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

Percentage Accurate: 70.8% → 96.5%

Time: 11.6s

Alternatives: 9

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: 70.8% 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: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-217}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          2.0
          (pow (exp (* 0.25 (- (log (- (- z) y)) (log (/ -1.0 x))))) 2.0))))
   (if (<= y -1.7e+35)
     t_0
     (if (<= y -9.5e-217)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y -5.5e-296) t_0 (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 2.0 * pow(exp((0.25 * (log((-z - y)) - log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -1.7e+35) {
		tmp = t_0;
	} else if (y <= -9.5e-217) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -5.5e-296) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (exp((0.25d0 * (log((-z - y)) - log(((-1.0d0) / x))))) ** 2.0d0)
    if (y <= (-1.7d+35)) then
        tmp = t_0
    else if (y <= (-9.5d-217)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-5.5d-296)) then
        tmp = t_0
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((-z - y)) - Math.log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -1.7e+35) {
		tmp = t_0;
	} else if (y <= -9.5e-217) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -5.5e-296) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 2.0 * math.pow(math.exp((0.25 * (math.log((-z - y)) - math.log((-1.0 / x))))), 2.0)
	tmp = 0
	if y <= -1.7e+35:
		tmp = t_0
	elif y <= -9.5e-217:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -5.5e-296:
		tmp = t_0
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-z) - y)) - log(Float64(-1.0 / x))))) ^ 2.0))
	tmp = 0.0
	if (y <= -1.7e+35)
		tmp = t_0;
	elseif (y <= -9.5e-217)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -5.5e-296)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 2.0 * (exp((0.25 * (log((-z - y)) - log((-1.0 / x))))) ^ 2.0);
	tmp = 0.0;
	if (y <= -1.7e+35)
		tmp = t_0;
	elseif (y <= -9.5e-217)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -5.5e-296)
		tmp = t_0;
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(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_] := Block[{t$95$0 = N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[((-z) - y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.7e+35], t$95$0, If[LessEqual[y, -9.5e-217], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.5e-296], t$95$0, N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7000000000000001e35 or -9.5000000000000001e-217 < y < -5.5000000000000004e-296

   1. Initial program 66.0%

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

      1. associate-+l+ 66.0%

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

      2. distribute-rgt-out 66.0%

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

      3. +-commutative 66.0%

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

   3. Simplified 66.0%

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

      1. distribute-rgt-in 66.0%

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

      2. associate-+r+ 66.0%

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

      3. *-commutative 66.0%

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

      4. distribute-lft-in 66.1%

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

      5. +-commutative 66.1%

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

      6. fma-udef 66.2%

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

      7. add-sqr-sqrt 65.9%

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

      8. pow2 65.9%

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

      9. pow1/2 66.2%

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

      10. sqrt-pow1 66.2%

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

      11. fma-udef 65.8%

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

      12. +-commutative 65.8%

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

      13. fma-def 66.2%

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

      14. metadata-eval 66.2%

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

   5. Applied egg-rr 66.2%

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

   6. Taylor expanded in x around -inf 44.4%

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

   7. Taylor expanded in x around 0 0.0%

      \[\leadsto 2 \cdot {\left(e^{0.25 \cdot \color{blue}{\left(\log \left(-1 \cdot y + -1 \cdot z\right) + -1 \cdot \left(\log -1 + -1 \cdot \log x\right)\right)}}\right)}^{2} \]
   8. Step-by-step derivation

      1. mul-1-neg 0.0%

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

      2. mul-1-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. log-div 44.4%

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

      5. unsub-neg 44.4%

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

      6. +-commutative 44.4%

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

      7. mul-1-neg 44.4%

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

      8. unsub-neg 44.4%

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

      9. mul-1-neg 44.4%

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

   9. Simplified 44.4%

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

   if -1.7000000000000001e35 < y < -9.5000000000000001e-217

   1. Initial program 87.0%

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

      1. associate-+l+ 87.0%

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

      2. distribute-rgt-out 87.0%

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

      3. +-commutative 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in x around inf 61.0%

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

   if -5.5000000000000004e-296 < y

   1. Initial program 67.9%

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

      1. associate-+l+ 67.9%

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

      2. distribute-rgt-out 67.9%

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

      3. +-commutative 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in z around inf 47.8%

