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

Average Error: 20.8 → 17.9

Time: 15.5s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999481199109:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r97479 = 2.0;
        double r97480 = x;
        double r97481 = sqrt(r97480);
        double r97482 = r97479 * r97481;
        double r97483 = y;
        double r97484 = z;
        double r97485 = t;
        double r97486 = r97484 * r97485;
        double r97487 = 3.0;
        double r97488 = r97486 / r97487;
        double r97489 = r97483 - r97488;
        double r97490 = cos(r97489);
        double r97491 = r97482 * r97490;
        double r97492 = a;
        double r97493 = b;
        double r97494 = r97493 * r97487;
        double r97495 = r97492 / r97494;
        double r97496 = r97491 - r97495;
        return r97496;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r97497 = y;
        double r97498 = z;
        double r97499 = t;
        double r97500 = r97498 * r97499;
        double r97501 = 3.0;
        double r97502 = r97500 / r97501;
        double r97503 = r97497 - r97502;
        double r97504 = cos(r97503);
        double r97505 = 0.9999999994811991;
        bool r97506 = r97504 <= r97505;
        double r97507 = 2.0;
        double r97508 = x;
        double r97509 = sqrt(r97508);
        double r97510 = r97507 * r97509;
        double r97511 = cos(r97497);
        double r97512 = cos(r97502);
        double r97513 = 3.0;
        double r97514 = pow(r97512, r97513);
        double r97515 = cbrt(r97514);
        double r97516 = cbrt(r97515);
        double r97517 = r97516 * r97516;
        double r97518 = cbrt(r97512);
        double r97519 = r97517 * r97518;
        double r97520 = r97511 * r97519;
        double r97521 = sin(r97497);
        double r97522 = sin(r97502);
        double r97523 = r97521 * r97522;
        double r97524 = r97520 + r97523;
        double r97525 = r97510 * r97524;
        double r97526 = a;
        double r97527 = b;
        double r97528 = r97527 * r97501;
        double r97529 = r97526 / r97528;
        double r97530 = r97525 - r97529;
        double r97531 = 1.0;
        double r97532 = 0.5;
        double r97533 = 2.0;
        double r97534 = pow(r97497, r97533);
        double r97535 = r97532 * r97534;
        double r97536 = r97531 - r97535;
        double r97537 = r97510 * r97536;
        double r97538 = r97537 - r97529;
        double r97539 = r97506 ? r97530 : r97538;
        return r97539;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	20.8
Target	18.6
Herbie	17.9

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999994811991

   1. Initial program 20.3

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

   3. Applied cos-diff 19.5

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 19.5

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)} + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\]
   6. Using strategy rm

   7. Applied add-cbrt-cube 19.5

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\color{blue}{\sqrt[3]{\left(\cos \left(\frac{z \cdot t}{3}\right) \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \cos \left(\frac{z \cdot t}{3}\right)}}}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\]

   8. Simplified 19.5

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sqrt[3]{\color{blue}{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\]
   9. Using strategy rm

   10. Applied add-cbrt-cube 19.5

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

   11. Simplified 19.5

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

   if 0.9999999994811991 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.7

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

   2. Taylor expanded around 0 15.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]
3. Recombined 2 regimes into one program.

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999481199109:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

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

  :herbie-target
  (if (< z -1.379333748723514e+129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))