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

Percentage Accurate: 70.1% → 96.3%

Time: 11.3s

Alternatives: 8

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: 70.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 69.3%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 60.6%

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

   10. Applied rewrites 66.1%

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

   if -4.999999999999985e-310 < y

   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 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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.8%

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

   10. Applied rewrites 70.5%

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

4. Final simplification 68.3%

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

Alternative 2: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(2 \cdot \sqrt{\frac{y}{z}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2e+23)
   (* y (* -2.0 (sqrt (/ x y))))
   (if (<= y 1.18e-7)
     (* 2.0 (sqrt (fma x y (* z (+ y x)))))
     (* z (* 2.0 (sqrt (/ y z)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e+23) {
		tmp = y * (-2.0 * sqrt((x / y)));
	} else if (y <= 1.18e-7) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = z * (2.0 * sqrt((y / z)));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2e+23)
		tmp = Float64(y * Float64(-2.0 * sqrt(Float64(x / y))));
	elseif (y <= 1.18e-7)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(z * Float64(2.0 * sqrt(Float64(y / 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, -2e+23], N[(y * N[(-2.0 * N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.18e-7], N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(z * N[(2.0 * N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9999999999999998e23

   1. Initial program 60.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 0.9%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 84.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 41.3%

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

   if -1.9999999999999998e23 < y < 1.18e-7

   1. Initial program 80.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.3

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 80.3

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

   4. Applied rewrites 80.3%

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

   if 1.18e-7 < y

   1. Initial program 53.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-*.f64 35.7

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 48.8%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(2 \cdot \sqrt{\frac{y}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(2 \cdot \sqrt{\frac{y}{z}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.18e-7)
   (* 2.0 (sqrt (fma x y (* z (+ y x)))))
   (* z (* 2.0 (sqrt (/ y z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.18e-7) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = z * (2.0 * sqrt((y / z)));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.18e-7)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(z * Float64(2.0 * sqrt(Float64(y / z))));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 1.18e-7], N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(z * N[(2.0 * N[Sqrt[N[(y / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.18e-7

   1. Initial program 72.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.9

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 72.9

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

   4. Applied rewrites 72.9%

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

   if 1.18e-7 < y

   1. Initial program 53.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-*.f64 35.7

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 48.8%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(2 \cdot \sqrt{\frac{y}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-297}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e-297)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e-297) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.1d-297) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e-297) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.1e-297:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e-297)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.1e-297)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2.1e-297], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.10000000000000013e-297

   1. Initial program 68.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 46.1

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

   5. Applied rewrites 46.1%

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

   if 2.10000000000000013e-297 < y

   1. Initial program 66.7%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. lower-*.f64 N/A

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

      2. lower-+.f64 48.3

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

   5. Applied rewrites 48.3%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-297}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e-265) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e-265) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.2d-265) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e-265) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 8.2e-265:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 8.2e-265)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.2e-265)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 8.2e-265], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.2e-265

   1. Initial program 67.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 46.7

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

   5. Applied rewrites 46.7%

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

   if 8.2e-265 < y

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 31.3

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

   5. Applied rewrites 31.3%

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

Alternative 6: 70.2% accurate, 1.2× speedup

Math

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

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 (fma x y (* z (+ y x))))))

C

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

Julia

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

Derivation

1. Initial program 67.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 67.6

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-rgt-out N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 67.6

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

4. Applied rewrites 67.6%

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

5. Final simplification 67.6%

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

Alternative 7: 67.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e-265) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-265) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.5d-265) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-265) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 8.5e-265:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e-265)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.5e-265)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 8.5e-265], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.4999999999999997e-265

   1. Initial program 67.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 23.7

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

   5. Applied rewrites 23.7%

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

   if 8.4999999999999997e-265 < y

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 31.3

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

   5. Applied rewrites 31.3%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

3. Taylor expanded in z around 0

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

   1. lower-*.f64 22.3

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

5. Applied rewrites 22.3%

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

6. Final simplification 22.3%

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

Developer Target 1: 83.1% 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 2024220 
(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)))))