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

Average Accuracy: 69.3% → 96.0%

Time: 15.6s

Precision: binary64

Cost: 19716

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-310)
   (* 2.0 (pow (* (cbrt y) (cbrt x)) 1.5))
   (* 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 tmp;
	if (y <= -1e-310) {
		tmp = 2.0 * pow((cbrt(y) * cbrt(x)), 1.5);
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	return tmp;
}

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 tmp;
	if (y <= -1e-310) {
		tmp = 2.0 * Math.pow((Math.cbrt(y) * Math.cbrt(x)), 1.5);
	} 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)
	tmp = 0.0
	if (y <= -1e-310)
		tmp = Float64(2.0 * (Float64(cbrt(y) * cbrt(x)) ^ 1.5));
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	end
	return 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_] := If[LessEqual[y, -1e-310], N[(2.0 * N[Power[N[(N[Power[y, 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	69.3%
Target	82.3%
Herbie	96.0%

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

Derivation

1. Split input into 2 regimes

2. if y < -9.999999999999969e-311

   1. Initial program 68.7%

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

   2. Simplified 68.7%

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

      [Start] 68.7	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 68.7	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in z around 0 66.5%

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

   4. Applied egg-rr 65.8%

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

      [Start] 66.5	\[ 2 \cdot \sqrt{y \cdot x} \]
      add-cube-cbrt [=>] 65.8	\[ 2 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}\right) \cdot \sqrt[3]{y \cdot x}}} \]
      sqrt-prod [=>] 65.8	\[ 2 \cdot \color{blue}{\left(\sqrt{\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}} \cdot \sqrt{\sqrt[3]{y \cdot x}}\right)} \]
      pow2 [=>] 65.8	\[ 2 \cdot \left(\sqrt{\color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{2}}} \cdot \sqrt{\sqrt[3]{y \cdot x}}\right) \]

   5. Simplified 65.8%

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

      [Start] 65.8	\[ 2 \cdot \left(\sqrt{{\left(\sqrt[3]{y \cdot x}\right)}^{2}} \cdot \sqrt{\sqrt[3]{y \cdot x}}\right) \]
      *-commutative [=>] 65.8	\[ 2 \cdot \color{blue}{\left(\sqrt{\sqrt[3]{y \cdot x}} \cdot \sqrt{{\left(\sqrt[3]{y \cdot x}\right)}^{2}}\right)} \]
      unpow1/2 [<=] 65.8	\[ 2 \cdot \left(\color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{0.5}} \cdot \sqrt{{\left(\sqrt[3]{y \cdot x}\right)}^{2}}\right) \]
      unpow2 [=>] 65.8	\[ 2 \cdot \left({\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot \sqrt{\color{blue}{\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}}}\right) \]
      rem-sqrt-square [=>] 65.8	\[ 2 \cdot \left({\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot \color{blue}{\left|\sqrt[3]{y \cdot x}\right|}\right) \]
      unpow1 [<=] 65.8	\[ 2 \cdot \left({\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot \left|\color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{1}}\right|\right) \]
      metadata-eval [<=] 65.8	\[ 2 \cdot \left({\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot \left|{\left(\sqrt[3]{y \cdot x}\right)}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|\right) \]
      pow-sqr [<=] 65.6	\[ 2 \cdot \left({\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot \left|\color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot {\left(\sqrt[3]{y \cdot x}\right)}^{0.5}}\right|\right) \]
      fabs-sqr [=>] 65.6	\[ 2 \cdot \left({\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot \color{blue}{\left({\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot {\left(\sqrt[3]{y \cdot x}\right)}^{0.5}\right)}\right) \]
      pow-sqr [=>] 65.8	\[ 2 \cdot \left({\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot \color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{\left(2 \cdot 0.5\right)}}\right) \]
      metadata-eval [=>] 65.8	\[ 2 \cdot \left({\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot {\left(\sqrt[3]{y \cdot x}\right)}^{\color{blue}{1}}\right) \]
      unpow1 [=>] 65.8	\[ 2 \cdot \left({\left(\sqrt[3]{y \cdot x}\right)}^{0.5} \cdot \color{blue}{\sqrt[3]{y \cdot x}}\right) \]
      pow-plus [=>] 65.8	\[ 2 \cdot \color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{\left(0.5 + 1\right)}} \]
      metadata-eval [=>] 65.8	\[ 2 \cdot {\left(\sqrt[3]{y \cdot x}\right)}^{\color{blue}{1.5}} \]

   6. Applied egg-rr 95.6%

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

      [Start] 65.8	\[ 2 \cdot {\left(\sqrt[3]{y \cdot x}\right)}^{1.5} \]
      cbrt-prod [=>] 95.6	\[ 2 \cdot {\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)}}^{1.5} \]

   if -9.999999999999969e-311 < y

   1. Initial program 69.8%

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

   2. Simplified 69.8%

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

      [Start] 69.8	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 69.8	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around 0 67.4%

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

   4. Applied egg-rr 96.4%

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

      [Start] 67.4	\[ 2 \cdot \sqrt{y \cdot z} \]
      sqrt-prod [=>] 96.4	\[ 2 \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{z}\right)} \]
      *-commutative [=>] 96.4	\[ 2 \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{y}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	71.5%
Cost	14921

\[\begin{array}{l} t_0 := \left(x \cdot z + y \cdot x\right) + y \cdot z\\ \mathbf{if}\;t_0 \leq 10^{-322} \lor \neg \left(t_0 \leq \infty\right):\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	71.5%
Cost	14665

\[\begin{array}{l} t_0 := \left(x \cdot z + y \cdot x\right) + y \cdot z\\ \mathbf{if}\;t_0 \leq 10^{-322} \lor \neg \left(t_0 \leq \infty\right):\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \end{array} \]

Alternative 3
Accuracy	69.2%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{-302}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 4
Accuracy	69.3%
Cost	7104

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

Alternative 5
Accuracy	69.3%
Cost	7104

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

Alternative 6
Accuracy	67.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-263}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 7
Accuracy	69.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 8
Accuracy	66.9%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 9
Accuracy	34.2%
Cost	6720

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

Reproduce

herbie shell --seed 2023130 
(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)))))