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

Percentage Accurate: 70.5% → 95.0%

Time: 32.9s

Alternatives: 13

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: 70.5% 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: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-305}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(0 - y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(y + x\right)}^{-0.5} \cdot {z}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.15e-305)
   (* 2.0 (pow (exp (* 0.25 (- (log (- (- 0.0 y) z)) (log (/ -1.0 x))))) 2.0))
   (* 2.0 (/ 1.0 (* (pow (+ y x) -0.5) (pow z -0.5))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.15e-305) {
		tmp = 2.0 * pow(exp((0.25 * (log(((0.0 - y) - z)) - log((-1.0 / x))))), 2.0);
	} else {
		tmp = 2.0 * (1.0 / (pow((y + x), -0.5) * pow(z, -0.5)));
	}
	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 <= 2.15d-305) then
        tmp = 2.0d0 * (exp((0.25d0 * (log(((0.0d0 - y) - z)) - log(((-1.0d0) / x))))) ** 2.0d0)
    else
        tmp = 2.0d0 * (1.0d0 / (((y + x) ** (-0.5d0)) * (z ** (-0.5d0))))
    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 <= 2.15e-305) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log(((0.0 - y) - z)) - Math.log((-1.0 / x))))), 2.0);
	} else {
		tmp = 2.0 * (1.0 / (Math.pow((y + x), -0.5) * Math.pow(z, -0.5)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.15e-305:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log(((0.0 - y) - z)) - math.log((-1.0 / x))))), 2.0)
	else:
		tmp = 2.0 * (1.0 / (math.pow((y + x), -0.5) * math.pow(z, -0.5)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.15e-305)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(0.0 - y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(1.0 / Float64((Float64(y + x) ^ -0.5) * (z ^ -0.5))));
	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 <= 2.15e-305)
		tmp = 2.0 * (exp((0.25 * (log(((0.0 - y) - z)) - log((-1.0 / x))))) ^ 2.0);
	else
		tmp = 2.0 * (1.0 / (((y + x) ^ -0.5) * (z ^ -0.5)));
	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, 2.15e-305], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(N[(0.0 - y), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[(N[Power[N[(y + x), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1500000000000001e-305

   1. Initial program 67.8%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 67.8%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 67.8%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)}^{\frac{1}{2}}\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\frac{1}{2}}\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{\color{blue}{2}}\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \color{blue}{2}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot y + z \cdot \left(x + y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(z \cdot \left(x + y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      14. metadata-eval 67.4%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \frac{1}{4}\right), 2\right)\right) \]

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\color{blue}{\left(e^{\frac{1}{4} \cdot \left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}, 2\right)\right) \]
   8. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\left(\frac{1}{4} \cdot \left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right), 2\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right), 2\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\log \left(-1 \cdot y + -1 \cdot z\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 2\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(-1 \cdot y + -1 \cdot z\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 2\right)\right) \]

      5. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{fma}\left(-1, y, -1 \cdot z\right)\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 2\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{fma}\left(-1, y, \mathsf{neg}\left(z\right)\right)\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 2\right)\right) \]

      7. fmm-undef N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(-1 \cdot y - z\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 2\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot y\right), z\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 2\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(y\right)\right), z\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 2\right)\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(y\right), z\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 2\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(y\right), z\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      12. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(y\right), z\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 2\right)\right) \]

      13. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(y\right), z\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{-1}{x}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      14. /-lowering-/.f64 40.8%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(y\right), z\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right)\right)\right)\right)\right), 2\right)\right) \]

   9. Simplified 40.8%

      \[\leadsto 2 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) + \left(-\log \left(\frac{-1}{x}\right)\right)\right)}\right)}}^{2} \]

   if 2.1500000000000001e-305 < y

   1. Initial program 67.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 67.7%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 67.7%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 41.5%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 41.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \left(x + y\right)}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \frac{x \cdot x - y \cdot y}{x - y}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{z \cdot \left(x \cdot x - y \cdot y\right)}{x - y}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{\left(x \cdot x - y \cdot y\right) \cdot z}{x - y}}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}\right)\right) \]

      6. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\left(x \cdot x - y \cdot y\right) \cdot z}{x - y}}\right)\right)\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x - y} \cdot z}\right)\right)\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\left(x + y\right) \cdot z}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{z \cdot \left(x + y\right)}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(z \cdot \left(x + y\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 40.1%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 40.1%

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

       1. inv-pow N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left(\sqrt{{\left(z \cdot \left(x + y\right)\right)}^{-1}}\right)\right)\right) \]

