;
	}
	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 <= (-5d-310)) then
        tmp = sqrt((x * y)) * 2.0d0
    else
        tmp = sqrt((z * y)) * 2.0d0
    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 <= -5e-310) {
		tmp = Math.sqrt((x * y)) * 2.0;
	} else {
		tmp = Math.sqrt((z * y)) * 2.0;
	}
	return tmp;
}

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 65.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.2

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

   5. Applied rewrites 20.2%

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

   if -4.999999999999985e-310 < y

   1. Initial program 66.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 26.2

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

   5. Applied rewrites 26.2%

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

4. Final simplification 22.8%

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

Alternative 12: 34.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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((x * y)) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in z around 0

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

   1. lower-sqrt.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 20.2

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

5. Applied rewrites 20.2%

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

6. Final simplification 20.2%

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

Developer Target 1: 83.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\\ \mathbf{if}\;z < 7.636950090573675 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z)))
          (* (pow z 0.25) (pow y 0.25)))))
   (if (< z 7.636950090573675e+176)
     (* 2.0 (sqrt (+ (* (+ x y) z) (* x y))))
     (* (* t_0 t_0) 2.0))))

C

double code(double x, double y, double z) {
	double t_0 = (0.25 * ((pow(y, -0.75) * (pow(z, -0.75) * x)) * (y + z))) + (pow(z, 0.25) * pow(y, 0.25));
	double tmp;
	if (z < 7.636950090573675e+176) {
		tmp = 2.0 * sqrt((((x + y) * z) + (x * y)));
	} else {
		tmp = (t_0 * t_0) * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.25d0 * (((y ** (-0.75d0)) * ((z ** (-0.75d0)) * x)) * (y + z))) + ((z ** 0.25d0) * (y ** 0.25d0))
    if (z < 7.636950090573675d+176) then
        tmp = 2.0d0 * sqrt((((x + y) * z) + (x * y)))
    else
        tmp = (t_0 * t_0) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.25 * ((Math.pow(y, -0.75) * (Math.pow(z, -0.75) * x)) * (y + z))) + (Math.pow(z, 0.25) * Math.pow(y, 0.25));
	double tmp;
	if (z < 7.636950090573675e+176) {
		tmp = 2.0 * Math.sqrt((((x + y) * z) + (x * y)));
	} else {
		tmp = (t_0 * t_0) * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.25 * ((math.pow(y, -0.75) * (math.pow(z, -0.75) * x)) * (y + z))) + (math.pow(z, 0.25) * math.pow(y, 0.25))
	tmp = 0
	if z < 7.636950090573675e+176:
		tmp = 2.0 * math.sqrt((((x + y) * z) + (x * y)))
	else:
		tmp = (t_0 * t_0) * 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.25 * Float64(Float64((y ^ -0.75) * Float64((z ^ -0.75) * x)) * Float64(y + z))) + Float64((z ^ 0.25) * (y ^ 0.25)))
	tmp = 0.0
	if (z < 7.636950090573675e+176)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x + y) * z) + Float64(x * y))));
	else
		tmp = Float64(Float64(t_0 * t_0) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.25 * (((y ^ -0.75) * ((z ^ -0.75) * x)) * (y + z))) + ((z ^ 0.25) * (y ^ 0.25));
	tmp = 0.0;
	if (z < 7.636950090573675e+176)
		tmp = 2.0 * sqrt((((x + y) * z) + (x * y)));
	else
		tmp = (t_0 * t_0) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.25 * N[(N[(N[Power[y, -0.75], $MachinePrecision] * N[(N[Power[z, -0.75], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[z, 0.25], $MachinePrecision] * N[Power[y, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, 7.636950090573675e+176], N[(2.0 * N[Sqrt[N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 2.0), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024308 
(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)))))