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

Percentage Accurate: 70.2% → 95.3%

Time: 13.3s

Alternatives: 11

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: 70.2% 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: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -40000000000000:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(0 - x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{z}\right)}^{-0.5} \cdot {\left(\frac{1}{y + x}\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -40000000000000.0)
   (* 2.0 (pow (exp (* 0.25 (- (log (- 0.0 x)) (log (/ -1.0 y))))) 2.0))
   (if (<= y 4.3e-271)
     (* 2.0 (sqrt (+ (* x z) (* y (+ x z)))))
     (* 2.0 (* (pow (/ 1.0 z) -0.5) (pow (/ 1.0 (+ y x)) -0.5))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -40000000000000.0) {
		tmp = 2.0 * pow(exp((0.25 * (log((0.0 - x)) - log((-1.0 / y))))), 2.0);
	} else if (y <= 4.3e-271) {
		tmp = 2.0 * sqrt(((x * z) + (y * (x + z))));
	} else {
		tmp = 2.0 * (pow((1.0 / z), -0.5) * pow((1.0 / (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 <= (-40000000000000.0d0)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log((0.0d0 - x)) - log(((-1.0d0) / y))))) ** 2.0d0)
    else if (y <= 4.3d-271) then
        tmp = 2.0d0 * sqrt(((x * z) + (y * (x + z))))
    else
        tmp = 2.0d0 * (((1.0d0 / z) ** (-0.5d0)) * ((1.0d0 / (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 <= -40000000000000.0) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((0.0 - x)) - Math.log((-1.0 / y))))), 2.0);
	} else if (y <= 4.3e-271) {
		tmp = 2.0 * Math.sqrt(((x * z) + (y * (x + z))));
	} else {
		tmp = 2.0 * (Math.pow((1.0 / z), -0.5) * Math.pow((1.0 / (y + x)), -0.5));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -40000000000000.0:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log((0.0 - x)) - math.log((-1.0 / y))))), 2.0)
	elif y <= 4.3e-271:
		tmp = 2.0 * math.sqrt(((x * z) + (y * (x + z))))
	else:
		tmp = 2.0 * (math.pow((1.0 / z), -0.5) * math.pow((1.0 / (y + x)), -0.5))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -40000000000000.0)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(0.0 - x)) - log(Float64(-1.0 / y))))) ^ 2.0));
	elseif (y <= 4.3e-271)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * z) + Float64(y * Float64(x + z)))));
	else
		tmp = Float64(2.0 * Float64((Float64(1.0 / z) ^ -0.5) * (Float64(1.0 / 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 <= -40000000000000.0)
		tmp = 2.0 * (exp((0.25 * (log((0.0 - x)) - log((-1.0 / y))))) ^ 2.0);
	elseif (y <= 4.3e-271)
		tmp = 2.0 * sqrt(((x * z) + (y * (x + z))));
	else
		tmp = 2.0 * (((1.0 / z) ^ -0.5) * ((1.0 / (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, -40000000000000.0], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e-271], N[(2.0 * N[Sqrt[N[(N[(x * z), $MachinePrecision] + N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(1.0 / z), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4e13

   1. Initial program 50.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right)}} \]
   4. 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 31.0%

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

   5. Simplified 31.0%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. metadata-eval 31.3%

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

   7. Applied egg-rr 31.3%

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

   8. Taylor expanded in y around -inf

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

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

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

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

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

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

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

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

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

      10. /-lowering-/.f64 46.2%

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

   10. Simplified 46.2%

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

   if -4e13 < y < 4.3e-271

   1. Initial program 85.3%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 85.2%

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

   4. Applied egg-rr 85.2%

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

   if 4.3e-271 < y

   1. Initial program 70.0%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}\right)}\right)\right) \]

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in z around -inf

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

      1. associate-/r* N/A

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

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

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

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

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

      4. distribute-lft-out N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 47.0%