       2. sqrt-pow1 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(z \cdot \left(x + y\right)\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(z \cdot \left(x + y\right)\right)}^{\frac{-1}{2}}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(z \cdot \left(x + y\right)\right)}^{\left(\frac{1}{2} \cdot \color{blue}{-1}\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(\left(x + y\right) \cdot z\right)}^{\left(\color{blue}{\frac{1}{2}} \cdot -1\right)}\right)\right)\right) \]

       6. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot \color{blue}{{z}^{\left(\frac{1}{2} \cdot -1\right)}}\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot {z}^{\frac{-1}{2}}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot {z}^{\left(\frac{-1}{\color{blue}{2}}\right)}\right)\right)\right) \]

       9. sqrt-pow1 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot \sqrt{{z}^{-1}}\right)\right)\right) \]

       10. inv-pow N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)}\right), \color{blue}{\left(\sqrt{\frac{1}{z}}\right)}\right)\right)\right) \]

       12. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x + y\right), \left(\frac{1}{2} \cdot -1\right)\right), \left(\sqrt{\color{blue}{\frac{1}{z}}}\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\frac{1}{2} \cdot -1\right)\right), \left(\sqrt{\frac{\color{blue}{1}}{z}}\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left(\sqrt{\frac{1}{\color{blue}{z}}}\right)\right)\right)\right) \]

       15. inv-pow N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left(\sqrt{{z}^{-1}}\right)\right)\right)\right) \]

       16. sqrt-pow1 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left({z}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left({z}^{\frac{-1}{2}}\right)\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left({z}^{\left(\frac{1}{2} \cdot \color{blue}{-1}\right)}\right)\right)\right)\right) \]

       19. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(z, \color{blue}{\left(\frac{1}{2} \cdot -1\right)}\right)\right)\right)\right) \]

       20. metadata-eval 45.6%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(z, \frac{-1}{2}\right)\right)\right)\right) \]

   11. Applied egg-rr 45.6%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{{\left(x + y\right)}^{-0.5} \cdot {z}^{-0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-305}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(0 - y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(y + x\right)}^{-0.5} \cdot {z}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot {\left(e^{\left(\log \left(\left(0 - y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.125}\right)}^{4}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(y + x\right)}^{-0.5} \cdot {z}^{-0.5}}\\ \end{array} \end{array} \]

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-293)
   (* 2.0 (pow (exp (* (- (log (- (- 0.0 y) z)) (log (/ -1.0 x))) 0.125)) 4.0))
   (* 2.0 (/ 1.0 (* (pow (+ y x) -0.5) (pow z -0.5))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e-293) {
		tmp = 2.0 * pow(exp(((log(((0.0 - y) - z)) - log((-1.0 / x))) * 0.125)), 4.0);
	} else {
		tmp = 2.0 * (1.0 / (pow((y + x), -0.5) * pow(z, -0.5)));
	}
	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-293) then
        tmp = 2.0d0 * (exp(((log(((0.0d0 - y) - z)) - log(((-1.0d0) / x))) * 0.125d0)) ** 4.0d0)
    else
        tmp = 2.0d0 * (1.0d0 / (((y + x) ** (-0.5d0)) * (z ** (-0.5d0))))
    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-293) {
		tmp = 2.0 * Math.pow(Math.exp(((Math.log(((0.0 - y) - z)) - Math.log((-1.0 / x))) * 0.125)), 4.0);
	} else {
		tmp = 2.0 * (1.0 / (Math.pow((y + x), -0.5) * Math.pow(z, -0.5)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.1e-293:
		tmp = 2.0 * math.pow(math.exp(((math.log(((0.0 - y) - z)) - math.log((-1.0 / x))) * 0.125)), 4.0)
	else:
		tmp = 2.0 * (1.0 / (math.pow((y + x), -0.5) * math.pow(z, -0.5)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e-293)
		tmp = Float64(2.0 * (exp(Float64(Float64(log(Float64(Float64(0.0 - y) - z)) - log(Float64(-1.0 / x))) * 0.125)) ^ 4.0));
	else
		tmp = Float64(2.0 * Float64(1.0 / Float64((Float64(y + x) ^ -0.5) * (z ^ -0.5))));
	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-293)
		tmp = 2.0 * (exp(((log(((0.0 - y) - z)) - log((-1.0 / x))) * 0.125)) ^ 4.0);
	else
		tmp = 2.0 * (1.0 / (((y + x) ^ -0.5) * (z ^ -0.5)));
	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-293], N[(2.0 * N[Power[N[Exp[N[(N[(N[Log[N[(N[(0.0 - y), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[(N[Power[N[(y + x), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.10000000000000005e-293

