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

Average Accuracy: 69.3% → 94.1%

Time: 17.3s

Precision: binary64

Cost: 26828

\[ \begin{array}{c}[x, y, z] = \mathsf{sort}([x, y, z])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot {\left(e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.16666666666666666}\right)}^{3}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-195}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + x \cdot z}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-256}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          2.0
          (pow
           (exp (* (- (log (- (- y) z)) (log (/ -1.0 x))) 0.16666666666666666))
           3.0))))
   (if (<= y -1.1e+49)
     t_0
     (if (<= y -2.6e-195)
       (* 2.0 (sqrt (+ (* y x) (* x z))))
       (if (<= y 5.1e-256) t_0 (* 2.0 (* (sqrt z) (sqrt y))))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = 2.0 * pow(exp(((log((-y - z)) - log((-1.0 / x))) * 0.16666666666666666)), 3.0);
	double tmp;
	if (y <= -1.1e+49) {
		tmp = t_0;
	} else if (y <= -2.6e-195) {
		tmp = 2.0 * sqrt(((y * x) + (x * z)));
	} else if (y <= 5.1e-256) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	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
    code = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end 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))) * 0.16666666666666666d0)) ** 3.0d0)
    if (y <= (-1.1d+49)) then
        tmp = t_0
    else if (y <= (-2.6d-195)) then
        tmp = 2.0d0 * sqrt(((y * x) + (x * z)))
    else if (y <= 5.1d-256) then
        tmp = t_0
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = 2.0 * Math.pow(Math.exp(((Math.log((-y - z)) - Math.log((-1.0 / x))) * 0.16666666666666666)), 3.0);
	double tmp;
	if (y <= -1.1e+49) {
		tmp = t_0;
	} else if (y <= -2.6e-195) {
		tmp = 2.0 * Math.sqrt(((y * x) + (x * z)));
	} else if (y <= 5.1e-256) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = 2.0 * math.pow(math.exp(((math.log((-y - z)) - math.log((-1.0 / x))) * 0.16666666666666666)), 3.0)
	tmp = 0
	if y <= -1.1e+49:
		tmp = t_0
	elif y <= -2.6e-195:
		tmp = 2.0 * math.sqrt(((y * x) + (x * z)))
	elif y <= 5.1e-256:
		tmp = t_0
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(2.0 * (exp(Float64(Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))) * 0.16666666666666666)) ^ 3.0))
	tmp = 0.0
	if (y <= -1.1e+49)
		tmp = t_0;
	elseif (y <= -2.6e-195)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * x) + Float64(x * z))));
	elseif (y <= 5.1e-256)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = 2.0 * (exp(((log((-y - z)) - log((-1.0 / x))) * 0.16666666666666666)) ^ 3.0);
	tmp = 0.0;
	if (y <= -1.1e+49)
		tmp = t_0;
	elseif (y <= -2.6e-195)
		tmp = 2.0 * sqrt(((y * x) + (x * z)));
	elseif (y <= 5.1e-256)
		tmp = t_0;
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	end
	tmp_2 = tmp;
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]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(2.0 * N[Power[N[Exp[N[(N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.1e+49], t$95$0, If[LessEqual[y, -2.6e-195], N[(2.0 * N[Sqrt[N[(N[(y * x), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.1e-256], t$95$0, N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	69.3%
Target	82.0%
Herbie	94.1%

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

Derivation

1. Split input into 3 regimes

2. if y < -1.1e49 or -2.6000000000000002e-195 < y < 5.10000000000000011e-256

   1. Initial program 38.3%

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

   2. Simplified 38.3%

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

      [Start] 38.3	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      associate-+l+ [=>] 38.3	\[ 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]
      +-commutative [=>] 38.3	\[ 2 \cdot \sqrt{\color{blue}{\left(x \cdot z + y \cdot z\right) + x \cdot y}} \]
      distribute-rgt-out [=>] 38.3	\[ 2 \cdot \sqrt{\color{blue}{z \cdot \left(x + y\right)} + x \cdot y} \]
      fma-def [=>] 38.3	\[ 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(z, x + y, x \cdot y\right)}} \]

   3. Applied egg-rr 34.8%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}^{0.16666666666666666}\right)}^{3}} \]
      Proof

      [Start] 38.3	\[ 2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)} \]
      add-cube-cbrt [=>] 37.6	\[ 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}}\right)} \]
      pow3 [=>] 37.6	\[ 2 \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}}\right)}^{3}} \]
      pow1/3 [=>] 34.8	\[ 2 \cdot {\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]
      pow1/2 [=>] 34.8	\[ 2 \cdot {\left({\color{blue}{\left({\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}^{0.5}\right)}}^{0.3333333333333333}\right)}^{3} \]
      pow-pow [=>] 34.8	\[ 2 \cdot {\color{blue}{\left({\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}}^{3} \]
      metadata-eval [=>] 34.8	\[ 2 \cdot {\left({\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3} \]

   4. Taylor expanded in x around -inf 85.6%

      \[\leadsto 2 \cdot {\color{blue}{\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot z + -1 \cdot y\right)\right)}\right)}}^{3} \]

   if -1.1e49 < y < -2.6000000000000002e-195

   1. Initial program 96.9%

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

   2. Simplified 96.9%

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

      [Start] 96.9	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 96.9	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 96.9%

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

   4. Taylor expanded in y around 0 96.9%

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

   if 5.10000000000000011e-256 < y

   1. Initial program 69.4%

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

   2. Simplified 69.4%

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

      [Start] 69.4	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 69.4	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around 0 67.2%

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

   4. Applied egg-rr 96.8%

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

      [Start] 67.2	\[ 2 \cdot \sqrt{y \cdot z} \]
      *-commutative [=>] 67.2	\[ 2 \cdot \sqrt{\color{blue}{z \cdot y}} \]
      sqrt-prod [=>] 96.8	\[ 2 \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{y}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot {\left(e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.16666666666666666}\right)}^{3}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-195}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + x \cdot z}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot {\left(e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.16666666666666666}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	93.0%
Cost	26564

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(\left(-z\right) - x\right) - \log \left(\frac{-1}{y}\right)\right)\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-233}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Accuracy	92.5%
Cost	26564

\[\begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-250}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Accuracy	82.7%
Cost	13892

\[\begin{array}{l} \mathbf{if}\;\left(y \cdot x + x \cdot z\right) + y \cdot z \leq 5 \cdot 10^{+307}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4
Accuracy	69.1%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 5
Accuracy	69.3%
Cost	7104

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

Alternative 6
Accuracy	69.3%
Cost	7104

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

Alternative 7
Accuracy	67.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 8
Accuracy	69.3%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 9
Accuracy	66.8%
Cost	6852

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

Alternative 10
Accuracy	35.3%
Cost	6720

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

Reproduce

herbie shell --seed 2023146 
(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)))))