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

Average Error: 19.9 → 5.0

Time: 8.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot {\left(e^{0.125}\right)}^{\left(\mathsf{fma}\left(-4, \log \left(\frac{-1}{x}\right), 4 \cdot \log \left(\left(-z\right) - y\right)\right)\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(e^{0.125}\right)}^{\left(\mathsf{fma}\left(4, \log \left(y + x\right), 4 \cdot \log z\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.6e+50)
   (*
    2.0
    (pow (exp 0.125) (fma -4.0 (log (/ -1.0 x)) (* 4.0 (log (- (- z) y))))))
   (if (<= y 2.6e+49)
     (* 2.0 (sqrt (fma x y (* z (+ y x)))))
     (* 2.0 (pow (exp 0.125) (fma 4.0 (log (+ y x)) (* 4.0 (log z))))))))

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 <= -2.6e+50) {
		tmp = 2.0 * pow(exp(0.125), fma(-4.0, log((-1.0 / x)), (4.0 * log((-z - y)))));
	} else if (y <= 2.6e+49) {
		tmp = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	} else {
		tmp = 2.0 * pow(exp(0.125), fma(4.0, log((y + x)), (4.0 * log(z))));
	}
	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 <= -2.6e+50)
		tmp = Float64(2.0 * (exp(0.125) ^ fma(-4.0, log(Float64(-1.0 / x)), Float64(4.0 * log(Float64(Float64(-z) - y))))));
	elseif (y <= 2.6e+49)
		tmp = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))));
	else
		tmp = Float64(2.0 * (exp(0.125) ^ fma(4.0, log(Float64(y + x)), Float64(4.0 * log(z)))));
	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, -2.6e+50], N[(2.0 * N[Power[N[Exp[0.125], $MachinePrecision], N[(-4.0 * N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(4.0 * N[Log[N[((-z) - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+49], N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[Exp[0.125], $MachinePrecision], N[(4.0 * N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + N[(4.0 * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	19.9
Target	11.6
Herbie	5.0

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

Derivation

1. Split input into 3 regimes

2. if y < -2.6000000000000002e50

   1. Initial program 46.1

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

   2. Simplified 46.1

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

   3. Applied egg-rr 46.2

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

   4. Applied egg-rr 46.2

      \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{0.25} \cdot \left(\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}^{0.75}}} \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}\right)} \]

   5. Applied egg-rr 46.2

      \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{0.25} \cdot \left(\sqrt[3]{\color{blue}{\left(\sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, x + y, x \cdot y\right)}}\right) \cdot \sqrt[3]{{\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}^{0.25}}}} \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}\right)} \]

   6. Taylor expanded in x around -inf 53.0

      \[\leadsto 2 \cdot \color{blue}{e^{0.125 \cdot \left(-4 \cdot \log \left(\frac{-1}{x}\right) + \log \left({\left(-1 \cdot z + -1 \cdot y\right)}^{4}\right)\right)}} \]

   7. Simplified 6.8

      \[\leadsto 2 \cdot \color{blue}{{\left(e^{0.125}\right)}^{\left(\mathsf{fma}\left(-4, \log \left(\frac{-1}{x}\right), 4 \cdot \log \left(\left(-z\right) - y\right)\right)\right)}} \]

   if -2.6000000000000002e50 < y < 2.59999999999999989e49

   1. Initial program 3.9

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

   2. Simplified 3.9

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

   if 2.59999999999999989e49 < y

   1. Initial program 45.8

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

   2. Simplified 45.8

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

   3. Applied egg-rr 45.9

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

   4. Applied egg-rr 45.9

      \[\leadsto 2 \cdot \sqrt{{\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{0.25} \cdot \left(\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(z, x + y, x \cdot y\right)\right)}^{0.75}}} \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}\right)} \]

   5. Taylor expanded in z around inf 54.7

      \[\leadsto 2 \cdot \color{blue}{e^{0.125 \cdot \left(\log \left({\left(y + x\right)}^{4}\right) + -4 \cdot \log \left(\frac{1}{z}\right)\right)}} \]

   6. Simplified 6.8

      \[\leadsto 2 \cdot \color{blue}{{\left(e^{0.125}\right)}^{\left(\mathsf{fma}\left(4, \log \left(y + x\right), -4 \cdot \left(-\log z\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot {\left(e^{0.125}\right)}^{\left(\mathsf{fma}\left(-4, \log \left(\frac{-1}{x}\right), 4 \cdot \log \left(\left(-z\right) - y\right)\right)\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(e^{0.125}\right)}^{\left(\mathsf{fma}\left(4, \log \left(y + x\right), 4 \cdot \log z\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(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)))))