   1. Initial program 67.8%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 67.8%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 67.8%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)}^{\frac{1}{2}}\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\frac{1}{2}}\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{\color{blue}{2}}\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \color{blue}{2}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot y + z \cdot \left(x + y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(z \cdot \left(x + y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

      14. metadata-eval 67.4%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \frac{1}{4}\right), 2\right)\right) \]

   6. Applied egg-rr 67.4%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{0.25}\right)}^{2}} \]
   7. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot {\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)}^{2}\right)\right) \]

      2. unpow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)}^{2} \cdot \color{blue}{{\left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)}^{2}}\right)\right) \]

      3. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left({\left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right)}^{\color{blue}{\left(2 + 2\right)}}\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}\right), \color{blue}{\left(2 + 2\right)}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot y + z \cdot \left(x + y\right)\right), \left(\frac{\frac{1}{4}}{2}\right)\right), \left(\color{blue}{2} + 2\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(z \cdot \left(x + y\right)\right)\right), \left(\frac{\frac{1}{4}}{2}\right)\right), \left(2 + 2\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right), \left(\frac{\frac{1}{4}}{2}\right)\right), \left(2 + 2\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right), \left(\frac{\frac{1}{4}}{2}\right)\right), \left(2 + 2\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\frac{\frac{1}{4}}{2}\right)\right), \left(2 + 2\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \frac{1}{8}\right), \left(2 + 2\right)\right)\right) \]

      11. metadata-eval 67.1%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right), \frac{1}{8}\right), 4\right)\right) \]

   8. Applied egg-rr 67.1%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(x \cdot y + z \cdot \left(x + y\right)\right)}^{0.125}\right)}^{4}} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\color{blue}{\left(e^{\frac{1}{8} \cdot \left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}, 4\right)\right) \]
   10. Step-by-step derivation

       1. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\left(\frac{1}{8} \cdot \left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right), 4\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right), 4\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\log \left(-1 \cdot y + -1 \cdot z\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 4\right)\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(-1 \cdot y + -1 \cdot z\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 4\right)\right) \]

       5. fma-define N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{fma}\left(-1, y, -1 \cdot z\right)\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 4\right)\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{fma}\left(-1, y, \mathsf{neg}\left(z\right)\right)\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 4\right)\right) \]

       7. fmm-undef N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(-1 \cdot y - z\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 4\right)\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot y\right), z\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 4\right)\right) \]

       9. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(y\right)\right), z\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 4\right)\right) \]

       10. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(y\right), z\right)\right), \left(-1 \cdot \log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 4\right)\right) \]

       11. mul-1-neg N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(y\right), z\right)\right), \left(\mathsf{neg}\left(\log \left(\frac{-1}{x}\right)\right)\right)\right)\right)\right), 4\right)\right) \]

       12. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(y\right), z\right)\right), \mathsf{neg.f64}\left(\log \left(\frac{-1}{x}\right)\right)\right)\right)\right), 4\right)\right) \]

       13. log-lowering-log.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(y\right), z\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{-1}{x}\right)\right)\right)\right)\right)\right), 4\right)\right) \]

       14. /-lowering-/.f64 41.9%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{+.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(y\right), z\right)\right), \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x\right)\right)\right)\right)\right)\right), 4\right)\right) \]

   11. Simplified 41.9%

       \[\leadsto 2 \cdot {\color{blue}{\left(e^{0.125 \cdot \left(\log \left(\left(-y\right) - z\right) + \left(-\log \left(\frac{-1}{x}\right)\right)\right)}\right)}}^{4} \]

   if 2.10000000000000005e-293 < y

   1. Initial program 67.6%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 67.7%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 67.7%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 40.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 40.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \left(x + y\right)}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \frac{x \cdot x - y \cdot y}{x - y}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{z \cdot \left(x \cdot x - y \cdot y\right)}{x - y}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{\left(x \cdot x - y \cdot y\right) \cdot z}{x - y}}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}\right)\right) \]

      6. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\left(x \cdot x - y \cdot y\right) \cdot z}{x - y}}\right)\right)\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x - y} \cdot z}\right)\right)\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\left(x + y\right) \cdot z}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{z \cdot \left(x + y\right)}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(z \cdot \left(x + y\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 40.2%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 40.2%

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

       1. inv-pow N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left(\sqrt{{\left(z \cdot \left(x + y\right)\right)}^{-1}}\right)\right)\right) \]

       2. sqrt-pow1 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(z \cdot \left(x + y\right)\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(z \cdot \left(x + y\right)\right)}^{\frac{-1}{2}}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(z \cdot \left(x + y\right)\right)}^{\left(\frac{1}{2} \cdot \color{blue}{-1}\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(\left(x + y\right) \cdot z\right)}^{\left(\color{blue}{\frac{1}{2}} \cdot -1\right)}\right)\right)\right) \]

