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

Percentage Accurate: 70.4% → 94.6%

Time: 18.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.4% 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: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+43}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) + \frac{y}{x} \cdot \frac{z}{y + z}\right)}\\ \mathbf{elif}\;y \leq 440:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot \left(\sqrt{\frac{y + x}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{{z}^{-3}}{y + x}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e+43)
   (*
    2.0
    (exp
     (*
      0.5
      (+ (- (log (- (- y) z)) (log (/ -1.0 x))) (* (/ y x) (/ z (+ y z)))))))
   (if (<= y 440.0)
     (* 2.0 (sqrt (fma x z (* y (+ z x)))))
     (*
      2.0
      (*
       z
       (+
        (sqrt (/ (+ y x) z))
        (* 0.5 (* x (* y (sqrt (/ (pow z -3.0) (+ y x))))))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e+43) {
		tmp = 2.0 * exp((0.5 * ((log((-y - z)) - log((-1.0 / x))) + ((y / x) * (z / (y + z))))));
	} else if (y <= 440.0) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = 2.0 * (z * (sqrt(((y + x) / z)) + (0.5 * (x * (y * sqrt((pow(z, -3.0) / (y + x))))))));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -2.7e+43)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))) + Float64(Float64(y / x) * Float64(z / Float64(y + z)))))));
	elseif (y <= 440.0)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(2.0 * Float64(z * Float64(sqrt(Float64(Float64(y + x) / z)) + Float64(0.5 * Float64(x * Float64(y * sqrt(Float64((z ^ -3.0) / 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, -2.7e+43], N[(2.0 * N[Exp[N[(0.5 * N[(N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] * N[(z / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 440.0], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z * N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(x * N[(y * N[Sqrt[N[(N[Power[z, -3.0], $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7000000000000002e43

   1. Initial program 55.0%

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

      1. distribute-lft-out 55.2%

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

      2. *-commutative 55.2%

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

   4. Applied egg-rr 55.2%

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

      1. pow1/2 55.2%

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

      2. pow-to-exp 51.1%

         \[\leadsto 2 \cdot \color{blue}{e^{\log \left(\left(y + z\right) \cdot x + y \cdot z\right) \cdot 0.5}} \]

      3. fma-define 51.2%

         \[\leadsto 2 \cdot e^{\log \color{blue}{\left(\mathsf{fma}\left(y + z, x, y \cdot z\right)\right)} \cdot 0.5} \]

   6. Applied egg-rr 51.2%

      \[\leadsto 2 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(y + z, x, y \cdot z\right)\right) \cdot 0.5}} \]

   7. Taylor expanded in x around -inf 34.9%

      \[\leadsto 2 \cdot e^{\color{blue}{0.5 \cdot \left(\log \left(-1 \cdot \left(y + z\right)\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right) + 0.5 \cdot \frac{y \cdot z}{x \cdot \left(y + z\right)}}} \]
   8. Step-by-step derivation

      1. distribute-lft-out 34.9%

         \[\leadsto 2 \cdot e^{\color{blue}{0.5 \cdot \left(\left(\log \left(-1 \cdot \left(y + z\right)\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right) + \frac{y \cdot z}{x \cdot \left(y + z\right)}\right)}} \]

      2. mul-1-neg 34.9%

         \[\leadsto 2 \cdot e^{0.5 \cdot \left(\left(\log \color{blue}{\left(-\left(y + z\right)\right)} + -1 \cdot \log \left(\frac{-1}{x}\right)\right) + \frac{y \cdot z}{x \cdot \left(y + z\right)}\right)} \]

      3. mul-1-neg 34.9%

         \[\leadsto 2 \cdot e^{0.5 \cdot \left(\left(\log \left(-\left(y + z\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{x}\right)\right)}\right) + \frac{y \cdot z}{x \cdot \left(y + z\right)}\right)} \]

