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

Percentage Accurate: 71.4% → 94.8%

Time: 9.6s

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: 71.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-189}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-279}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 2.0 (exp (* (- (log (- (- y) z)) (log (/ -1.0 x))) 0.5)))))
   (if (<= y -7.2e+49)
     t_0
     (if (<= y -1.05e-189)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y 2.6e-279) t_0 (* 2.0 (* (sqrt z) (sqrt y))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 2.0 * exp(((log((-y - z)) - log((-1.0 / x))) * 0.5));
	double tmp;
	if (y <= -7.2e+49) {
		tmp = t_0;
	} else if (y <= -1.05e-189) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= 2.6e-279) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * exp(((log((-y - z)) - log(((-1.0d0) / x))) * 0.5d0))
    if (y <= (-7.2d+49)) then
        tmp = t_0
    else if (y <= (-1.05d-189)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= 2.6d-279) then
        tmp = t_0
    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 t_0 = 2.0 * Math.exp(((Math.log((-y - z)) - Math.log((-1.0 / x))) * 0.5));
	double tmp;
	if (y <= -7.2e+49) {
		tmp = t_0;
	} else if (y <= -1.05e-189) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= 2.6e-279) {
		tmp = t_0;
	} 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):
	t_0 = 2.0 * math.exp(((math.log((-y - z)) - math.log((-1.0 / x))) * 0.5))
	tmp = 0
	if y <= -7.2e+49:
		tmp = t_0
	elif y <= -1.05e-189:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= 2.6e-279:
		tmp = t_0
	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)
	t_0 = Float64(2.0 * exp(Float64(Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))) * 0.5)))
	tmp = 0.0
	if (y <= -7.2e+49)
		tmp = t_0;
	elseif (y <= -1.05e-189)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= 2.6e-279)
		tmp = t_0;
	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)
	t_0 = 2.0 * exp(((log((-y - z)) - log((-1.0 / x))) * 0.5));
	tmp = 0.0;
	if (y <= -7.2e+49)
		tmp = t_0;
	elseif (y <= -1.05e-189)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= 2.6e-279)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(2.0 * N[Exp[N[(N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e+49], t$95$0, If[LessEqual[y, -1.05e-189], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e-279], t$95$0, N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999993e49 or -1.05000000000000008e-189 < y < 2.6000000000000002e-279

   1. Initial program 57.8%

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

      1. distribute-lft-out 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in x around inf 38.4%

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

      1. pow1/2 38.5%

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

      2. metadata-eval 38.5%

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

      3. pow-to-exp 36.0%

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

      4. *-commutative 36.0%

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

      5. metadata-eval 36.0%

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

   6. Applied egg-rr 36.0%

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

   7. Taylor expanded in x around -inf 39.8%

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

      1. +-commutative 39.8%

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

      2. mul-1-neg 39.8%

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

      3. unsub-neg 39.8%

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

      4. distribute-lft-in 39.8%

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

      5. mul-1-neg 39.8%

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

      6. unsub-neg 39.8%

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

      7. mul-1-neg 39.8%

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

   9. Simplified 39.8%

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

   if -7.19999999999999993e49 < y < -1.05000000000000008e-189

   1. Initial program 95.7%

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

      1. distribute-lft-out 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in x around inf 68.0%

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

   if 2.6000000000000002e-279 < y

   1. Initial program 69.9%

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

      1. distribute-lft-out 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in x around 0 20.5%

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

      1. sqrt-prod 35.4%

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

   6. Applied egg-rr 35.4%

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

      1. *-commutative 35.4%

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

   8. Simplified 35.4%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-189}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-279}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-235}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e+49)
   (* 2.0 (exp (* 0.5 (- (log (- x)) (log (/ -1.0 y))))))
   (if (<= y 9e-235)
     (* 2.0 (sqrt (fma x y (* 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 <= -7.2e+49) {
		tmp = 2.0 * exp((0.5 * (log(-x) - log((-1.0 / y)))));
	} else if (y <= 9e-235) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e+49)
		tmp = Float64(2.0 * exp(Float64(0.5 * Float64(log(Float64(-x)) - log(Float64(-1.0 / y))))));
	elseif (y <= 9e-235)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -7.2e+49], N[(2.0 * N[Exp[N[(0.5 * N[(N[Log[(-x)], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-235], N[(2.0 * N[Sqrt[N[(x * y + 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 3 regimes

2. if y < -7.19999999999999993e49

   1. Initial program 54.2%

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

      1. distribute-lft-out 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in x around inf 25.7%