       6. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot \color{blue}{{z}^{\left(\frac{1}{2} \cdot -1\right)}}\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot {z}^{\frac{-1}{2}}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot {z}^{\left(\frac{-1}{\color{blue}{2}}\right)}\right)\right)\right) \]

       9. sqrt-pow1 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot \sqrt{{z}^{-1}}\right)\right)\right) \]

       10. inv-pow N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)}\right), \color{blue}{\left(\sqrt{\frac{1}{z}}\right)}\right)\right)\right) \]

       12. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x + y\right), \left(\frac{1}{2} \cdot -1\right)\right), \left(\sqrt{\color{blue}{\frac{1}{z}}}\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\frac{1}{2} \cdot -1\right)\right), \left(\sqrt{\frac{\color{blue}{1}}{z}}\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left(\sqrt{\frac{1}{\color{blue}{z}}}\right)\right)\right)\right) \]

       15. inv-pow N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left(\sqrt{{z}^{-1}}\right)\right)\right)\right) \]

       16. sqrt-pow1 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left({z}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left({z}^{\frac{-1}{2}}\right)\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left({z}^{\left(\frac{1}{2} \cdot \color{blue}{-1}\right)}\right)\right)\right)\right) \]

       19. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(z, \color{blue}{\left(\frac{1}{2} \cdot -1\right)}\right)\right)\right)\right) \]

       20. metadata-eval 46.6%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(z, \frac{-1}{2}\right)\right)\right)\right) \]

   11. Applied egg-rr 46.6%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{{\left(x + y\right)}^{-0.5} \cdot {z}^{-0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot {\left(e^{\left(\log \left(\left(0 - y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.125}\right)}^{4}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(y + x\right)}^{-0.5} \cdot {z}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+25}:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot \sqrt{\frac{z}{y}} + x \cdot \sqrt{\frac{z}{y \cdot \left(y \cdot y\right)}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.15e+25)
   (* 2.0 (sqrt (+ (+ (* y x) (* z x)) (* y z))))
   (* y (+ (* 2.0 (sqrt (/ z y))) (* x (sqrt (/ z (* y (* y y)))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.15e+25) {
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = y * ((2.0 * sqrt((z / y))) + (x * sqrt((z / (y * (y * 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 <= 2.15d+25) then
        tmp = 2.0d0 * sqrt((((y * x) + (z * x)) + (y * z)))
    else
        tmp = y * ((2.0d0 * sqrt((z / y))) + (x * sqrt((z / (y * (y * 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 <= 2.15e+25) {
		tmp = 2.0 * Math.sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = y * ((2.0 * Math.sqrt((z / y))) + (x * Math.sqrt((z / (y * (y * y))))));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.15e+25:
		tmp = 2.0 * math.sqrt((((y * x) + (z * x)) + (y * z)))
	else:
		tmp = y * ((2.0 * math.sqrt((z / y))) + (x * math.sqrt((z / (y * (y * y))))))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.15e+25)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(y * x) + Float64(z * x)) + Float64(y * z))));
	else
		tmp = Float64(y * Float64(Float64(2.0 * sqrt(Float64(z / y))) + Float64(x * sqrt(Float64(z / Float64(y * Float64(y * 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 <= 2.15e+25)
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	else
		tmp = y * ((2.0 * sqrt((z / y))) + (x * sqrt((z / (y * (y * 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, 2.15e+25], N[(2.0 * N[Sqrt[N[(N[(N[(y * x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(2.0 * N[Sqrt[N[(z / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(x * N[Sqrt[N[(z / N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.14999999999999999e25

   1. Initial program 73.6%

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

   if 2.14999999999999999e25 < y

   1. Initial program 46.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 46.5%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 46.5%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 24.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 24.3%

