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

Average Error: 19.7 → 4.6

Time: 5.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ t_1 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ t_2 := z \cdot \left(y + x\right)\\ \mathbf{if}\;y \leq -3.0121278070499345 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.3866893327893407 \cdot 10^{-224}:\\ \;\;\;\;2 \cdot \left|\sqrt{\mathsf{fma}\left(x, y, {\left(\sqrt[3]{t_2}\right)}^{3}\right)}\right|\\ \mathbf{elif}\;y \leq 5.248403633372559 \cdot 10^{-266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1764325343930836 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.965577108647369 \cdot 10^{+21}:\\ \;\;\;\;2 \cdot \left|\sqrt{t_2}\right|\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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) z)) (log (/ -1.0 x))))) 2.0)))
        (t_1
         (* 2.0 (pow (exp (* 0.25 (- (log (+ y x)) (log (/ 1.0 z))))) 2.0)))
        (t_2 (* z (+ y x))))
   (if (<= y -3.0121278070499345e+44)
     t_0
     (if (<= y -1.3866893327893407e-224)
       (* 2.0 (fabs (sqrt (fma x y (pow (cbrt t_2) 3.0)))))
       (if (<= y 5.248403633372559e-266)
         t_0
         (if (<= y 1.1764325343930836e-207)
           t_1
           (if (<= y 6.965577108647369e+21)
             (* 2.0 (fabs (sqrt t_2)))
             t_1)))))))

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 - z)) - log((-1.0 / x))))), 2.0);
	double t_1 = 2.0 * pow(exp((0.25 * (log((y + x)) - log((1.0 / z))))), 2.0);
	double t_2 = z * (y + x);
	double tmp;
	if (y <= -3.0121278070499345e+44) {
		tmp = t_0;
	} else if (y <= -1.3866893327893407e-224) {
		tmp = 2.0 * fabs(sqrt(fma(x, y, pow(cbrt(t_2), 3.0))));
	} else if (y <= 5.248403633372559e-266) {
		tmp = t_0;
	} else if (y <= 1.1764325343930836e-207) {
		tmp = t_1;
	} else if (y <= 6.965577108647369e+21) {
		tmp = 2.0 * fabs(sqrt(t_2));
	} else {
		tmp = t_1;
	}
	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(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0))
	t_1 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(y + x)) - log(Float64(1.0 / z))))) ^ 2.0))
	t_2 = Float64(z * Float64(y + x))
	tmp = 0.0
	if (y <= -3.0121278070499345e+44)
		tmp = t_0;
	elseif (y <= -1.3866893327893407e-224)
		tmp = Float64(2.0 * abs(sqrt(fma(x, y, (cbrt(t_2) ^ 3.0)))));
	elseif (y <= 5.248403633372559e-266)
		tmp = t_0;
	elseif (y <= 1.1764325343930836e-207)
		tmp = t_1;
	elseif (y <= 6.965577108647369e+21)
		tmp = Float64(2.0 * abs(sqrt(t_2)));
	else
		tmp = t_1;
	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) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $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 + x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.0121278070499345e+44], t$95$0, If[LessEqual[y, -1.3866893327893407e-224], N[(2.0 * N[Abs[N[Sqrt[N[(x * y + N[Power[N[Power[t$95$2, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.248403633372559e-266], t$95$0, If[LessEqual[y, 1.1764325343930836e-207], t$95$1, If[LessEqual[y, 6.965577108647369e+21], N[(2.0 * N[Abs[N[Sqrt[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.7
Target	11.6
Herbie	4.6

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

Derivation

1. Split input into 4 regimes

2. if y < -3.0121278070499345e44 or -1.3866893327893407e-224 < y < 5.2484036333725592e-266

   1. Initial program 41.2

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

   2. Simplified 41.2

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

   3. Applied egg-rr 41.4

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

   4. Taylor expanded in x around -inf 8.5

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

   if -3.0121278070499345e44 < y < -1.3866893327893407e-224

   1. Initial program 2.6

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

   2. Simplified 2.6

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

   3. Applied egg-rr 3.0

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

   4. Applied egg-rr 2.6

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

   5. Applied egg-rr 2.6

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

   if 5.2484036333725592e-266 < y < 1.17643253439308359e-207 or 6965577108647368980000 < y

   1. Initial program 37.6

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

   2. Simplified 37.6

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

   3. Applied egg-rr 37.7

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

   4. Taylor expanded in z around inf 6.5

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

   if 1.17643253439308359e-207 < y < 6965577108647368980000

   1. Initial program 1.1

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

   2. Simplified 1.1

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

   3. Applied egg-rr 1.5

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

   4. Applied egg-rr 1.1

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

   5. Taylor expanded in z around inf 1.1

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

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.0121278070499345 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -1.3866893327893407 \cdot 10^{-224}:\\ \;\;\;\;2 \cdot \left|\sqrt{\mathsf{fma}\left(x, y, {\left(\sqrt[3]{z \cdot \left(y + x\right)}\right)}^{3}\right)}\right|\\ \mathbf{elif}\;y \leq 5.248403633372559 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 1.1764325343930836 \cdot 10^{-207}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 6.965577108647369 \cdot 10^{+21}:\\ \;\;\;\;2 \cdot \left|\sqrt{z \cdot \left(y + x\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))))