      4. times-frac 36.7%

         \[\leadsto 2 \cdot e^{0.5 \cdot \left(\left(\log \left(-\left(y + z\right)\right) + \left(-\log \left(\frac{-1}{x}\right)\right)\right) + \color{blue}{\frac{y}{x} \cdot \frac{z}{y + z}}\right)} \]

   9. Simplified 36.7%

      \[\leadsto 2 \cdot e^{\color{blue}{0.5 \cdot \left(\left(\log \left(-\left(y + z\right)\right) + \left(-\log \left(\frac{-1}{x}\right)\right)\right) + \frac{y}{x} \cdot \frac{z}{y + z}\right)}} \]

   if -2.7000000000000002e43 < y < 440

   1. Initial program 80.4%

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

      1. +-commutative 80.4%

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

      2. associate-+r+ 80.4%

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

      3. *-commutative 80.4%

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

      4. +-commutative 80.4%

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

      5. +-commutative 80.4%

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

      6. *-commutative 80.4%

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

      7. *-commutative 80.4%

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

      8. associate-+l+ 80.4%

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

      9. associate-+l+ 80.4%

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

      10. *-commutative 80.4%

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

      11. *-commutative 80.4%

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

      12. fma-define 80.4%

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

      13. distribute-lft-out 80.4%

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

   3. Simplified 80.4%

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

   if 440 < y

   1. Initial program 52.7%

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

      1. +-commutative 52.7%

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

      2. *-commutative 52.7%

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

      3. +-commutative 52.7%

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

      4. *-commutative 52.7%

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

      5. associate-+l+ 52.7%

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

      6. *-commutative 52.7%

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

      7. *-commutative 52.7%

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

      8. +-commutative 52.7%

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

      9. fma-define 52.7%

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

      10. +-commutative 52.7%

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

      11. distribute-lft-out 52.8%

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

   3. Simplified 52.8%

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

      1. fma-undefine 52.7%

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

      2. distribute-rgt-in 52.7%

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

      3. associate-+l+ 52.7%

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

      4. add-sqr-sqrt 52.4%

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

      5. pow2 52.4%

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

      6. pow1/2 52.4%

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

      7. sqrt-pow1 52.5%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      8. associate-+l+ 52.5%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      9. distribute-rgt-in 52.5%

         \[\leadsto 2 \cdot {\left({\left(x \cdot y + \color{blue}{z \cdot \left(x + y\right)}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      10. fma-undefine 52.6%

          \[\leadsto 2 \cdot {\left({\color{blue}{\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      11. metadata-eval 52.6%

          \[\leadsto 2 \cdot {\left({\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   6. Applied egg-rr 52.6%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in z around inf 42.5%

      \[\leadsto 2 \cdot \color{blue}{\left(z \cdot \left(\sqrt{\frac{x + y}{z}} + 0.5 \cdot \left(\left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*l* 48.7%

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

   9. Simplified 48.7%

      \[\leadsto 2 \cdot \color{blue}{\left(z \cdot \left(\sqrt{\frac{x + y}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 48.7%

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

       2. associate-/r* 48.7%

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

       3. pow-flip 48.7%

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

       4. metadata-eval 48.7%

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

   11. Applied egg-rr 48.7%

       \[\leadsto 2 \cdot \left(z \cdot \left(\sqrt{\frac{x + y}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{{z}^{-3}}{x + y}}\right)}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lft-identity 48.7%