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

   8. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(2 \cdot \sqrt{\frac{z}{y}} + x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(2 \cdot \sqrt{\frac{z}{y}}\right), \color{blue}{\left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{\frac{z}{y}}\right)\right), \left(\color{blue}{x} \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\frac{z}{y}\right)\right)\right), \left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \left(x \cdot \sqrt{\frac{z}{{y}^{3}}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\sqrt{\frac{z}{{y}^{3}}}\right)}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\frac{z}{{y}^{3}}\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left({y}^{3}\right)\right)\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \left(y \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \left({y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 47.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 47.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+25}:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot \sqrt{\frac{z}{y}} + x \cdot \sqrt{\frac{z}{y \cdot \left(y \cdot y\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(y + x\right)}^{-0.5} \cdot {z}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.8e+56)
   (* 2.0 (sqrt (+ (+ (* y x) (* z x)) (* y z))))
   (* 2.0 (/ 1.0 (* (pow (+ y x) -0.5) (pow z -0.5))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.8e+56) {
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = 2.0 * (1.0 / (pow((y + x), -0.5) * pow(z, -0.5)));
	}
	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 <= 6.8d+56) then
        tmp = 2.0d0 * sqrt((((y * x) + (z * x)) + (y * z)))
    else
        tmp = 2.0d0 * (1.0d0 / (((y + x) ** (-0.5d0)) * (z ** (-0.5d0))))
    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 <= 6.8e+56) {
		tmp = 2.0 * Math.sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = 2.0 * (1.0 / (Math.pow((y + x), -0.5) * Math.pow(z, -0.5)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 6.8e+56:
		tmp = 2.0 * math.sqrt((((y * x) + (z * x)) + (y * z)))
	else:
		tmp = 2.0 * (1.0 / (math.pow((y + x), -0.5) * math.pow(z, -0.5)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6.8e+56)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(y * x) + Float64(z * x)) + Float64(y * z))));
	else
		tmp = Float64(2.0 * Float64(1.0 / Float64((Float64(y + x) ^ -0.5) * (z ^ -0.5))));
	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 <= 6.8e+56)
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	else
		tmp = 2.0 * (1.0 / (((y + x) ^ -0.5) * (z ^ -0.5)));
	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, 6.8e+56], N[(2.0 * N[Sqrt[N[(N[(N[(y * x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[(N[Power[N[(y + x), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.80000000000000002e56

   1. Initial program 73.8%

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

   if 6.80000000000000002e56 < y

   1. Initial program 42.8%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 43.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 43.0%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 20.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 20.3%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \left(x + y\right)}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \frac{x \cdot x - y \cdot y}{x - y}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{z \cdot \left(x \cdot x - y \cdot y\right)}{x - y}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{\left(x \cdot x - y \cdot y\right) \cdot z}{x - y}}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}\right)\right) \]

      6. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\left(x \cdot x - y \cdot y\right) \cdot z}{x - y}}\right)\right)\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x - y} \cdot z}\right)\right)\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\left(x + y\right) \cdot z}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{z \cdot \left(x + y\right)}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(z \cdot \left(x + y\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 20.3%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 20.3%

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

       1. inv-pow N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left(\sqrt{{\left(z \cdot \left(x + y\right)\right)}^{-1}}\right)\right)\right) \]

       2. sqrt-pow1 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(z \cdot \left(x + y\right)\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(z \cdot \left(x + y\right)\right)}^{\frac{-1}{2}}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(z \cdot \left(x + y\right)\right)}^{\left(\frac{1}{2} \cdot \color{blue}{-1}\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(\left(x + y\right) \cdot z\right)}^{\left(\color{blue}{\frac{1}{2}} \cdot -1\right)}\right)\right)\right) \]

       6. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot \color{blue}{{z}^{\left(\frac{1}{2} \cdot -1\right)}}\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot {z}^{\frac{-1}{2}}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot {z}^{\left(\frac{-1}{\color{blue}{2}}\right)}\right)\right)\right) \]

       9. sqrt-pow1 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot \sqrt{{z}^{-1}}\right)\right)\right) \]

       10. inv-pow N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)} \cdot \sqrt{\frac{1}{z}}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left({\left(x + y\right)}^{\left(\frac{1}{2} \cdot -1\right)}\right), \color{blue}{\left(\sqrt{\frac{1}{z}}\right)}\right)\right)\right) \]

       12. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x + y\right), \left(\frac{1}{2} \cdot -1\right)\right), \left(\sqrt{\color{blue}{\frac{1}{z}}}\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\frac{1}{2} \cdot -1\right)\right), \left(\sqrt{\frac{\color{blue}{1}}{z}}\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left(\sqrt{\frac{1}{\color{blue}{z}}}\right)\right)\right)\right) \]

       15. inv-pow N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left(\sqrt{{z}^{-1}}\right)\right)\right)\right) \]

       16. sqrt-pow1 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left({z}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left({z}^{\frac{-1}{2}}\right)\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \left({z}^{\left(\frac{1}{2} \cdot \color{blue}{-1}\right)}\right)\right)\right)\right) \]

       19. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(z, \color{blue}{\left(\frac{1}{2} \cdot -1\right)}\right)\right)\right)\right) \]

       20. metadata-eval 52.2%

           \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(z, \frac{-1}{2}\right)\right)\right)\right) \]