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

   7. Simplified 47.0%

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

      1. pow1/2 N/A

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

      2. pow-flip N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. div-inv N/A

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

      9. unpow-prod-down N/A

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

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

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

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

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

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

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 56.0%

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

   9. Applied egg-rr 56.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -40000000000000:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(0 - x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{z}\right)}^{-0.5} \cdot {\left(\frac{1}{y + x}\right)}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{x}}{y + z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{z}\right)}^{-0.5} \cdot {\left(\frac{1}{y + x}\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e-296)
   (* 2.0 (pow (/ (/ 1.0 x) (+ y z)) -0.5))
   (* 2.0 (* (pow (/ 1.0 z) -0.5) (pow (/ 1.0 (+ y x)) -0.5)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e-296) {
		tmp = 2.0 * pow(((1.0 / x) / (y + z)), -0.5);
	} else {
		tmp = 2.0 * (pow((1.0 / z), -0.5) * pow((1.0 / (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 <= 2.8d-296) then
        tmp = 2.0d0 * (((1.0d0 / x) / (y + z)) ** (-0.5d0))
    else
        tmp = 2.0d0 * (((1.0d0 / z) ** (-0.5d0)) * ((1.0d0 / (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 <= 2.8e-296) {
		tmp = 2.0 * Math.pow(((1.0 / x) / (y + z)), -0.5);
	} else {
		tmp = 2.0 * (Math.pow((1.0 / z), -0.5) * Math.pow((1.0 / (y + x)), -0.5));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.8e-296:
		tmp = 2.0 * math.pow(((1.0 / x) / (y + z)), -0.5)
	else:
		tmp = 2.0 * (math.pow((1.0 / z), -0.5) * math.pow((1.0 / (y + x)), -0.5))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.8e-296)
		tmp = Float64(2.0 * (Float64(Float64(1.0 / x) / Float64(y + z)) ^ -0.5));
	else
		tmp = Float64(2.0 * Float64((Float64(1.0 / z) ^ -0.5) * (Float64(1.0 / 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 <= 2.8e-296)
		tmp = 2.0 * (((1.0 / x) / (y + z)) ^ -0.5);
	else
		tmp = 2.0 * (((1.0 / z) ^ -0.5) * ((1.0 / (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, 2.8e-296], N[(2.0 * N[Power[N[(N[(1.0 / x), $MachinePrecision] / N[(y + z), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(1.0 / z), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7999999999999999e-296

   1. Initial program 67.6%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 67.6%

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

   4. Applied egg-rr 67.6%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 34.1%

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

      1. inv-pow N/A

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

      2. pow1/2 N/A

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

      3. clear-num N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. metadata-eval N/A

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

   8. Applied egg-rr 67.5%

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

   9. Taylor expanded in x around inf

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

       1. associate-/r* N/A

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

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

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

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

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

       4. +-lowering-+.f64 50.6%

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

   11. Simplified 50.6%

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

   if 2.7999999999999999e-296 < y

   1. Initial program 70.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}\right)}\right)\right) \]

   4. Applied egg-rr 70.6%

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

   5. Taylor expanded in z around -inf

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

      1. associate-/r* N/A

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

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

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

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

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

      4. distribute-lft-out N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 47.9%

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

   7. Simplified 47.9%

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

      1. pow1/2 N/A

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

      2. pow-flip N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. div-inv N/A

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

      9. unpow-prod-down N/A

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

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

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

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

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

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

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

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

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 55.5%

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

   9. Applied egg-rr 55.5%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{x}}{y + z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{z}\right)}^{-0.5} \cdot {\left(\frac{1}{y + x}\right)}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{x}}{y + z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(y + x\right)}^{0.5}}{{z}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e-296)
   (* 2.0 (pow (/ (/ 1.0 x) (+ y z)) -0.5))
   (* 2.0 (/ (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 <= 2.8e-296) {
		tmp = 2.0 * pow(((1.0 / x) / (y + z)), -0.5);
	} else {
		tmp = 2.0 * (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 <= 2.8d-296) then
        tmp = 2.0d0 * (((1.0d0 / x) / (y + z)) ** (-0.5d0))
    else
        tmp = 2.0d0 * (((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 <= 2.8e-296) {
		tmp = 2.0 * Math.pow(((1.0 / x) / (y + z)), -0.5);
	} else {
		tmp = 2.0 * (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 <= 2.8e-296:
		tmp = 2.0 * math.pow(((1.0 / x) / (y + z)), -0.5)
	else:
		tmp = 2.0 * (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 <= 2.8e-296)
		tmp = Float64(2.0 * (Float64(Float64(1.0 / x) / Float64(y + z)) ^ -0.5));
	else
		tmp = Float64(2.0 * 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 <= 2.8e-296)
		tmp = 2.0 * (((1.0 / x) / (y + z)) ^ -0.5);
	else
		tmp = 2.0 * (((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, 2.8e-296], N[(2.0 * N[Power[N[(N[(1.0 / x), $MachinePrecision] / N[(y + z), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(y + x), $MachinePrecision], 0.5], $MachinePrecision] / N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7999999999999999e-296

