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

Average Accuracy: 69.7% → 94.3%

Time: 13.1s

Precision: binary64

Cost: 20228

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \log \left(\left(-z\right) - x\right) + -0.5 \cdot \log \left(\frac{-1}{y}\right)}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot {\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.4e+60)
   (* 2.0 (exp (+ (* 0.5 (log (- (- z) x))) (* -0.5 (log (/ -1.0 y))))))
   (if (<= y 8.6e-277)
     (* 2.0 (pow (fma x (+ y z) (* y z)) 0.5))
     (* 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 <= -3.4e+60) {
		tmp = 2.0 * exp(((0.5 * log((-z - x))) + (-0.5 * log((-1.0 / y)))));
	} else if (y <= 8.6e-277) {
		tmp = 2.0 * pow(fma(x, (y + z), (y * z)), 0.5);
	} 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)
	tmp = 0.0
	if (y <= -3.4e+60)
		tmp = Float64(2.0 * exp(Float64(Float64(0.5 * log(Float64(Float64(-z) - x))) + Float64(-0.5 * log(Float64(-1.0 / y))))));
	elseif (y <= 8.6e-277)
		tmp = Float64(2.0 * (fma(x, Float64(y + z), Float64(y * z)) ^ 0.5));
	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, -3.4e+60], N[(2.0 * N[Exp[N[(N[(0.5 * N[Log[N[((-z) - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e-277], N[(2.0 * N[Power[N[(x * N[(y + z), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	69.7%
Target	82.8%
Herbie	94.3%

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

Derivation

1. Split input into 3 regimes

2. if y < -3.4e60

   1. Initial program 47.5%

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

   2. Simplified 47.5%

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

      [Start] 47.5	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 47.5	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 11.8%

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

      [Start] 47.5	\[ 2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z} \]
      pow1/2 [=>] 47.5	\[ 2 \cdot \color{blue}{{\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{0.5}} \]
      metadata-eval [<=] 47.5	\[ 2 \cdot {\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{\color{blue}{\left(0.25 + 0.25\right)}} \]
      metadata-eval [<=] 47.5	\[ 2 \cdot {\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{\left(\color{blue}{0.5 \cdot 0.5} + 0.25\right)} \]
      metadata-eval [<=] 47.5	\[ 2 \cdot {\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{\left(0.5 \cdot 0.5 + \color{blue}{0.5 \cdot 0.5}\right)} \]
      pow-prod-up [<=] 47.2	\[ 2 \cdot \color{blue}{\left({\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{\left(0.5 \cdot 0.5\right)} \cdot {\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{\left(0.5 \cdot 0.5\right)}\right)} \]
      pow-prod-down [=>] 11.8	\[ 2 \cdot \color{blue}{{\left(\left(x \cdot \left(y + z\right) + y \cdot z\right) \cdot \left(x \cdot \left(y + z\right) + y \cdot z\right)\right)}^{\left(0.5 \cdot 0.5\right)}} \]
      pow2 [=>] 11.8	\[ 2 \cdot {\color{blue}{\left({\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{2}\right)}}^{\left(0.5 \cdot 0.5\right)} \]
      fma-def [=>] 11.8	\[ 2 \cdot {\left({\color{blue}{\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}}^{2}\right)}^{\left(0.5 \cdot 0.5\right)} \]
      metadata-eval [=>] 11.8	\[ 2 \cdot {\left({\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{2}\right)}^{\color{blue}{0.25}} \]

   4. Taylor expanded in y around -inf 43.1%

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

   5. Simplified 86.2%

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

      [Start] 43.1	\[ 2 \cdot e^{0.25 \cdot \left(\log \left({\left(-1 \cdot z + -1 \cdot x\right)}^{2}\right) + -2 \cdot \log \left(\frac{-1}{y}\right)\right)} \]
      distribute-lft-in [=>] 43.1	\[ 2 \cdot e^{\color{blue}{0.25 \cdot \log \left({\left(-1 \cdot z + -1 \cdot x\right)}^{2}\right) + 0.25 \cdot \left(-2 \cdot \log \left(\frac{-1}{y}\right)\right)}} \]
      log-pow [=>] 86.2	\[ 2 \cdot e^{0.25 \cdot \color{blue}{\left(2 \cdot \log \left(-1 \cdot z + -1 \cdot x\right)\right)} + 0.25 \cdot \left(-2 \cdot \log \left(\frac{-1}{y}\right)\right)} \]
      associate-*r* [=>] 86.2	\[ 2 \cdot e^{\color{blue}{\left(0.25 \cdot 2\right) \cdot \log \left(-1 \cdot z + -1 \cdot x\right)} + 0.25 \cdot \left(-2 \cdot \log \left(\frac{-1}{y}\right)\right)} \]
      metadata-eval [=>] 86.2	\[ 2 \cdot e^{\color{blue}{0.5} \cdot \log \left(-1 \cdot z + -1 \cdot x\right) + 0.25 \cdot \left(-2 \cdot \log \left(\frac{-1}{y}\right)\right)} \]
      mul-1-neg [=>] 86.2	\[ 2 \cdot e^{0.5 \cdot \log \left(-1 \cdot z + \color{blue}{\left(-x\right)}\right) + 0.25 \cdot \left(-2 \cdot \log \left(\frac{-1}{y}\right)\right)} \]
      unsub-neg [=>] 86.2	\[ 2 \cdot e^{0.5 \cdot \log \color{blue}{\left(-1 \cdot z - x\right)} + 0.25 \cdot \left(-2 \cdot \log \left(\frac{-1}{y}\right)\right)} \]
      mul-1-neg [=>] 86.2	\[ 2 \cdot e^{0.5 \cdot \log \left(\color{blue}{\left(-z\right)} - x\right) + 0.25 \cdot \left(-2 \cdot \log \left(\frac{-1}{y}\right)\right)} \]
      associate-*r* [=>] 86.2	\[ 2 \cdot e^{0.5 \cdot \log \left(\left(-z\right) - x\right) + \color{blue}{\left(0.25 \cdot -2\right) \cdot \log \left(\frac{-1}{y}\right)}} \]
      metadata-eval [=>] 86.2	\[ 2 \cdot e^{0.5 \cdot \log \left(\left(-z\right) - x\right) + \color{blue}{-0.5} \cdot \log \left(\frac{-1}{y}\right)} \]

