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

Percentage Accurate: 70.1% → 94.2%

Time: 12.5s

Alternatives: 9

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: 94.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+16}:\\ \;\;\;\;\left(-y\right) \cdot \mathsf{fma}\left(2, \sqrt{\frac{x + z}{y}}, \left(-x \cdot z\right) \cdot \sqrt{\frac{1}{\left(x + z\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}}\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-232}:\\ \;\;\;\;2 \cdot \sqrt{\left(x \cdot z + y \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e+16)
   (*
    (- y)
    (fma
     2.0
     (sqrt (/ (+ x z) y))
     (* (- (* x z)) (sqrt (/ 1.0 (* (+ x z) (* y (* y y))))))))
   (if (<= y 4.6e-232)
     (* 2.0 (sqrt (+ (+ (* x z) (* y x)) (* y 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 <= -2.7e+16) {
		tmp = -y * fma(2.0, sqrt(((x + z) / y)), (-(x * z) * sqrt((1.0 / ((x + z) * (y * (y * y)))))));
	} else if (y <= 4.6e-232) {
		tmp = 2.0 * sqrt((((x * z) + (y * x)) + (y * 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 <= -2.7e+16)
		tmp = Float64(Float64(-y) * fma(2.0, sqrt(Float64(Float64(x + z) / y)), Float64(Float64(-Float64(x * z)) * sqrt(Float64(1.0 / Float64(Float64(x + z) * Float64(y * Float64(y * y))))))));
	elseif (y <= 4.6e-232)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(x * z) + Float64(y * x)) + Float64(y * 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, -2.7e+16], N[((-y) * N[(2.0 * N[Sqrt[N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision] + N[((-N[(x * z), $MachinePrecision]) * N[Sqrt[N[(1.0 / N[(N[(x + z), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e-232], N[(2.0 * N[Sqrt[N[(N[(N[(x * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + N[(y * 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 < -2.7e16

   1. Initial program 51.0%

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

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

      4. lift-*.f64 N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-+ N/A

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

   4. Applied egg-rr 51.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. un-div-inv N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 51.3

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

   9. Simplified 43.7%

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

   10. Taylor expanded in y around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. mul-1-neg N/A

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

       14. lower-neg.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-sqrt.f64 N/A

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

       17. lower-/.f64 N/A

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

       18. lower-*.f64 N/A

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

   12. Simplified 80.4%

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

   if -2.7e16 < y < 4.6e-232

   1. Initial program 83.0%

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

   if 4.6e-232 < y

   1. Initial program 64.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 36.5

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

   5. Simplified 36.5%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)}} \]
   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 53.4

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

   7. Applied egg-rr 53.4%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+16}:\\ \;\;\;\;\left(-y\right) \cdot \mathsf{fma}\left(2, \sqrt{\frac{x + z}{y}}, \left(-x \cdot z\right) \cdot \sqrt{\frac{1}{\left(x + z\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}}\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-232}:\\ \;\;\;\;2 \cdot \sqrt{\left(x \cdot z + y \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-275}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\frac{x \cdot z}{y \cdot \left(\left(x + z\right) \cdot \left(x + z\right)\right)} + \frac{-1}{x + z}}}{\sqrt{-y}}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -9.2e-275)
   (/
    2.0
    (/
     (sqrt (+ (/ (* x z) (* y (* (+ x z) (+ x z)))) (/ -1.0 (+ x z))))
     (sqrt (- y))))
   (* (* 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 <= -9.2e-275) {
		tmp = 2.0 / (sqrt((((x * z) / (y * ((x + z) * (x + z)))) + (-1.0 / (x + z)))) / sqrt(-y));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((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 <= (-9.2d-275)) then
        tmp = 2.0d0 / (sqrt((((x * z) / (y * ((x + z) * (x + z)))) + ((-1.0d0) / (x + z)))) / sqrt(-y))
    else
        tmp = (2.0d0 * sqrt(z)) * sqrt((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 <= -9.2e-275) {
		tmp = 2.0 / (Math.sqrt((((x * z) / (y * ((x + z) * (x + z)))) + (-1.0 / (x + z)))) / Math.sqrt(-y));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.sqrt((y + x));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -9.2e-275:
		tmp = 2.0 / (math.sqrt((((x * z) / (y * ((x + z) * (x + z)))) + (-1.0 / (x + z)))) / math.sqrt(-y))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.sqrt((y + x))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -9.2e-275)
		tmp = Float64(2.0 / Float64(sqrt(Float64(Float64(Float64(x * z) / Float64(y * Float64(Float64(x + z) * Float64(x + z)))) + Float64(-1.0 / Float64(x + z)))) / sqrt(Float64(-y))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(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 <= -9.2e-275)
		tmp = 2.0 / (sqrt((((x * z) / (y * ((x + z) * (x + z)))) + (-1.0 / (x + z)))) / sqrt(-y));
	else
		tmp = (2.0 * sqrt(z)) * sqrt((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, -9.2e-275], N[(2.0 / N[(N[Sqrt[N[(N[(N[(x * z), $MachinePrecision] / N[(y * N[(N[(x + z), $MachinePrecision] * N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-y)], $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 2 regimes