   1. Initial program 67.6%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 67.6%

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

   4. Applied egg-rr 67.6%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 34.1%

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

      1. inv-pow N/A

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

      2. pow1/2 N/A

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

      3. clear-num N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. metadata-eval N/A

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

   8. Applied egg-rr 67.5%

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

   9. Taylor expanded in x around inf

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

       1. associate-/r* N/A

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

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

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

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

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

       4. +-lowering-+.f64 50.6%

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

   11. Simplified 50.6%

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

   if 2.7999999999999999e-296 < y

   1. Initial program 70.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\sqrt{\frac{1}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(\frac{1}{\sqrt{\color{blue}{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\left(x \cdot y + x \cdot z\right) \cdot \left(x \cdot y + x \cdot z\right) + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - \left(x \cdot y + x \cdot z\right) \cdot \left(y \cdot z\right)\right)}{{\left(x \cdot y + x \cdot z\right)}^{3} + {\left(y \cdot z\right)}^{3}}}\right)}\right)\right) \]

   4. Applied egg-rr 70.6%

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

   5. Taylor expanded in z around -inf

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

      1. associate-/r* N/A

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

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

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

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

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

      4. distribute-lft-out N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 47.9%

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

   7. Simplified 47.9%

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

      1. sqrt-div N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. mul-1-neg N/A

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

      5. div-inv N/A

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

      6. mul-1-neg N/A

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

      7. frac-2neg N/A

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

      8. sqrt-div N/A

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

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

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

      10. pow1/2 N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. inv-pow N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. pow-lowering-pow.f64 55.4%

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

   9. Applied egg-rr 55.4%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{x}}{y + z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(y + x\right)}^{0.5}}{{z}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{x}}{y + z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e-296)
   (* 2.0 (pow (/ (/ 1.0 x) (+ y z)) -0.5))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e-296) {
		tmp = 2.0 * pow(((1.0 / x) / (y + z)), -0.5);
	} 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 <= 2.8d-296) then
        tmp = 2.0d0 * (((1.0d0 / x) / (y + z)) ** (-0.5d0))
    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 <= 2.8e-296) {
		tmp = 2.0 * Math.pow(((1.0 / x) / (y + z)), -0.5);
	} 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 <= 2.8e-296:
		tmp = 2.0 * math.pow(((1.0 / x) / (y + z)), -0.5)
	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 <= 2.8e-296)
		tmp = Float64(2.0 * (Float64(Float64(1.0 / x) / Float64(y + z)) ^ -0.5));
	else
		tmp = Float64(2.0 * Float64(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 <= 2.8e-296)
		tmp = 2.0 * (((1.0 / x) / (y + z)) ^ -0.5);
	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, 2.8e-296], N[(2.0 * N[Power[N[(N[(1.0 / x), $MachinePrecision] / N[(y + z), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7999999999999999e-296

   1. Initial program 67.6%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 67.6%

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

   4. Applied egg-rr 67.6%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 34.1%

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

      1. inv-pow N/A

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

      2. pow1/2 N/A

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

      3. clear-num N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. metadata-eval N/A

