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

Average Error: 20.2 → 17.3

Time: 30.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -8.109587613107832301864398073451134304215 \cdot 10^{304} \lor \neg \left(z \cdot t \le 1.852243384787185415512611130974845637804 \cdot 10^{297}\right):\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot {y}^{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{z}{\frac{3}{t}}\right) \cdot \left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \left(2 \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt{x}\right)\right)\right) + \left(\left(\sqrt{x} \cdot \sqrt[3]{{\left(\cos \left(\left(z \cdot 0.3333333333333333148296162562473909929395\right) \cdot t\right)\right)}^{3}}\right) \cdot \cos y\right) \cdot 2\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r675602 = 2.0;
        double r675603 = x;
        double r675604 = sqrt(r675603);
        double r675605 = r675602 * r675604;
        double r675606 = y;
        double r675607 = z;
        double r675608 = t;
        double r675609 = r675607 * r675608;
        double r675610 = 3.0;
        double r675611 = r675609 / r675610;
        double r675612 = r675606 - r675611;
        double r675613 = cos(r675612);
        double r675614 = r675605 * r675613;
        double r675615 = a;
        double r675616 = b;
        double r675617 = r675616 * r675610;
        double r675618 = r675615 / r675617;
        double r675619 = r675614 - r675618;
        return r675619;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r675620 = z;
        double r675621 = t;
        double r675622 = r675620 * r675621;
        double r675623 = -8.109587613107832e+304;
        bool r675624 = r675622 <= r675623;
        double r675625 = 1.8522433847871854e+297;
        bool r675626 = r675622 <= r675625;
        double r675627 = !r675626;
        bool r675628 = r675624 || r675627;
        double r675629 = 1.0;
        double r675630 = 0.5;
        double r675631 = y;
        double r675632 = 2.0;
        double r675633 = pow(r675631, r675632);
        double r675634 = r675630 * r675633;
        double r675635 = r675629 - r675634;
        double r675636 = x;
        double r675637 = sqrt(r675636);
        double r675638 = 2.0;
        double r675639 = r675637 * r675638;
        double r675640 = r675635 * r675639;
        double r675641 = a;
        double r675642 = b;
        double r675643 = 3.0;
        double r675644 = r675642 * r675643;
        double r675645 = r675641 / r675644;
        double r675646 = r675640 - r675645;
        double r675647 = r675643 / r675621;
        double r675648 = r675620 / r675647;
        double r675649 = sin(r675648);
        double r675650 = sin(r675631);
        double r675651 = cbrt(r675650);
        double r675652 = r675651 * r675651;
        double r675653 = r675651 * r675637;
        double r675654 = r675638 * r675653;
        double r675655 = r675652 * r675654;
        double r675656 = r675649 * r675655;
        double r675657 = 0.3333333333333333;
        double r675658 = r675620 * r675657;
        double r675659 = r675658 * r675621;
        double r675660 = cos(r675659);
        double r675661 = 3.0;
        double r675662 = pow(r675660, r675661);
        double r675663 = cbrt(r675662);
        double r675664 = r675637 * r675663;
        double r675665 = cos(r675631);
        double r675666 = r675664 * r675665;
        double r675667 = r675666 * r675638;
        double r675668 = r675656 + r675667;
        double r675669 = r675668 - r675645;
        double r675670 = r675628 ? r675646 : r675669;
        return r675670;
}

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	20.2
Target	18.2
Herbie	17.3

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

Derivation

1. Split input into 2 regimes

2. if (* z t) < -8.109587613107832e+304 or 1.8522433847871854e+297 < (* z t)

   1. Initial program 63.1

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

   2. Taylor expanded around 0 43.5

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

   3. Simplified 43.5

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

   if -8.109587613107832e+304 < (* z t) < 1.8522433847871854e+297

   1. Initial program 14.1

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

   3. Applied cos-diff 13.6

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

   4. Applied distribute-lft-in 13.6

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

   5. Simplified 13.6

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

   6. Simplified 13.6

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

   7. Taylor expanded around inf 13.6

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

   8. Simplified 13.6

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

   10. Applied add-cube-cbrt 13.6

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

   11. Applied associate-*l* 13.6

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

   12. Simplified 13.6

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

   14. Applied add-cbrt-cube 13.6

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

   15. Simplified 13.6

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

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -8.109587613107832301864398073451134304215 \cdot 10^{304} \lor \neg \left(z \cdot t \le 1.852243384787185415512611130974845637804 \cdot 10^{297}\right):\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot {y}^{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{z}{\frac{3}{t}}\right) \cdot \left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \left(2 \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt{x}\right)\right)\right) + \left(\left(\sqrt{x} \cdot \sqrt[3]{{\left(\cos \left(\left(z \cdot 0.3333333333333333148296162562473909929395\right) \cdot t\right)\right)}^{3}}\right) \cdot \cos y\right) \cdot 2\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))