   if -3.4e60 < y < 8.59999999999999981e-277

   1. Initial program 80.5%

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

   2. Simplified 80.5%

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

      [Start] 80.5	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 80.5	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 80.5%

      \[\leadsto 2 \cdot \color{blue}{{\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{0.5}} \]
      Step-by-step derivation

      [Start] 80.5	\[ 2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z} \]
      pow1/2 [=>] 80.5	\[ 2 \cdot \color{blue}{{\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{0.5}} \]
      fma-def [=>] 80.5	\[ 2 \cdot {\color{blue}{\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}}^{0.5} \]

   if 8.59999999999999981e-277 < y

   1. Initial program 71.3%

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

   2. Simplified 71.3%

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

      [Start] 71.3	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 71.3	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 40.8%

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

      [Start] 71.3	\[ 2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z} \]
      pow1/2 [=>] 71.3	\[ 2 \cdot \color{blue}{{\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{0.5}} \]
      metadata-eval [<=] 71.3	\[ 2 \cdot {\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{\color{blue}{\left(0.25 + 0.25\right)}} \]
      metadata-eval [<=] 71.3	\[ 2 \cdot {\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{\left(\color{blue}{0.5 \cdot 0.5} + 0.25\right)} \]
      metadata-eval [<=] 71.3	\[ 2 \cdot {\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{\left(0.5 \cdot 0.5 + \color{blue}{0.5 \cdot 0.5}\right)} \]
      pow-prod-up [<=] 70.8	\[ 2 \cdot \color{blue}{\left({\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{\left(0.5 \cdot 0.5\right)} \cdot {\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{\left(0.5 \cdot 0.5\right)}\right)} \]
      pow-prod-down [=>] 40.8	\[ 2 \cdot \color{blue}{{\left(\left(x \cdot \left(y + z\right) + y \cdot z\right) \cdot \left(x \cdot \left(y + z\right) + y \cdot z\right)\right)}^{\left(0.5 \cdot 0.5\right)}} \]
      pow2 [=>] 40.8	\[ 2 \cdot {\color{blue}{\left({\left(x \cdot \left(y + z\right) + y \cdot z\right)}^{2}\right)}}^{\left(0.5 \cdot 0.5\right)} \]
      fma-def [=>] 40.8	\[ 2 \cdot {\left({\color{blue}{\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}}^{2}\right)}^{\left(0.5 \cdot 0.5\right)} \]
      metadata-eval [=>] 40.8	\[ 2 \cdot {\left({\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{2}\right)}^{\color{blue}{0.25}} \]

   4. Taylor expanded in x around 0 21.5%

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

   5. Applied egg-rr 41.4%

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

      [Start] 21.5	\[ 2 \cdot {\left({\left(y \cdot z\right)}^{2}\right)}^{0.25} \]
      pow-pow [=>] 31.4	\[ 2 \cdot \color{blue}{{\left(y \cdot z\right)}^{\left(2 \cdot 0.25\right)}} \]
      metadata-eval [=>] 31.4	\[ 2 \cdot {\left(y \cdot z\right)}^{\color{blue}{0.5}} \]
      pow1/2 [<=] 31.4	\[ 2 \cdot \color{blue}{\sqrt{y \cdot z}} \]
      *-commutative [=>] 31.4	\[ 2 \cdot \sqrt{\color{blue}{z \cdot y}} \]
      sqrt-prod [=>] 41.4	\[ 2 \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{y}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \log \left(\left(-z\right) - x\right) + -0.5 \cdot \log \left(\frac{-1}{y}\right)}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot {\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.3%
Cost	13572

\[\begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot {\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Accuracy	83.1%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Accuracy	69.7%
Cost	7104

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

Alternative 4
Accuracy	68.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot {\left(y \cdot x\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 5
Accuracy	69.3%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 6
Accuracy	67.4%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot {\left(y \cdot x\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 7
Accuracy	67.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 8
Accuracy	35.4%
Cost	6720

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

Reproduce

herbie shell --seed 2023157 
(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)))))