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

Average Error: 20.5 → 18.1

Time: 11.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9981936348282990367764000438910443335772:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r905440 = 2.0;
        double r905441 = x;
        double r905442 = sqrt(r905441);
        double r905443 = r905440 * r905442;
        double r905444 = y;
        double r905445 = z;
        double r905446 = t;
        double r905447 = r905445 * r905446;
        double r905448 = 3.0;
        double r905449 = r905447 / r905448;
        double r905450 = r905444 - r905449;
        double r905451 = cos(r905450);
        double r905452 = r905443 * r905451;
        double r905453 = a;
        double r905454 = b;
        double r905455 = r905454 * r905448;
        double r905456 = r905453 / r905455;
        double r905457 = r905452 - r905456;
        return r905457;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r905458 = y;
        double r905459 = z;
        double r905460 = t;
        double r905461 = r905459 * r905460;
        double r905462 = 3.0;
        double r905463 = r905461 / r905462;
        double r905464 = r905458 - r905463;
        double r905465 = cos(r905464);
        double r905466 = 0.998193634828299;
        bool r905467 = r905465 <= r905466;
        double r905468 = 2.0;
        double r905469 = x;
        double r905470 = sqrt(r905469);
        double r905471 = r905468 * r905470;
        double r905472 = cos(r905458);
        double r905473 = cos(r905463);
        double r905474 = r905472 * r905473;
        double r905475 = r905471 * r905474;
        double r905476 = sin(r905458);
        double r905477 = 0.3333333333333333;
        double r905478 = r905460 * r905459;
        double r905479 = r905477 * r905478;
        double r905480 = sin(r905479);
        double r905481 = r905476 * r905480;
        double r905482 = r905471 * r905481;
        double r905483 = cbrt(r905482);
        double r905484 = r905483 * r905483;
        double r905485 = r905484 * r905483;
        double r905486 = r905475 + r905485;
        double r905487 = a;
        double r905488 = b;
        double r905489 = r905488 * r905462;
        double r905490 = r905487 / r905489;
        double r905491 = r905486 - r905490;
        double r905492 = 1.0;
        double r905493 = 0.5;
        double r905494 = 2.0;
        double r905495 = pow(r905458, r905494);
        double r905496 = r905493 * r905495;
        double r905497 = r905492 - r905496;
        double r905498 = r905471 * r905497;
        double r905499 = r905498 - r905490;
        double r905500 = r905467 ? r905491 : r905499;
        return r905500;
}

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	20.5
Target	18.4
Herbie	18.1

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.998193634828299

   1. Initial program 19.8

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

   3. Applied cos-diff 19.2

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

   4. Applied distribute-lft-in 19.2

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

   5. Taylor expanded around inf 19.2

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) - \frac{a}{b \cdot 3}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 19.2

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \color{blue}{\left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}}\right) - \frac{a}{b \cdot 3}\]

   if 0.998193634828299 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.6

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

   2. Taylor expanded around 0 16.5

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9981936348282990367764000438910443335772:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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))))