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

Average Error: 20.6 → 18.3

Time: 22.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9988949060685164971218341634084936231375:\\ \;\;\;\;\frac{\left({\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \cos y\right)}^{3} + {\left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)}^{3}\right) \cdot \left(2 \cdot \sqrt{x}\right)}{\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \cos y\right) \cdot \left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \cos y\right) + \left(\left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \cos y\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right)} - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r484458 = 2.0;
        double r484459 = x;
        double r484460 = sqrt(r484459);
        double r484461 = r484458 * r484460;
        double r484462 = y;
        double r484463 = z;
        double r484464 = t;
        double r484465 = r484463 * r484464;
        double r484466 = 3.0;
        double r484467 = r484465 / r484466;
        double r484468 = r484462 - r484467;
        double r484469 = cos(r484468);
        double r484470 = r484461 * r484469;
        double r484471 = a;
        double r484472 = b;
        double r484473 = r484472 * r484466;
        double r484474 = r484471 / r484473;
        double r484475 = r484470 - r484474;
        return r484475;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r484476 = y;
        double r484477 = z;
        double r484478 = t;
        double r484479 = r484477 * r484478;
        double r484480 = 3.0;
        double r484481 = r484479 / r484480;
        double r484482 = r484476 - r484481;
        double r484483 = cos(r484482);
        double r484484 = 0.9988949060685165;
        bool r484485 = r484483 <= r484484;
        double r484486 = 0.3333333333333333;
        double r484487 = r484478 * r484477;
        double r484488 = r484486 * r484487;
        double r484489 = cos(r484488);
        double r484490 = cos(r484476);
        double r484491 = r484489 * r484490;
        double r484492 = 3.0;
        double r484493 = pow(r484491, r484492);
        double r484494 = sin(r484476);
        double r484495 = sin(r484481);
        double r484496 = r484494 * r484495;
        double r484497 = pow(r484496, r484492);
        double r484498 = r484493 + r484497;
        double r484499 = 2.0;
        double r484500 = x;
        double r484501 = sqrt(r484500);
        double r484502 = r484499 * r484501;
        double r484503 = r484498 * r484502;
        double r484504 = r484491 * r484491;
        double r484505 = r484496 * r484496;
        double r484506 = sin(r484488);
        double r484507 = r484494 * r484506;
        double r484508 = r484491 * r484507;
        double r484509 = r484505 - r484508;
        double r484510 = r484504 + r484509;
        double r484511 = r484503 / r484510;
        double r484512 = a;
        double r484513 = b;
        double r484514 = r484513 * r484480;
        double r484515 = r484512 / r484514;
        double r484516 = r484511 - r484515;
        double r484517 = 1.0;
        double r484518 = 0.5;
        double r484519 = 2.0;
        double r484520 = pow(r484476, r484519);
        double r484521 = r484518 * r484520;
        double r484522 = r484517 - r484521;
        double r484523 = r484502 * r484522;
        double r484524 = r484523 - r484515;
        double r484525 = r484485 ? r484516 : r484524;
        return r484525;
}

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	20.6
Target	19.0
Herbie	18.3

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9988949060685165

   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 19.7

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

   4. Simplified 19.7

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

   5. Taylor expanded around inf 19.7

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

   7. Applied flip3-+ 19.7

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

   8. Applied associate-*r/ 19.7

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

   9. Simplified 19.7

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

   10. Taylor expanded around inf 19.7

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

   if 0.9988949060685165 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.1

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

   2. Taylor expanded around 0 16.1

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

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9988949060685164971218341634084936231375:\\ \;\;\;\;\frac{\left({\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \cos y\right)}^{3} + {\left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)}^{3}\right) \cdot \left(2 \cdot \sqrt{x}\right)}{\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \cos y\right) \cdot \left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \cos y\right) + \left(\left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \cos y\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right)} - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.333333333333333315 z) t)))) (/ (/ a 3) b)) (if (< z 3.51629061355598715e106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.333333333333333315 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

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