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

Average Error: 20.0 → 4.5

Time: 8.3s

Precision: binary64

\[[x, y, z] = \mathsf{sort}([x, y, z]) \\]

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ t_1 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-\left(y + z\right)\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{if}\;y \leq -4.0550781064396446 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.353237923677478 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, z, y \cdot x\right) + z \cdot x}\\ \mathbf{elif}\;y \leq 1.2873572104499456 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.748072096169368 \cdot 10^{-179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4758287118476183 \cdot 10^{+79}:\\ \;\;\;\;2 \cdot \sqrt{{\left(\sqrt{\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (* 0.25 (- (log (+ y x)) (log (/ 1.0 z))))) 2.0)))
        (t_1
         (*
          2.0
          (pow (exp (* 0.25 (- (log (- (+ y z))) (log (/ -1.0 x))))) 2.0))))
   (if (<= y -4.0550781064396446e+40)
     t_1
     (if (<= y -7.353237923677478e-180)
       (* 2.0 (sqrt (+ (fma y z (* y x)) (* z x))))
       (if (<= y 1.2873572104499456e-282)
         t_1
         (if (<= y 2.748072096169368e-179)
           t_0
           (if (<= y 1.4758287118476183e+79)
             (* 2.0 (sqrt (pow (sqrt (fma y z (* (+ y z) x))) 2.0)))
             t_0)))))))

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((0.25 * (log((y + x)) - log((1.0 / z))))), 2.0);
	double t_1 = 2.0 * pow(exp((0.25 * (log(-(y + z)) - log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -4.0550781064396446e+40) {
		tmp = t_1;
	} else if (y <= -7.353237923677478e-180) {
		tmp = 2.0 * sqrt((fma(y, z, (y * x)) + (z * x)));
	} else if (y <= 1.2873572104499456e-282) {
		tmp = t_1;
	} else if (y <= 2.748072096169368e-179) {
		tmp = t_0;
	} else if (y <= 1.4758287118476183e+79) {
		tmp = 2.0 * sqrt(pow(sqrt(fma(y, z, ((y + z) * x))), 2.0));
	} else {
		tmp = t_0;
	}
	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(0.25 * Float64(log(Float64(y + x)) - log(Float64(1.0 / z))))) ^ 2.0))
	t_1 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(-Float64(y + z))) - log(Float64(-1.0 / x))))) ^ 2.0))
	tmp = 0.0
	if (y <= -4.0550781064396446e+40)
		tmp = t_1;
	elseif (y <= -7.353237923677478e-180)
		tmp = Float64(2.0 * sqrt(Float64(fma(y, z, Float64(y * x)) + Float64(z * x))));
	elseif (y <= 1.2873572104499456e-282)
		tmp = t_1;
	elseif (y <= 2.748072096169368e-179)
		tmp = t_0;
	elseif (y <= 1.4758287118476183e+79)
		tmp = Float64(2.0 * sqrt((sqrt(fma(y, z, Float64(Float64(y + z) * x))) ^ 2.0)));
	else
		tmp = t_0;
	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[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[(-N[(y + z), $MachinePrecision])], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.0550781064396446e+40], t$95$1, If[LessEqual[y, -7.353237923677478e-180], N[(2.0 * N[Sqrt[N[(N[(y * z + N[(y * x), $MachinePrecision]), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2873572104499456e-282], t$95$1, If[LessEqual[y, 2.748072096169368e-179], t$95$0, If[LessEqual[y, 1.4758287118476183e+79], N[(2.0 * N[Sqrt[N[Power[N[Sqrt[N[(y * z + N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.0
Target	11.5
Herbie	4.5

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

Derivation

1. Split input into 4 regimes

2. if y < -4.0550781064396446e40 or -7.35323792367747765e-180 < y < 1.2873572104499456e-282

   1. Initial program 36.5

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

   2. Applied egg-rr 36.7

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

   3. Taylor expanded in x around -inf 7.1

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

   if -4.0550781064396446e40 < y < -7.35323792367747765e-180

   1. Initial program 1.1

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

   2. Applied egg-rr 1.1

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

   3. Applied egg-rr 1.1

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

   if 1.2873572104499456e-282 < y < 2.74807209616936804e-179 or 1.47582871184761831e79 < y

   1. Initial program 42.5

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

   2. Applied egg-rr 42.6

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

   3. Taylor expanded in z around inf 6.6

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

   if 2.74807209616936804e-179 < y < 1.47582871184761831e79

   1. Initial program 3.2

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

   2. Applied egg-rr 6.8

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

   3. Applied egg-rr 3.2

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

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.0550781064396446 \cdot 10^{+40}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-\left(y + z\right)\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -7.353237923677478 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, z, y \cdot x\right) + z \cdot x}\\ \mathbf{elif}\;y \leq 1.2873572104499456 \cdot 10^{-282}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-\left(y + z\right)\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 2.748072096169368 \cdot 10^{-179}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 1.4758287118476183 \cdot 10^{+79}:\\ \;\;\;\;2 \cdot \sqrt{{\left(\sqrt{\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))))