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

Percentage Accurate: 70.9% → 97.5%

Time: 11.5s

Alternatives: 10

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: 70.9% 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: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, x + z, x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \sqrt{z}}{\sqrt{\frac{1}{y + x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -9e+30)
   (* y (* -2.0 (sqrt (/ x y))))
   (if (<= y 1.9e-281)
     (* 2.0 (sqrt (fma y (+ x z) (* x z))))
     (/ (* 2.0 (sqrt z)) (sqrt (/ 1.0 (+ y x)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -9e+30) {
		tmp = y * (-2.0 * sqrt((x / y)));
	} else if (y <= 1.9e-281) {
		tmp = 2.0 * sqrt(fma(y, (x + z), (x * z)));
	} else {
		tmp = (2.0 * sqrt(z)) / sqrt((1.0 / (y + x)));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -9e+30)
		tmp = Float64(y * Float64(-2.0 * sqrt(Float64(x / y))));
	elseif (y <= 1.9e-281)
		tmp = Float64(2.0 * sqrt(fma(y, Float64(x + z), Float64(x * z))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) / sqrt(Float64(1.0 / Float64(y + x))));
	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, -9e+30], N[(y * N[(-2.0 * N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e-281], N[(2.0 * N[Sqrt[N[(y * N[(x + z), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.9999999999999999e30

   1. Initial program 45.9%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. 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. Simplified 0.6%

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

   6. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 97.7

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

   8. Simplified 97.7%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 97.7

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

   11. Simplified 97.7%

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

   if -8.9999999999999999e30 < y < 1.89999999999999988e-281

   1. Initial program 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 93.6

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

   4. Applied egg-rr 93.6%

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

   if 1.89999999999999988e-281 < 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 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 67.6

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

   5. Simplified 67.6%

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

      1. lift-+.f64 N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 99.4

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.4

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

   7. Applied egg-rr 99.4%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.4

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

   9. Applied egg-rr 99.4%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. metadata-eval N/A

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

       4. flip3-+ N/A

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

       5. clear-num N/A

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

       6. sqrt-div N/A

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

       7. clear-num N/A

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

       8. flip3-+ N/A

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

       9. lift-+.f64 N/A

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

       10. remove-double-div N/A

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

       11. metadata-eval N/A

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

       12. sqrt-div N/A

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

       13. lift-/.f64 N/A

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

       14. lift-sqrt.f64 N/A

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

       15. un-div-inv N/A

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

       16. lower-/.f64 99.5

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

   11. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\frac{2 \cdot \sqrt{z}}{\sqrt{\frac{1}{x + y}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, x + z, x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \sqrt{z}}{\sqrt{\frac{1}{y + x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, x + z, x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -9e+30)
   (* y (* -2.0 (sqrt (/ x y))))
   (if (<= y 2.1e-289)
     (* 2.0 (sqrt (fma y (+ x z) (* x z))))
     (* (* 2.0 (sqrt z)) (sqrt (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -9e+30) {
		tmp = y * (-2.0 * sqrt((x / y)));
	} else if (y <= 2.1e-289) {
		tmp = 2.0 * sqrt(fma(y, (x + z), (x * z)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((y + x));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -9e+30)
		tmp = Float64(y * Float64(-2.0 * sqrt(Float64(x / y))));
	elseif (y <= 2.1e-289)
		tmp = Float64(2.0 * sqrt(fma(y, Float64(x + z), Float64(x * z))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(Float64(y + x)));
	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, -9e+30], N[(y * N[(-2.0 * N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-289], N[(2.0 * N[Sqrt[N[(y * N[(x + z), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.9999999999999999e30

   1. Initial program 39.7%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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. Simplified 0.6%

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

   6. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 98.3

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

   8. Simplified 98.3%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 98.3

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

   11. Simplified 98.3%

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

   if -8.9999999999999999e30 < y < 2.0999999999999998e-289

   1. Initial program 94.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 94.1

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

   4. Applied egg-rr 94.1%

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

   if 2.0999999999999998e-289 < y

   1. Initial program 70.2%

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

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

      2. lower-+.f64 70.4

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

   5. Simplified 70.4%

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

      1. lift-+.f64 N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 99.4

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.4

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

   7. Applied egg-rr 99.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(-2 \cdot \sqrt{\frac{x}{y}}\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, x + z, x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 83.5% 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 2024218 
(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)))))