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

   13. Simplified 48.7%

       \[\leadsto 2 \cdot \left(z \cdot \left(\sqrt{\frac{x + y}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \color{blue}{\sqrt{\frac{{z}^{-3}}{x + y}}}\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+43}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) + \frac{y}{x} \cdot \frac{z}{y + z}\right)}\\ \mathbf{elif}\;y \leq 440:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot \left(\sqrt{\frac{y + x}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{{z}^{-3}}{y + x}}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7000:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot \left(\sqrt{\frac{y + x}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{{z}^{-3}}{y + x}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 7000.0)
   (* 2.0 (sqrt (fma x z (* y (+ z x)))))
   (*
    2.0
    (*
     z
     (+
      (sqrt (/ (+ y x) z))
      (* 0.5 (* x (* y (sqrt (/ (pow z -3.0) (+ y x)))))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 7000.0) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = 2.0 * (z * (sqrt(((y + x) / z)) + (0.5 * (x * (y * sqrt((pow(z, -3.0) / (y + x))))))));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 7000.0)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(2.0 * Float64(z * Float64(sqrt(Float64(Float64(y + x) / z)) + Float64(0.5 * Float64(x * Float64(y * sqrt(Float64((z ^ -3.0) / 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, 7000.0], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(z * N[(N[Sqrt[N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(x * N[(y * N[Sqrt[N[(N[Power[z, -3.0], $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7e3

   1. Initial program 73.1%

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

      1. +-commutative 73.1%

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

      2. associate-+r+ 73.1%

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

      3. *-commutative 73.1%

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

      4. +-commutative 73.1%

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

      5. +-commutative 73.1%

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

      6. *-commutative 73.1%

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

      7. *-commutative 73.1%

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

      8. associate-+l+ 73.1%

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

      9. associate-+l+ 73.1%

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

      10. *-commutative 73.1%

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

      11. *-commutative 73.1%

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

      12. fma-define 73.1%

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

      13. distribute-lft-out 73.2%

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

   3. Simplified 73.2%

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

   if 7e3 < y

   1. Initial program 52.7%

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

      1. +-commutative 52.7%

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

      2. *-commutative 52.7%

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

      3. +-commutative 52.7%

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

      4. *-commutative 52.7%

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

      5. associate-+l+ 52.7%

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

      6. *-commutative 52.7%

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

      7. *-commutative 52.7%

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

      8. +-commutative 52.7%

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

      9. fma-define 52.7%

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

      10. +-commutative 52.7%

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

      11. distribute-lft-out 52.8%

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

   3. Simplified 52.8%

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

      1. fma-undefine 52.7%

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

      2. distribute-rgt-in 52.7%

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

      3. associate-+l+ 52.7%

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

      4. add-sqr-sqrt 52.4%

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

      5. pow2 52.4%

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

      6. pow1/2 52.4%

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

      7. sqrt-pow1 52.5%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      8. associate-+l+ 52.5%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      9. distribute-rgt-in 52.5%

         \[\leadsto 2 \cdot {\left({\left(x \cdot y + \color{blue}{z \cdot \left(x + y\right)}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      10. fma-undefine 52.6%

          \[\leadsto 2 \cdot {\left({\color{blue}{\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]

      11. metadata-eval 52.6%

          \[\leadsto 2 \cdot {\left({\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   6. Applied egg-rr 52.6%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in z around inf 42.5%

      \[\leadsto 2 \cdot \color{blue}{\left(z \cdot \left(\sqrt{\frac{x + y}{z}} + 0.5 \cdot \left(\left(x \cdot y\right) \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*l* 48.7%

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

   9. Simplified 48.7%

      \[\leadsto 2 \cdot \color{blue}{\left(z \cdot \left(\sqrt{\frac{x + y}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{1}{{z}^{3} \cdot \left(x + y\right)}}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 48.7%

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

       2. associate-/r* 48.7%

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

       3. pow-flip 48.7%

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

       4. metadata-eval 48.7%

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

   11. Applied egg-rr 48.7%

       \[\leadsto 2 \cdot \left(z \cdot \left(\sqrt{\frac{x + y}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{{z}^{-3}}{x + y}}\right)}\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lft-identity 48.7%