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

   8. Applied egg-rr 67.5%

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

   9. Taylor expanded in x around inf

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

       1. associate-/r* N/A

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

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

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

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

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

       4. +-lowering-+.f64 50.6%

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

   11. Simplified 50.6%

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

   if 2.7999999999999999e-296 < y

   1. Initial program 70.8%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 70.8%

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

   4. Applied egg-rr 70.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. *-lowering-*.f64 23.9%

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

   7. Simplified 23.9%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

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

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 37.6%

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

   9. Applied egg-rr 37.6%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{x}}{y + z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot z + y \cdot x\right) + y \cdot z \leq 2 \cdot 10^{+298}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{y}}{x + z}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ (+ (* x z) (* y x)) (* y z)) 2e+298)
   (* 2.0 (sqrt (+ (* x z) (* y (+ x z)))))
   (* 2.0 (pow (/ (/ 1.0 y) (+ x z)) -0.5))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((((x * z) + (y * x)) + (y * z)) <= 2e+298) {
		tmp = 2.0 * sqrt(((x * z) + (y * (x + z))));
	} else {
		tmp = 2.0 * pow(((1.0 / y) / (x + 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 ((((x * z) + (y * x)) + (y * z)) <= 2d+298) then
        tmp = 2.0d0 * sqrt(((x * z) + (y * (x + z))))
    else
        tmp = 2.0d0 * (((1.0d0 / y) / (x + 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 ((((x * z) + (y * x)) + (y * z)) <= 2e+298) {
		tmp = 2.0 * Math.sqrt(((x * z) + (y * (x + z))));
	} else {
		tmp = 2.0 * Math.pow(((1.0 / y) / (x + z)), -0.5);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (((x * z) + (y * x)) + (y * z)) <= 2e+298:
		tmp = 2.0 * math.sqrt(((x * z) + (y * (x + z))))
	else:
		tmp = 2.0 * math.pow(((1.0 / y) / (x + z)), -0.5)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x * z) + Float64(y * x)) + Float64(y * z)) <= 2e+298)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * z) + Float64(y * Float64(x + z)))));
	else
		tmp = Float64(2.0 * (Float64(Float64(1.0 / y) / Float64(x + 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 ((((x * z) + (y * x)) + (y * z)) <= 2e+298)
		tmp = 2.0 * sqrt(((x * z) + (y * (x + z))));
	else
		tmp = 2.0 * (((1.0 / y) / (x + 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[N[(N[(N[(x * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision], 2e+298], N[(2.0 * N[Sqrt[N[(N[(x * z), $MachinePrecision] + N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(1.0 / y), $MachinePrecision] / N[(x + z), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)) < 1.9999999999999999e298

   1. Initial program 96.5%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 96.5%

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

   4. Applied egg-rr 96.5%

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

   if 1.9999999999999999e298 < (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))

   1. Initial program 10.9%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 11.0%

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

   4. Applied egg-rr 11.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 0.1%

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

      1. inv-pow N/A

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

      2. pow1/2 N/A

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

      3. clear-num N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. metadata-eval N/A

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

   8. Applied egg-rr 10.9%

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

   9. Taylor expanded in y around inf

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

       1. associate-/r* N/A

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

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

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

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

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

       4. +-commutative N/A

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

       5. +-lowering-+.f64 16.0%

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

   11. Simplified 16.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot z + y \cdot x\right) + y \cdot z \leq 2 \cdot 10^{+298}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{y}}{x + z}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-299}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{x}}{y + z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e-299)
   (* 2.0 (pow (/ (/ 1.0 x) (+ y z)) -0.5))
   (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e-299) {
		tmp = 2.0 * pow(((1.0 / x) / (y + z)), -0.5);
	} 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 <= (-1.3d-299)) then
        tmp = 2.0d0 * (((1.0d0 / x) / (y + z)) ** (-0.5d0))
    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 <= -1.3e-299) {
		tmp = 2.0 * Math.pow(((1.0 / x) / (y + z)), -0.5);
	} 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 <= -1.3e-299:
		tmp = 2.0 * math.pow(((1.0 / x) / (y + z)), -0.5)
	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 <= -1.3e-299)
		tmp = Float64(2.0 * (Float64(Float64(1.0 / x) / Float64(y + z)) ^ -0.5));
	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 <= -1.3e-299)
		tmp = 2.0 * (((1.0 / x) / (y + z)) ^ -0.5);
	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, -1.3e-299], N[(2.0 * N[Power[N[(N[(1.0 / x), $MachinePrecision] / N[(y + z), $MachinePrecision]), $MachinePrecision], -0.5], $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 < -1.2999999999999999e-299