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

      1. +-commutative 47.8%

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

   6. Simplified 47.8%

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

      1. *-commutative 47.8%

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

      2. sqrt-prod 50.6%

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

   8. Applied egg-rr 50.6%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+35}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-217}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 2: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-217}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (* 2.0 (pow (exp (* 0.25 (- (log (- y)) (log (/ -1.0 x))))) 2.0))))
   (if (<= y -1.6e+35)
     t_0
     (if (<= y -9.5e-217)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y -2.9e-285) t_0 (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 2.0 * pow(exp((0.25 * (log(-y) - log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -1.6e+35) {
		tmp = t_0;
	} else if (y <= -9.5e-217) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -2.9e-285) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (exp((0.25d0 * (log(-y) - log(((-1.0d0) / x))))) ** 2.0d0)
    if (y <= (-1.6d+35)) then
        tmp = t_0
    else if (y <= (-9.5d-217)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-2.9d-285)) then
        tmp = t_0
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 2.0 * Math.pow(Math.exp((0.25 * (Math.log(-y) - Math.log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -1.6e+35) {
		tmp = t_0;
	} else if (y <= -9.5e-217) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -2.9e-285) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 2.0 * math.pow(math.exp((0.25 * (math.log(-y) - math.log((-1.0 / x))))), 2.0)
	tmp = 0
	if y <= -1.6e+35:
		tmp = t_0
	elif y <= -9.5e-217:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -2.9e-285:
		tmp = t_0
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(-y)) - log(Float64(-1.0 / x))))) ^ 2.0))
	tmp = 0.0
	if (y <= -1.6e+35)
		tmp = t_0;
	elseif (y <= -9.5e-217)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -2.9e-285)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 2.0 * (exp((0.25 * (log(-y) - log((-1.0 / x))))) ^ 2.0);
	tmp = 0.0;
	if (y <= -1.6e+35)
		tmp = t_0;
	elseif (y <= -9.5e-217)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -2.9e-285)
		tmp = t_0;
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(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_] := Block[{t$95$0 = N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[(-y)], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+35], t$95$0, If[LessEqual[y, -9.5e-217], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.9e-285], t$95$0, N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.59999999999999991e35 or -9.5000000000000001e-217 < y < -2.9e-285

   1. Initial program 64.7%

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

      1. associate-+l+ 64.7%

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

      2. distribute-rgt-out 64.7%

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

      3. +-commutative 64.7%

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

   3. Simplified 64.7%

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

      1. distribute-rgt-in 64.7%

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

      2. associate-+r+ 64.7%

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

      3. *-commutative 64.7%

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

      4. distribute-lft-in 64.8%

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

      5. +-commutative 64.8%

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

      6. fma-udef 64.9%

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

      7. add-sqr-sqrt 64.6%

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

      8. pow2 64.6%

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

      9. pow1/2 64.9%

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

      10. sqrt-pow1 64.9%

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

      11. fma-udef 64.5%

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

      12. +-commutative 64.5%

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

      13. fma-def 64.9%

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

      14. metadata-eval 64.9%

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

   5. Applied egg-rr 64.9%

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

   6. Taylor expanded in x around -inf 42.6%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. distribute-lft-out 0.0%

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

      4. mul-1-neg 0.0%

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

      5. log-rec 0.0%

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

      6. remove-double-neg 0.0%

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

      7. associate-+l+ 0.0%

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

      8. log-prod 34.3%

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

      9. *-commutative 34.3%

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

      10. +-commutative 34.3%

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

      11. mul-1-neg 34.3%

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

      12. unsub-neg 34.3%

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

      13. mul-1-neg 34.3%

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

   9. Simplified 34.3%

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

   if -1.59999999999999991e35 < y < -9.5000000000000001e-217

   1. Initial program 87.0%

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

      1. associate-+l+ 87.0%

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

      2. distribute-rgt-out 87.0%

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

      3. +-commutative 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in x around inf 61.0%

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

   if -2.9e-285 < y

   1. Initial program 68.6%

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

      1. associate-+l+ 68.6%

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

      2. distribute-rgt-out 68.6%

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

      3. +-commutative 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in z around inf 49.0%

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

      1. +-commutative 49.0%

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

   6. Simplified 49.0%

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

      1. *-commutative 49.0%

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

      2. sqrt-prod 49.4%

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

   8. Applied egg-rr 49.4%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+35}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-217}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 3: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 3.1e-293)
   (* 2.0 (sqrt (* x (+ y z))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.1e-293) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.1d-293) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.1e-293) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 3.1e-293:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 3.1e-293)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.1e-293)
		tmp = 2.0 * sqrt((x * (y + z)));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 3.1e-293], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.09999999999999983e-293

   1. Initial program 73.5%

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

      1. associate-+l+ 73.5%

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

      2. distribute-rgt-out 73.5%

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

      3. +-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in x around inf 50.9%

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

   if 3.09999999999999983e-293 < y

   1. Initial program 68.4%

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

      1. associate-+l+ 68.4%

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

      2. distribute-rgt-out 68.4%

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

      3. +-commutative 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in z around inf 47.1%