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

   13. Simplified 48.7%

       \[\leadsto 2 \cdot \left(z \cdot \left(\sqrt{\frac{x + y}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \color{blue}{\sqrt{\frac{{z}^{-3}}{x + y}}}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7000:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(z \cdot \left(\sqrt{\frac{y + x}{z}} + 0.5 \cdot \left(x \cdot \left(y \cdot \sqrt{\frac{{z}^{-3}}{y + x}}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 72000000:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, z + x\right)} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 72000000.0)
   (* 2.0 (sqrt (fma x z (* y (+ z x)))))
   (* 2.0 (* (sqrt (fma x (/ z y) (+ z x))) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 72000000.0) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = 2.0 * (sqrt(fma(x, (z / y), (z + x))) * sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 72000000.0)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(fma(x, Float64(z / y), Float64(z + x))) * sqrt(y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 72000000.0], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(x * N[(z / y), $MachinePrecision] + N[(z + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.2e7

   1. Initial program 73.1%

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

      1. +-commutative 73.1%

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

      2. associate-+r+ 73.1%

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

      3. *-commutative 73.1%

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

      4. +-commutative 73.1%

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

      5. +-commutative 73.1%

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

      6. *-commutative 73.1%

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

      7. *-commutative 73.1%

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

      8. associate-+l+ 73.1%

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

      9. associate-+l+ 73.1%

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

      10. *-commutative 73.1%

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

      11. *-commutative 73.1%

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

      12. fma-define 73.1%

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

      13. distribute-lft-out 73.2%

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

   3. Simplified 73.2%

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

   if 7.2e7 < y

   1. Initial program 52.7%

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

      1. +-commutative 52.7%

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

      2. *-commutative 52.7%

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

      3. +-commutative 52.7%

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

      4. *-commutative 52.7%

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

      5. associate-+l+ 52.7%

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

      6. *-commutative 52.7%

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

      7. *-commutative 52.7%

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

      8. +-commutative 52.7%

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

      9. fma-define 52.7%

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

      10. +-commutative 52.7%

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

      11. distribute-lft-out 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y around inf 52.6%

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

      1. associate-+r+ 52.6%

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

      2. +-commutative 52.6%

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

      3. associate-/l* 52.7%

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

   7. Simplified 52.7%

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

      1. pow1/2 52.7%

         \[\leadsto 2 \cdot \color{blue}{{\left(y \cdot \left(\left(z + x\right) + x \cdot \frac{z}{y}\right)\right)}^{0.5}} \]

      2. *-commutative 52.7%

         \[\leadsto 2 \cdot {\color{blue}{\left(\left(\left(z + x\right) + x \cdot \frac{z}{y}\right) \cdot y\right)}}^{0.5} \]

      3. unpow-prod-down 96.8%

         \[\leadsto 2 \cdot \color{blue}{\left({\left(\left(z + x\right) + x \cdot \frac{z}{y}\right)}^{0.5} \cdot {y}^{0.5}\right)} \]

      4. pow1/2 96.8%

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

      5. +-commutative 96.8%

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

      6. fma-define 96.8%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, z + x\right)}} \cdot {y}^{0.5}\right) \]

      7. pow1/2 96.8%

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

   9. Applied egg-rr 96.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 72000000:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{\mathsf{fma}\left(x, \frac{z}{y}, z + x\right)} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + \mathsf{fma}\left(y, \frac{z}{x}, z\right)} \cdot \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x 4.6e+36)
   (* 2.0 (sqrt (fma x z (* y (+ z x)))))
   (* 2.0 (* (sqrt (+ y (fma y (/ z x) z))) (sqrt x)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.6e+36) {
		tmp = 2.0 * sqrt(fma(x, z, (y * (z + x))));
	} else {
		tmp = 2.0 * (sqrt((y + fma(y, (z / x), z))) * sqrt(x));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= 4.6e+36)
		tmp = Float64(2.0 * sqrt(fma(x, z, Float64(y * Float64(z + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + fma(y, Float64(z / x), z))) * sqrt(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[x, 4.6e+36], N[(2.0 * N[Sqrt[N[(x * z + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + N[(y * N[(z / x), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.59999999999999993e36