   1. Initial program 65.8%

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

      1. *-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

      11. +-lowering-+.f64 65.9%

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

   4. Applied egg-rr 65.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 32.0%

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

      1. inv-pow N/A

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

      2. pow1/2 N/A

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

      3. clear-num N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. metadata-eval N/A

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

   8. Applied egg-rr 65.7%

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

   9. Taylor expanded in x around inf

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

       1. associate-/r* N/A

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

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

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

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

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

       4. +-lowering-+.f64 47.9%

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

   11. Simplified 47.9%

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

   if -1.2999999999999999e-299 < y

   1. Initial program 72.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{z \cdot \left(x + y\right)}} \]
   4. 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. +-lowering-+.f64 49.3%

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

   5. Simplified 49.3%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-299}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{x}}{y + z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-289) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* z (+ y x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-289) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((z * (y + x)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-289)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((z * (y + x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-289) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -5e-289:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-289)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-289)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((z * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -5e-289], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000029e-289

   1. Initial program 66.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right)}} \]
   4. 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 46.0%

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

   5. Simplified 46.0%

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

   if -5.00000000000000029e-289 < y

   1. Initial program 72.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{z \cdot \left(x + y\right)}} \]
   4. 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. +-lowering-+.f64 49.3%

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

   5. Simplified 49.3%

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

4. Final simplification 47.7%

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

Alternative 8: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e-296) (* 2.0 (sqrt (* x (+ y z)))) (* 2.0 (sqrt (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e-296) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * sqrt((y * z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.8d-296) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * sqrt((y * z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e-296) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * Math.sqrt((y * z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.8e-296:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * math.sqrt((y * z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.8e-296)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * sqrt(Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.8e-296)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * sqrt((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2.8e-296], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7999999999999999e-296

   1. Initial program 67.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right)}} \]
   4. 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 48.9%

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

   5. Simplified 48.9%

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

   if 2.7999999999999999e-296 < y

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \sqrt{y \cdot z}} \]
   4. 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 23.9%

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

   5. Simplified 23.9%

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

Alternative 9: 68.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 66.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot y}} \]
   4. 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. *-lowering-*.f64 26.4%

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

   5. Simplified 26.4%

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

   if -4.999999999999985e-310 < y

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \sqrt{y \cdot z}} \]
   4. 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 23.2%

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

   5. Simplified 23.2%

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

4. Final simplification 24.8%

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

Alternative 10: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ 2 \cdot {\left(\frac{\frac{1}{y}}{x + z}\right)}^{-0.5} \end{array} \]

FPCore

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

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return 2.0 * pow(((1.0 / y) / (x + z)), -0.5);
}

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 * (((1.0d0 / y) / (x + z)) ** (-0.5d0))
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return 2.0 * Math.pow(((1.0 / y) / (x + z)), -0.5);
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 2.0 * math.pow(((1.0 / y) / (x + z)), -0.5)

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(2.0 * (Float64(Float64(1.0 / y) / Float64(x + z)) ^ -0.5))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = 2.0 * (((1.0 / y) / (x + z)) ^ -0.5);
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[Power[N[(N[(1.0 / y), $MachinePrecision] / N[(x + z), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.1%

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

   1. *-commutative N/A

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

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

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

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

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. distribute-lft-out N/A

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

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

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

   11. +-lowering-+.f64 69.1%

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

4. Applied egg-rr 69.1%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

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

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

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

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

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

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

6. Applied egg-rr 34.9%

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

   1. inv-pow N/A

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

   2. pow1/2 N/A

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

   3. clear-num N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. flip-+ N/A

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

   7. metadata-eval N/A

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

8. Applied egg-rr 69.0%

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

9. Taylor expanded in y around inf

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

    1. associate-/r* N/A

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

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

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

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

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

    4. +-commutative N/A

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

    5. +-lowering-+.f64 47.7%

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

11. Simplified 47.7%

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

12. Final simplification 47.7%

    \[\leadsto 2 \cdot {\left(\frac{\frac{1}{y}}{x + z}\right)}^{-0.5} \]
13. Add Preprocessing

Alternative 11: 35.4% 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 69.1%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot y}} \]
4. 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. *-lowering-*.f64 25.5%

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

5. Simplified 25.5%

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

6. Final simplification 25.5%

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

Developer Target 1: 82.2% 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 2024191 
(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)))))