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

Average Error: 19.6 → 10.3

Time: 11.1s

Precision: binary64

Cost: 14920

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(x \cdot y + x \cdot z\right) + y \cdot z\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;2 \cdot \left(\sqrt{x + y} \cdot \sqrt{z}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y, x \cdot \left(y + z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

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 (+ (+ (* x y) (* x z)) (* y z))))
   (if (<= t_0 5e-324)
     (* 2.0 (* (sqrt (+ x y)) (sqrt z)))
     (if (<= t_0 5e+302)
       (* 2.0 (sqrt (fma z y (* x (+ y z)))))
       (* 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 = ((x * y) + (x * z)) + (y * z);
	double tmp;
	if (t_0 <= 5e-324) {
		tmp = 2.0 * (sqrt((x + y)) * sqrt(z));
	} else if (t_0 <= 5e+302) {
		tmp = 2.0 * sqrt(fma(z, y, (x * (y + z))));
	} else {
		tmp = 2.0 * (sqrt(z) * 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(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))
	tmp = 0.0
	if (t_0 <= 5e-324)
		tmp = Float64(2.0 * Float64(sqrt(Float64(x + y)) * sqrt(z)));
	elseif (t_0 <= 5e+302)
		tmp = Float64(2.0 * sqrt(fma(z, y, Float64(x * Float64(y + z)))));
	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_] := Block[{t$95$0 = N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-324], N[(2.0 * N[(N[Sqrt[N[(x + y), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(2.0 * N[Sqrt[N[(z * y + N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	19.6
Target	11.2
Herbie	10.3

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)) < 4.94066e-324

   1. Initial program 60.4

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

   2. Simplified 60.4

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

      [Start] 60.4	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 60.4	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in z around inf 60.4

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

   4. Applied egg-rr 35.6

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

   if 4.94066e-324 < (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z)) < 5e302

   1. Initial program 0.3

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

   2. Simplified 0.3

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

      [Start] 0.3	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 0.3	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 0.3

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

   if 5e302 < (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))

   1. Initial program 62.4

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

   2. Simplified 62.4

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

      [Start] 62.4	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 62.4	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in z around inf 62.5

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

   4. Taylor expanded in x around 0 62.5

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

   5. Simplified 62.5

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

      [Start] 62.5	\[ 2 \cdot \sqrt{y \cdot z} \]
      *-commutative [<=] 62.5	\[ 2 \cdot \sqrt{\color{blue}{z \cdot y}} \]

   6. Applied egg-rr 32.1

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y + x \cdot z\right) + y \cdot z \leq 5 \cdot 10^{-324}:\\ \;\;\;\;2 \cdot \left(\sqrt{x + y} \cdot \sqrt{z}\right)\\ \mathbf{elif}\;\left(x \cdot y + x \cdot z\right) + y \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y, x \cdot \left(y + z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	14664

\[\begin{array}{l} t_0 := \left(x \cdot y + x \cdot z\right) + y \cdot z\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;2 \cdot \left(\sqrt{x + y} \cdot \sqrt{z}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	10.9
Cost	13252

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

Alternative 3
Error	19.6
Cost	7104

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

Alternative 4
Error	20.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(x + y\right)}\\ \end{array} \]

Alternative 5
Error	19.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(x + y\right)}\\ \end{array} \]

Alternative 6
Error	21.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 7
Error	41.7
Cost	6720

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

Reproduce

herbie shell --seed 2022364 
(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)))))