2. if y < -9.19999999999999959e-275

   1. Initial program 66.1%

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

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

      4. lift-*.f64 N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-+ N/A

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

   4. Applied egg-rr 66.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. un-div-inv N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 66.1

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

   6. Applied egg-rr 66.1%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

   9. Simplified 40.6%

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

       1. lift-neg.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-fma.f64 N/A

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

       10. frac-2neg N/A

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

   11. Applied egg-rr 55.1%

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

   if -9.19999999999999959e-275 < y

   1. Initial program 69.2%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 46.2

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

   5. Simplified 46.2%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)}} \]
   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 52.0

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

   7. Applied egg-rr 52.0%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-275}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\frac{x \cdot z}{y \cdot \left(\left(x + z\right) \cdot \left(x + z\right)\right)} + \frac{-1}{x + z}}}{\sqrt{-y}}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-247}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-247)
   (* 2.0 (sqrt (* x (+ y 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 <= 2e-247) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt((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 <= 2d-247) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = (2.0d0 * sqrt(z)) * sqrt((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 <= 2e-247) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.sqrt((y + x));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2e-247:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.sqrt((y + x))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-247)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(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 <= 2e-247)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = (2.0 * sqrt(z)) * sqrt((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, 2e-247], N[(2.0 * N[Sqrt[N[(x * N[(y + 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 2 regimes

2. if y < 2e-247

   1. Initial program 68.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 51.4

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

   5. Simplified 51.4%

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

   if 2e-247 < y

   1. Initial program 66.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 40.0

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

   5. Simplified 40.0%

      \[\leadsto 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)}} \]
   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 54.1

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

   7. Applied egg-rr 54.1%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-247}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-247}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(2 \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e-247)
   (* 2.0 (sqrt (* x (+ y z))))
   (* (sqrt z) (* 2.0 (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e-247) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = sqrt(z) * (2.0 * 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 <= 2.1d-247) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = sqrt(z) * (2.0d0 * 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 <= 2.1e-247) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = Math.sqrt(z) * (2.0 * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.1e-247:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = math.sqrt(z) * (2.0 * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e-247)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(sqrt(z) * Float64(2.0 * 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 <= 2.1e-247)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = sqrt(z) * (2.0 * 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, 2.1e-247], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(2.0 * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.10000000000000014e-247

   1. Initial program 68.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 51.4

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

   5. Simplified 51.4%

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

   if 2.10000000000000014e-247 < y

   1. Initial program 66.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 27.5

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

   5. Simplified 27.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 44.8

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

   7. Applied egg-rr 44.8%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-247}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(2 \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, x, y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-294)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (sqrt (fma z x (* y z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-294) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt(fma(z, x, (y * z)));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-294)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(fma(z, x, 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, 5e-294], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * x + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000003e-294

   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 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 48.3

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

   5. Simplified 48.3%

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

   if 5.0000000000000003e-294 < y

   1. Initial program 68.7%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 44.7

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

   5. Simplified 44.7%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 44.7

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

   7. Applied egg-rr 44.7%

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

Alternative 6: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-294) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-294) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5d-294) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-294) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 5e-294:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-294)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e-294)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 5e-294], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000003e-294

   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 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 48.3

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

   5. Simplified 48.3%

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

   if 5.0000000000000003e-294 < y

   1. Initial program 68.7%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 44.7

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

   5. Simplified 44.7%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-247}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e-247) (* 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 <= 2.1e-247) {
		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 <= 2.1d-247) 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 <= 2.1e-247) {
		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 <= 2.1e-247:
		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 <= 2.1e-247)
		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 <= 2.1e-247)
		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, 2.1e-247], 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 < 2.10000000000000014e-247

   1. Initial program 68.5%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 51.4

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

   5. Simplified 51.4%

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

   if 2.10000000000000014e-247 < y

   1. Initial program 66.4%

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   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 27.5

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

   5. Simplified 27.5%

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

Alternative 8: 68.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-311) (* 2.0 (sqrt (* y x))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-311) {
		tmp = 2.0 * sqrt((y * x));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-311)) then
        tmp = 2.0d0 * sqrt((y * x))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-311) {
		tmp = 2.0 * Math.sqrt((y * x));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -5e-311:
		tmp = 2.0 * math.sqrt((y * x))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-311)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-311)
		tmp = 2.0 * sqrt((y * x));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -5e-311], 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 < -5.00000000000023e-311

   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 0

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

      1. lower-*.f64 27.3

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

   5. Simplified 27.3%

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

   if -5.00000000000023e-311 < y

   1. Initial program 68.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 25.1

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

   5. Simplified 25.1%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.0% 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 26.3

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

5. Simplified 26.3%

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

6. Final simplification 26.3%

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

Developer Target 1: 82.9% 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 2024210 
(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)))))