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

      1. +-commutative 47.1%

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

   6. Simplified 47.1%

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

      1. *-commutative 47.1%

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

      2. sqrt-prod 48.5%

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

   8. Applied egg-rr 48.5%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-293}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 4: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+29}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 6.2e+29)
   (* 2.0 (sqrt (+ (* y x) (* z (+ y x)))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e+29) {
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6.2d+29) then
        tmp = 2.0d0 * sqrt(((y * x) + (z * (y + x))))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e+29) {
		tmp = 2.0 * Math.sqrt(((y * x) + (z * (y + x))));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 6.2e+29:
		tmp = 2.0 * math.sqrt(((y * x) + (z * (y + x))))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 6.2e+29)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(z * Float64(y + x)))));
	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 <= 6.2e+29)
		tmp = 2.0 * sqrt(((y * x) + (z * (y + x))));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 6.2e+29], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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 < 6.1999999999999998e29

   1. Initial program 76.7%

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

      1. associate-+l+ 76.7%

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

      2. distribute-rgt-out 76.7%

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

      3. +-commutative 76.7%

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

   3. Simplified 76.7%

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

   if 6.1999999999999998e29 < y

   1. Initial program 50.6%

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

      1. associate-+l+ 50.6%

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

      2. distribute-rgt-out 50.6%

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

      3. +-commutative 50.6%

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

   3. Simplified 50.6%

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

      1. distribute-rgt-in 50.6%

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

      2. associate-+r+ 50.6%

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

      3. *-commutative 50.6%

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

      4. distribute-lft-in 50.6%

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

      5. +-commutative 50.6%

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

      6. fma-udef 50.8%

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

      7. add-cbrt-cube 38.1%

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

      8. pow1/3 35.7%

         \[\leadsto 2 \cdot \color{blue}{{\left(\left(\sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)} \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\right)}^{0.3333333333333333}} \]

      9. add-sqr-sqrt 35.7%

         \[\leadsto 2 \cdot {\left(\color{blue}{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)} \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\right)}^{0.3333333333333333} \]

      10. pow1 35.7%

          \[\leadsto 2 \cdot {\left(\color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{1}} \cdot \sqrt{\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)}\right)}^{0.3333333333333333} \]

      11. pow1/2 36.2%

          \[\leadsto 2 \cdot {\left({\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{1} \cdot \color{blue}{{\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      12. pow-prod-up 36.2%

          \[\leadsto 2 \cdot {\color{blue}{\left({\left(\mathsf{fma}\left(x, z, y \cdot \left(x + z\right)\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      13. fma-udef 35.5%

          \[\leadsto 2 \cdot {\left({\color{blue}{\left(x \cdot z + y \cdot \left(x + z\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      14. +-commutative 35.5%

          \[\leadsto 2 \cdot {\left({\color{blue}{\left(y \cdot \left(x + z\right) + x \cdot z\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      15. fma-def 36.2%

          \[\leadsto 2 \cdot {\left({\color{blue}{\left(\mathsf{fma}\left(y, x + z, x \cdot z\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      16. metadata-eval 36.2%

          \[\leadsto 2 \cdot {\left({\left(\mathsf{fma}\left(y, x + z, x \cdot z\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 36.2%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y, x + z, x \cdot z\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 38.6%

         \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(y, x + z, x \cdot z\right)\right)}^{1.5}}} \]

      2. fma-def 37.9%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(y \cdot \left(x + z\right) + x \cdot z\right)}}^{1.5}} \]

      3. +-commutative 37.9%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(x \cdot z + y \cdot \left(x + z\right)\right)}}^{1.5}} \]

      4. +-commutative 37.9%

         \[\leadsto 2 \cdot \sqrt[3]{{\left(x \cdot z + y \cdot \color{blue}{\left(z + x\right)}\right)}^{1.5}} \]

      5. distribute-lft-in 37.9%

         \[\leadsto 2 \cdot \sqrt[3]{{\left(x \cdot z + \color{blue}{\left(y \cdot z + y \cdot x\right)}\right)}^{1.5}} \]

      6. *-commutative 37.9%

         \[\leadsto 2 \cdot \sqrt[3]{{\left(x \cdot z + \left(y \cdot z + \color{blue}{x \cdot y}\right)\right)}^{1.5}} \]

      7. associate-+l+ 37.9%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(\left(x \cdot z + y \cdot z\right) + x \cdot y\right)}}^{1.5}} \]

      8. distribute-rgt-in 37.9%

         \[\leadsto 2 \cdot \sqrt[3]{{\left(\color{blue}{z \cdot \left(x + y\right)} + x \cdot y\right)}^{1.5}} \]