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

      1. pow1/2 25.8%

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

      2. metadata-eval 25.8%

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

      3. pow-to-exp 24.1%

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

      4. *-commutative 24.1%

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

      5. metadata-eval 24.1%

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

   6. Applied egg-rr 24.1%

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

   7. Taylor expanded in y around -inf 40.4%

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

      1. +-commutative 40.4%

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

      2. mul-1-neg 40.4%

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

      3. unsub-neg 40.4%

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

      4. mul-1-neg 40.4%

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

   9. Simplified 40.4%

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

   if -7.19999999999999993e49 < y < 8.9999999999999996e-235

   1. Initial program 84.2%

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

      1. associate-+l+ 84.2%

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

      2. fma-def 84.2%

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

      3. distribute-rgt-out 84.2%

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

   3. Simplified 84.2%

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

   if 8.9999999999999996e-235 < y

   1. Initial program 67.5%

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

      1. distribute-lft-out 67.4%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in x around 0 21.7%

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

      1. sqrt-prod 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. *-commutative 38.3%

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

   8. Simplified 38.3%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-235}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-235}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y + x, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 9e-235)
   (* 2.0 (sqrt (fma z (+ y x) (* 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 <= 9e-235) {
		tmp = 2.0 * sqrt(fma(z, (y + x), (y * x)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 9e-235)
		tmp = Float64(2.0 * sqrt(fma(z, Float64(y + x), Float64(y * x))));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 9e-235], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision] + N[(y * x), $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 < 8.9999999999999996e-235

   1. Initial program 72.5%

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

      1. associate-+l+ 72.5%

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

      2. +-commutative 72.5%

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

      3. distribute-rgt-out 72.5%

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

      4. fma-def 72.5%

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

   3. Simplified 72.5%

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

   if 8.9999999999999996e-235 < y

   1. Initial program 67.5%

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

      1. distribute-lft-out 67.4%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in x around 0 21.7%

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

      1. sqrt-prod 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. *-commutative 38.3%

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

   8. Simplified 38.3%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-235}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y + x, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-235}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 9e-235)
   (* 2.0 (sqrt (+ (* x (+ y z)) (* y z))))
   (* 2.0 (* (sqrt z) (sqrt y)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e-235) {
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9d-235) then
        tmp = 2.0d0 * sqrt(((x * (y + z)) + (y * z)))
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e-235) {
		tmp = 2.0 * Math.sqrt(((x * (y + z)) + (y * z)));
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 9e-235:
		tmp = 2.0 * math.sqrt(((x * (y + z)) + (y * z)))
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 9e-235)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * Float64(y + z)) + Float64(y * z))));
	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 <= 9e-235)
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 9e-235], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $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 < 8.9999999999999996e-235

   1. Initial program 72.5%

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

      1. distribute-lft-out 72.5%

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

   3. Simplified 72.5%

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

   if 8.9999999999999996e-235 < y

   1. Initial program 67.5%

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

      1. distribute-lft-out 67.4%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in x around 0 21.7%

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

      1. sqrt-prod 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. *-commutative 38.3%

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

   8. Simplified 38.3%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-235}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 5: 71.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

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

   1. distribute-lft-out 70.4%

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

3. Simplified 70.4%

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

4. Final simplification 70.4%

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

Alternative 6: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

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-278) (* 2.0 (sqrt (* y x))) (* 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-278) {
		tmp = 2.0 * sqrt((y * x));
	} 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-278)) then
        tmp = 2.0d0 * sqrt((y * x))
    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-278) {
		tmp = 2.0 * Math.sqrt((y * x));
	} 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-278:
		tmp = 2.0 * math.sqrt((y * x))
	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-278)
		tmp = Float64(2.0 * sqrt(Float64(y * x)));
	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-278)
		tmp = 2.0 * sqrt((y * x));
	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-278], N[(2.0 * N[Sqrt[N[(y * x), $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 < -1.2999999999999999e-278

   1. Initial program 70.0%

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

      1. distribute-lft-out 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in z around 0 25.2%

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

   if -1.2999999999999999e-278 < y

   1. Initial program 70.7%

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

      1. distribute-lft-out 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in z around inf 46.6%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 7: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-291}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-291) (* 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 <= 2e-291) {
		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 <= 2d-291) 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 <= 2e-291) {
		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 <= 2e-291:
		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 <= 2e-291)
		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 <= 2e-291)
		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, 2e-291], 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 < 1.99999999999999992e-291

   1. Initial program 70.1%

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

      1. distribute-lft-out 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in x around inf 47.4%

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

   if 1.99999999999999992e-291 < y

   1. Initial program 70.7%

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

      1. distribute-lft-out 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in z around inf 44.6%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-291}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 8: 69.2% accurate, 1.1× speedup

Math

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

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 71.4%

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

      1. distribute-lft-out 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in z around 0 24.2%

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

   if -4.999999999999985e-310 < y

   1. Initial program 69.4%

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

      1. distribute-lft-out 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 19.6%

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

4. Final simplification 21.9%

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

Alternative 9: 36.9% 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 70.4%

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

   1. distribute-lft-out 70.4%

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

3. Simplified 70.4%

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

4. Taylor expanded in z around 0 25.2%

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

5. Final simplification 25.2%

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

Developer target: 83.9% 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 2023185 
(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)))))