   1. Initial program 75.0%

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

      1. +-commutative 75.0%

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

      2. associate-+r+ 75.0%

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

      3. *-commutative 75.0%

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

      4. +-commutative 75.0%

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

      5. +-commutative 75.0%

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

      6. *-commutative 75.0%

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

      7. *-commutative 75.0%

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

      8. associate-+l+ 75.0%

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

      9. associate-+l+ 75.0%

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

      10. *-commutative 75.0%

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

      11. *-commutative 75.0%

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

      12. fma-define 75.0%

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

      13. distribute-lft-out 75.1%

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

   3. Simplified 75.1%

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

   if 4.59999999999999993e36 < x

   1. Initial program 44.8%

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

      1. +-commutative 44.8%

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

      2. *-commutative 44.8%

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

      3. +-commutative 44.8%

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

      4. *-commutative 44.8%

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

      5. associate-+l+ 44.8%

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

      6. *-commutative 44.8%

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

      7. *-commutative 44.8%

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

      8. +-commutative 44.8%

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

      9. fma-define 44.8%

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

      10. +-commutative 44.8%

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

      11. distribute-lft-out 45.0%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in x around inf 44.9%

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

      1. *-commutative 44.9%

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

      2. sqrt-prod 86.4%

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

      3. +-commutative 86.4%

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

      4. associate-/l* 96.5%

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

      5. fma-define 96.5%

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

   7. Applied egg-rr 96.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + \mathsf{fma}\left(y, \frac{z}{x}, z\right)} \cdot \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. +-commutative 67.3%

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

   2. associate-+r+ 67.3%

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

   3. *-commutative 67.3%

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

   4. +-commutative 67.3%

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

   5. +-commutative 67.3%

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

   6. *-commutative 67.3%

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

   7. *-commutative 67.3%

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

   8. associate-+l+ 67.3%

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

   9. associate-+l+ 67.3%

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

   10. *-commutative 67.3%

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

   11. *-commutative 67.3%

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

   12. fma-define 67.4%

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

   13. distribute-lft-out 67.4%

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

3. Simplified 67.4%

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

5. Final simplification 67.4%

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

Alternative 6: 70.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. +-commutative 67.3%

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

   2. *-commutative 67.3%

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

   3. +-commutative 67.3%

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

   4. *-commutative 67.3%

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

   5. associate-+l+ 67.3%

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

   6. *-commutative 67.3%

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

   7. *-commutative 67.3%

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

   8. +-commutative 67.3%

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

   9. fma-define 67.3%

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

   10. +-commutative 67.3%

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

   11. distribute-lft-out 67.4%

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

3. Simplified 67.4%

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

5. Final simplification 67.4%

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

Alternative 7: 70.5% accurate, 1.0× speedup

Math

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

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 z) x) (* y z)))))

C

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

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 + z) * x) + (y * z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. distribute-lft-out 67.3%

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

   2. *-commutative 67.3%

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

4. Applied egg-rr 67.3%

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

Alternative 8: 70.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. associate-+l+ 67.3%

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

   2. *-commutative 67.3%

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

   3. *-commutative 67.3%

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

   4. *-commutative 67.3%

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

   5. +-commutative 67.3%

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

   6. +-commutative 67.3%

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

   7. associate-+l+ 67.3%

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

   8. *-commutative 67.3%

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

   9. *-commutative 67.3%

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

   10. +-commutative 67.3%

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

   11. +-commutative 67.3%

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

   12. *-commutative 67.3%

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

   13. associate-+l+ 67.3%

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

   14. *-commutative 67.3%

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

   15. *-commutative 67.3%

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

   16. +-commutative 67.3%

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

3. Simplified 67.3%

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

5. Final simplification 67.3%

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

Developer Target 1: 82.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024179 
(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)))))