      9. fma-def 38.6%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}}^{1.5}} \]

      10. +-commutative 38.6%

          \[\leadsto 2 \cdot \sqrt[3]{{\left(\mathsf{fma}\left(z, \color{blue}{y + x}, x \cdot y\right)\right)}^{1.5}} \]

      11. *-commutative 38.6%

          \[\leadsto 2 \cdot \sqrt[3]{{\left(\mathsf{fma}\left(z, y + x, \color{blue}{y \cdot x}\right)\right)}^{1.5}} \]

   7. Simplified 38.6%

      \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(z, y + x, y \cdot x\right)\right)}^{1.5}}} \]

   8. Taylor expanded in x around 0 23.7%

      \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(y \cdot z\right)}}^{1.5}} \]
   9. Step-by-step derivation

      1. *-commutative 23.7%

         \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(z \cdot y\right)}}^{1.5}} \]

   10. Simplified 23.7%

       \[\leadsto 2 \cdot \sqrt[3]{{\color{blue}{\left(z \cdot y\right)}}^{1.5}} \]
   11. Step-by-step derivation

       1. pow1/3 22.4%

          \[\leadsto 2 \cdot \color{blue}{{\left({\left(z \cdot y\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 31.0%

          \[\leadsto 2 \cdot \color{blue}{{\left(z \cdot y\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 31.0%

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

       4. pow1/2 30.6%

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

       5. sqrt-prod 53.0%

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

   12. Applied egg-rr 53.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+29}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 5: 70.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. associate-+l+ 71.2%

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

   2. distribute-rgt-out 71.2%

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

   3. +-commutative 71.2%

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

3. Simplified 71.2%

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

4. Final simplification 71.2%

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

Alternative 6: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3e-285) (* 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 <= -3e-285) {
		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 <= (-3d-285)) 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 <= -3e-285) {
		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 <= -3e-285:
		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 <= -3e-285)
		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 <= -3e-285)
		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, -3e-285], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000003e-285

   1. Initial program 73.6%

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

      1. associate-+l+ 73.6%

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

      2. distribute-rgt-out 73.6%

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

      3. +-commutative 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in x around inf 49.3%

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

   if -3.00000000000000003e-285 < y

   1. Initial program 68.6%

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

      1. associate-+l+ 68.6%

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

      2. distribute-rgt-out 68.6%

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

      3. +-commutative 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in x around 0 24.8%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 7: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3.15e-285)
   (* 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 <= -3.15e-285) {
		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 <= (-3.15d-285)) 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 <= -3.15e-285) {
		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 <= -3.15e-285:
		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 <= -3.15e-285)
		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 <= -3.15e-285)
		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, -3.15e-285], 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 < -3.14999999999999994e-285

   1. Initial program 73.6%

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

      1. associate-+l+ 73.6%

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

      2. distribute-rgt-out 73.6%

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

      3. +-commutative 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in x around inf 49.3%

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

   if -3.14999999999999994e-285 < y

   1. Initial program 68.6%

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

      1. associate-+l+ 68.6%

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

      2. distribute-rgt-out 68.6%

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

      3. +-commutative 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in z around inf 49.0%

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

      1. +-commutative 49.0%

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

   6. Simplified 49.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 8: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -3e-285) (* 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 <= -3e-285) {
		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 <= (-3d-285)) 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 <= -3e-285) {
		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 <= -3e-285:
		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 <= -3e-285)
		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 <= -3e-285)
		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, -3e-285], N[(2.0 * N[Sqrt[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000003e-285

   1. Initial program 73.6%

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

      1. associate-+l+ 73.6%

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

      2. distribute-rgt-out 73.6%

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

      3. +-commutative 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in z around 0 29.9%

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

      1. *-commutative 29.9%

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

   6. Simplified 29.9%

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

   if -3.00000000000000003e-285 < y

   1. Initial program 68.6%

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

      1. associate-+l+ 68.6%

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

      2. distribute-rgt-out 68.6%

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

      3. +-commutative 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in x around 0 24.8%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 9: 35.8% 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 71.2%

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

   1. associate-+l+ 71.2%

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

   2. distribute-rgt-out 71.2%

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

   3. +-commutative 71.2%

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

3. Simplified 71.2%

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

4. Taylor expanded in z around 0 25.3%

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

   1. *-commutative 25.3%

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

6. Simplified 25.3%

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

7. Final simplification 25.3%

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

Developer target: 82.9% 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 2023331 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< z 7.636950090573675e+176) (* 2.0 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25))) (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25)))) 2.0))

  (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))