   11. Applied egg-rr 52.2%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{{\left(x + y\right)}^{-0.5} \cdot {z}^{-0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(y + x\right)}^{-0.5} \cdot {z}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot {\left(y + x\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.6e-27)
   (* 2.0 (sqrt (+ (+ (* y x) (* z x)) (* y z))))
   (* (* 2.0 (sqrt z)) (pow (+ y x) 0.5))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.6e-27) {
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = (2.0 * sqrt(z)) * pow((y + x), 0.5);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.6d-27) then
        tmp = 2.0d0 * sqrt((((y * x) + (z * x)) + (y * z)))
    else
        tmp = (2.0d0 * sqrt(z)) * ((y + x) ** 0.5d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.6e-27) {
		tmp = 2.0 * Math.sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.pow((y + x), 0.5);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 4.6e-27:
		tmp = 2.0 * math.sqrt((((y * x) + (z * x)) + (y * z)))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.pow((y + x), 0.5)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.6e-27)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(y * x) + Float64(z * x)) + Float64(y * z))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * (Float64(y + x) ^ 0.5));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.6e-27)
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	else
		tmp = (2.0 * sqrt(z)) * ((y + x) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 4.6e-27], N[(2.0 * N[Sqrt[N[(N[(N[(y * x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Power[N[(y + x), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5999999999999999e-27

   1. Initial program 73.7%

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

   if 4.5999999999999999e-27 < y

   1. Initial program 51.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 51.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 51.2%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 23.5%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 23.5%

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

      1. pow1/2 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

         \[\leadsto 2 \cdot {\left(z \cdot \left(x + y\right)\right)}^{\left(\frac{1}{4} \cdot \color{blue}{2}\right)} \]

      4. unpow-prod-down N/A

         \[\leadsto 2 \cdot \left({z}^{\left(\frac{1}{4} \cdot 2\right)} \cdot \color{blue}{{\left(x + y\right)}^{\left(\frac{1}{4} \cdot 2\right)}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(2 \cdot {z}^{\left(\frac{1}{4} \cdot 2\right)}\right) \cdot \color{blue}{{\left(x + y\right)}^{\left(\frac{1}{4} \cdot 2\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot {z}^{\left(\frac{1}{4} \cdot 2\right)}\right), \color{blue}{\left({\left(x + y\right)}^{\left(\frac{1}{4} \cdot 2\right)}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({z}^{\left(\frac{1}{4} \cdot 2\right)}\right)\right), \left({\color{blue}{\left(x + y\right)}}^{\left(\frac{1}{4} \cdot 2\right)}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({z}^{\frac{1}{2}}\right)\right), \left({\left(x + y\right)}^{\left(\frac{1}{4} \cdot 2\right)}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{z}\right)\right), \left({\left(x + \color{blue}{y}\right)}^{\left(\frac{1}{4} \cdot 2\right)}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \left({\left(x + \color{blue}{y}\right)}^{\left(\frac{1}{4} \cdot 2\right)}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \mathsf{pow.f64}\left(\left(x + y\right), \color{blue}{\left(\frac{1}{4} \cdot 2\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\frac{1}{4}} \cdot 2\right)\right)\right) \]

      13. metadata-eval 49.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, y\right), \frac{1}{2}\right)\right) \]

   9. Applied egg-rr 49.4%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot {\left(y + x\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.8e+56)
   (* 2.0 (sqrt (+ (+ (* y x) (* z x)) (* y z))))
   (* (* 2.0 (sqrt z)) (sqrt y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.8e+56) {
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = (2.0 * sqrt(z)) * sqrt(y);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6.8d+56) then
        tmp = 2.0d0 * sqrt((((y * x) + (z * x)) + (y * z)))
    else
        tmp = (2.0d0 * sqrt(z)) * sqrt(y)
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.8e+56) {
		tmp = 2.0 * Math.sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = (2.0 * Math.sqrt(z)) * Math.sqrt(y);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 6.8e+56:
		tmp = 2.0 * math.sqrt((((y * x) + (z * x)) + (y * z)))
	else:
		tmp = (2.0 * math.sqrt(z)) * math.sqrt(y)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6.8e+56)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(y * x) + Float64(z * x)) + Float64(y * z))));
	else
		tmp = Float64(Float64(2.0 * sqrt(z)) * sqrt(y));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.8e+56)
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	else
		tmp = (2.0 * sqrt(z)) * sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 6.8e+56], N[(2.0 * N[Sqrt[N[(N[(N[(y * x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.80000000000000002e56

   1. Initial program 73.8%

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

   if 6.80000000000000002e56 < y

   1. Initial program 42.8%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 43.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 43.0%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 20.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 20.3%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{y}\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 20.3%

