Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, D

Average Error: 1.2 → 0.3

Time: 7.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

↓

\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\]

C

double f(double x, double y, double z, double t) {
        double r140361 = 1.0;
        double r140362 = 3.0;
        double r140363 = r140361 / r140362;
        double r140364 = x;
        double r140365 = y;
        double r140366 = 27.0;
        double r140367 = r140365 * r140366;
        double r140368 = r140364 / r140367;
        double r140369 = r140362 * r140368;
        double r140370 = z;
        double r140371 = 2.0;
        double r140372 = r140370 * r140371;
        double r140373 = r140369 / r140372;
        double r140374 = t;
        double r140375 = sqrt(r140374);
        double r140376 = r140373 * r140375;
        double r140377 = acos(r140376);
        double r140378 = r140363 * r140377;
        return r140378;
}

↓

double f(double x, double y, double z, double t) {
        double r140379 = 1.0;
        double r140380 = 3.0;
        double r140381 = cbrt(r140380);
        double r140382 = r140381 * r140381;
        double r140383 = r140379 / r140382;
        double r140384 = 1.0;
        double r140385 = r140384 / r140381;
        double r140386 = x;
        double r140387 = y;
        double r140388 = 27.0;
        double r140389 = r140387 * r140388;
        double r140390 = r140386 / r140389;
        double r140391 = r140380 * r140390;
        double r140392 = z;
        double r140393 = 2.0;
        double r140394 = r140392 * r140393;
        double r140395 = r140391 / r140394;
        double r140396 = t;
        double r140397 = sqrt(r140396);
        double r140398 = r140395 * r140397;
        double r140399 = acos(r140398);
        double r140400 = r140385 * r140399;
        double r140401 = r140383 * r140400;
        return r140401;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.2
Target	1.2
Herbie	0.3

\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

Derivation

1. Initial program 1.2

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
2. Using strategy rm

3. Applied add-cube-cbrt 1.2

   \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

4. Applied *-un-lft-identity 1.2

   \[\leadsto \frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

5. Applied times-frac 0.3

   \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{3}}\right)} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

6. Applied associate-*l* 0.3

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)}\]

7. Final simplification 0.3

   \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27) (* y z)) (/ (sqrt t) (/ 2 3)))) 3)

  (* (/ 1 3) (acos (* (/ (* 3 (/ x (* y 27))) (* z 2)) (sqrt t)))))