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

Average Error: 20.1 → 17.5

Time: 21.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999420662742917187415514490567147731781:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r486483 = 2.0;
        double r486484 = x;
        double r486485 = sqrt(r486484);
        double r486486 = r486483 * r486485;
        double r486487 = y;
        double r486488 = z;
        double r486489 = t;
        double r486490 = r486488 * r486489;
        double r486491 = 3.0;
        double r486492 = r486490 / r486491;
        double r486493 = r486487 - r486492;
        double r486494 = cos(r486493);
        double r486495 = r486486 * r486494;
        double r486496 = a;
        double r486497 = b;
        double r486498 = r486497 * r486491;
        double r486499 = r486496 / r486498;
        double r486500 = r486495 - r486499;
        return r486500;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r486501 = y;
        double r486502 = z;
        double r486503 = t;
        double r486504 = r486502 * r486503;
        double r486505 = 3.0;
        double r486506 = r486504 / r486505;
        double r486507 = r486501 - r486506;
        double r486508 = cos(r486507);
        double r486509 = 0.9999420662742917;
        bool r486510 = r486508 <= r486509;
        double r486511 = 2.0;
        double r486512 = x;
        double r486513 = sqrt(r486512);
        double r486514 = r486511 * r486513;
        double r486515 = cos(r486501);
        double r486516 = cos(r486506);
        double r486517 = cbrt(r486516);
        double r486518 = r486517 * r486517;
        double r486519 = r486518 * r486517;
        double r486520 = r486515 * r486519;
        double r486521 = r486514 * r486520;
        double r486522 = sin(r486501);
        double r486523 = cbrt(r486522);
        double r486524 = r486523 * r486523;
        double r486525 = sin(r486506);
        double r486526 = r486523 * r486525;
        double r486527 = r486524 * r486526;
        double r486528 = r486514 * r486527;
        double r486529 = r486521 + r486528;
        double r486530 = a;
        double r486531 = b;
        double r486532 = r486531 * r486505;
        double r486533 = r486530 / r486532;
        double r486534 = r486529 - r486533;
        double r486535 = 1.0;
        double r486536 = 0.5;
        double r486537 = 2.0;
        double r486538 = pow(r486501, r486537);
        double r486539 = r486536 * r486538;
        double r486540 = r486535 - r486539;
        double r486541 = r486514 * r486540;
        double r486542 = r486541 - r486533;
        double r486543 = r486510 ? r486534 : r486542;
        return r486543;
}

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	20.1
Target	17.9
Herbie	17.5

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999420662742917

   1. Initial program 19.3

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

   3. Applied cos-diff 18.7

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

   4. Applied distribute-lft-in 18.7

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

   6. Applied add-cube-cbrt 18.7

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

   7. Applied associate-*l* 18.7

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

   9. Applied add-cube-cbrt 18.7

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

   if 0.9999420662742917 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.4

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

   2. Taylor expanded around 0 15.5

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

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999420662742917187415514490567147731781:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(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))))