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

       1. pow1/2 N/A

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

       2. unpow-prod-down N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot {z}^{\frac{1}{2}}\right), \color{blue}{\left({y}^{\frac{1}{2}}\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({z}^{\frac{1}{2}}\right)\right), \left({\color{blue}{y}}^{\frac{1}{2}}\right)\right) \]

       6. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{z}\right)\right), \left({y}^{\frac{1}{2}}\right)\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \left({y}^{\frac{1}{2}}\right)\right) \]

       8. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \left(\sqrt{y}\right)\right) \]

       9. sqrt-lowering-sqrt.f64 48.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(z\right)\right), \mathsf{sqrt.f64}\left(y\right)\right) \]

   12. Applied egg-rr 48.6%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{z}\right) \cdot \sqrt{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\sqrt{\frac{\frac{1}{y + x}}{z}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 8.8e+56)
   (* 2.0 (sqrt (+ (+ (* y x) (* z x)) (* y z))))
   (* 2.0 (/ 1.0 (sqrt (/ (/ 1.0 (+ y x)) z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.8e+56) {
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = 2.0 * (1.0 / sqrt(((1.0 / (y + x)) / z)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.8d+56) then
        tmp = 2.0d0 * sqrt((((y * x) + (z * x)) + (y * z)))
    else
        tmp = 2.0d0 * (1.0d0 / sqrt(((1.0d0 / (y + x)) / z)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.8e+56) {
		tmp = 2.0 * Math.sqrt((((y * x) + (z * x)) + (y * z)));
	} else {
		tmp = 2.0 * (1.0 / Math.sqrt(((1.0 / (y + x)) / z)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 8.8e+56:
		tmp = 2.0 * math.sqrt((((y * x) + (z * x)) + (y * z)))
	else:
		tmp = 2.0 * (1.0 / math.sqrt(((1.0 / (y + x)) / z)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 8.8e+56)
		tmp = Float64(2.0 * sqrt(Float64(Float64(Float64(y * x) + Float64(z * x)) + Float64(y * z))));
	else
		tmp = Float64(2.0 * Float64(1.0 / sqrt(Float64(Float64(1.0 / Float64(y + x)) / z))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.8e+56)
		tmp = 2.0 * sqrt((((y * x) + (z * x)) + (y * z)));
	else
		tmp = 2.0 * (1.0 / sqrt(((1.0 / (y + x)) / z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 8.8e+56], N[(2.0 * N[Sqrt[N[(N[(N[(y * x), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[Sqrt[N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.80000000000000063e56

   1. Initial program 73.8%

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

   if 8.80000000000000063e56 < y

   1. Initial program 42.8%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 43.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 43.0%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 20.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 20.3%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \left(x + y\right)}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \frac{x \cdot x - y \cdot y}{x - y}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{z \cdot \left(x \cdot x - y \cdot y\right)}{x - y}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{\left(x \cdot x - y \cdot y\right) \cdot z}{x - y}}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}\right)\right) \]

      6. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\left(x \cdot x - y \cdot y\right) \cdot z}{x - y}}\right)\right)\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x - y} \cdot z}\right)\right)\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\left(x + y\right) \cdot z}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{z \cdot \left(x + y\right)}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(z \cdot \left(x + y\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 20.3%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 20.3%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\left(x + y\right) \cdot z}\right)\right)\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{x + y}}{z}\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{x + y}\right), z\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x + y\right)\right), z\right)\right)\right)\right) \]

       5. +-lowering-+.f64 26.9%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), z\right)\right)\right)\right) \]

   11. Applied egg-rr 26.9%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot \sqrt{\left(y \cdot x + z \cdot x\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\sqrt{\frac{\frac{1}{y + x}}{z}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.02 \cdot 10^{+91}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\sqrt{\frac{\frac{1}{y + x}}{z}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.02e+91)
   (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
   (* 2.0 (/ 1.0 (sqrt (/ (/ 1.0 (+ y x)) z))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.02e+91) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (1.0 / sqrt(((1.0 / (y + x)) / 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 (z <= 1.02d+91) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    else
        tmp = 2.0d0 * (1.0d0 / sqrt(((1.0d0 / (y + x)) / 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 (z <= 1.02e+91) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (1.0 / Math.sqrt(((1.0 / (y + x)) / z)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= 1.02e+91:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	else:
		tmp = 2.0 * (1.0 / math.sqrt(((1.0 / (y + x)) / z)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 1.02e+91)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(1.0 / sqrt(Float64(Float64(1.0 / Float64(y + x)) / 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 (z <= 1.02e+91)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	else
		tmp = 2.0 * (1.0 / sqrt(((1.0 / (y + x)) / 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[z, 1.02e+91], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[Sqrt[N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.01999999999999992e91

   1. Initial program 74.2%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 74.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 74.2%

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

   if 1.01999999999999992e91 < z

   1. Initial program 38.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 38.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 38.2%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 38.4%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 38.4%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \left(x + y\right)}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{z \cdot \frac{x \cdot x - y \cdot y}{x - y}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{z \cdot \left(x \cdot x - y \cdot y\right)}{x - y}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{\left(x \cdot x - y \cdot y\right) \cdot z}{x - y}}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}\right)\right) \]

      6. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{x - y}{\left(x \cdot x - y \cdot y\right) \cdot z}\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\left(x \cdot x - y \cdot y\right) \cdot z}{x - y}}\right)\right)\right)\right) \]

      11. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x - y} \cdot z}\right)\right)\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\left(x + y\right) \cdot z}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{z \cdot \left(x + y\right)}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(z \cdot \left(x + y\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 38.2%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 38.2%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\left(x + y\right) \cdot z}\right)\right)\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{x + y}}{z}\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{x + y}\right), z\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x + y\right)\right), z\right)\right)\right)\right) \]

       5. +-lowering-+.f64 45.8%

          \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), z\right)\right)\right)\right) \]

   11. Applied egg-rr 45.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.02 \cdot 10^{+91}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\sqrt{\frac{\frac{1}{y + x}}{z}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-292}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-292) (* 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 <= -4e-292) {
		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 <= (-4d-292)) 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 <= -4e-292) {
		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 <= -4e-292:
		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 <= -4e-292)
		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 <= -4e-292)
		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, -4e-292], 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 < -4.0000000000000002e-292

   1. Initial program 67.1%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 67.1%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 67.1%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot \left(y + z\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot \left(y + z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(y + z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 38.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, z\right)\right)\right)\right) \]

   7. Simplified 38.9%

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

   if -4.0000000000000002e-292 < y

   1. Initial program 68.3%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 68.4%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 68.4%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{z \cdot \left(x + y\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(z \cdot \left(x + y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \left(y + x\right)\right)\right)\right) \]

      5. +-lowering-+.f64 42.8%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(y, x\right)\right)\right)\right) \]

   7. Simplified 42.8%

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

Alternative 10: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.05e-298) (* 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 <= -3.05e-298) {
		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 <= (-3.05d-298)) 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 <= -3.05e-298) {
		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 <= -3.05e-298:
		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 <= -3.05e-298)
		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 <= -3.05e-298)
		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, -3.05e-298], 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 < -3.05000000000000006e-298

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 67.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 67.3%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot \left(y + z\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot \left(y + z\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(y + z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 39.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, z\right)\right)\right)\right) \]

   7. Simplified 39.3%

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

   if -3.05000000000000006e-298 < y

   1. Initial program 68.1%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 68.1%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot z\right)\right)\right) \]

      3. *-lowering-*.f64 19.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, z\right)\right)\right) \]

   7. Simplified 19.0%

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

Alternative 11: 70.6% 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 67.7%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f64 67.7%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

3. Simplified 67.7%

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

5. Final simplification 67.7%

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

Alternative 12: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.05e-298) (* 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 <= -3.05e-298) {
		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 <= (-3.05d-298)) 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 <= -3.05e-298) {
		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 <= -3.05e-298:
		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 <= -3.05e-298)
		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 <= -3.05e-298)
		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, -3.05e-298], 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 < -3.05000000000000006e-298

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 67.3%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 67.3%

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

   5. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot y}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 19.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, x\right)\right)\right) \]

   7. Simplified 19.9%

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

   if -3.05000000000000006e-298 < y

   1. Initial program 68.1%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

   3. Simplified 68.1%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{y \cdot z}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot z\right)\right)\right) \]

      3. *-lowering-*.f64 19.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, z\right)\right)\right) \]

   7. Simplified 19.0%

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

Alternative 13: 36.0% accurate, 1.1× 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 67.7%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x \cdot y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(x \cdot z + y \cdot z\right)\right)\right)\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \left(x + y\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(x + y\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f64 67.7%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(x, y\right)\right)\right)\right)\right) \]

3. Simplified 67.7%

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

5. Taylor expanded in z around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(\sqrt{x \cdot y}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(x \cdot y\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(y \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 24.0%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(y, x\right)\right)\right) \]

7. Simplified 24.0%

   \[\leadsto \color{blue}{2 \cdot \sqrt{y \cdot x}} \]
8. 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 2024141 
(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)))))