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

Average Error: 20.2 → 17.3

Time: 30.1s

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 r659620 = 2.0;
        double r659621 = x;
        double r659622 = sqrt(r659621);
        double r659623 = r659620 * r659622;
        double r659624 = y;
        double r659625 = z;
        double r659626 = t;
        double r659627 = r659625 * r659626;
        double r659628 = 3.0;
        double r659629 = r659627 / r659628;
        double r659630 = r659624 - r659629;
        double r659631 = cos(r659630);
        double r659632 = r659623 * r659631;
        double r659633 = a;
        double r659634 = b;
        double r659635 = r659634 * r659628;
        double r659636 = r659633 / r659635;
        double r659637 = r659632 - r659636;
        return r659637;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r659638 = z;
        double r659639 = t;
        double r659640 = r659638 * r659639;
        double r659641 = -8.109587613107832e+304;
        bool r659642 = r659640 <= r659641;
        double r659643 = 1.8522433847871854e+297;
        bool r659644 = r659640 <= r659643;
        double r659645 = !r659644;
        bool r659646 = r659642 || r659645;
        double r659647 = 1.0;
        double r659648 = 0.5;
        double r659649 = y;
        double r659650 = 2.0;
        double r659651 = pow(r659649, r659650);
        double r659652 = r659648 * r659651;
        double r659653 = r659647 - r659652;
        double r659654 = x;
        double r659655 = sqrt(r659654);
        double r659656 = 2.0;
        double r659657 = r659655 * r659656;
        double r659658 = r659653 * r659657;
        double r659659 = a;
        double r659660 = b;
        double r659661 = 3.0;
        double r659662 = r659660 * r659661;
        double r659663 = r659659 / r659662;
        double r659664 = r659658 - r659663;
        double r659665 = r659661 / r659639;
        double r659666 = r659638 / r659665;
        double r659667 = sin(r659666);
        double r659668 = sin(r659649);
        double r659669 = cbrt(r659668);
        double r659670 = r659669 * r659669;
        double r659671 = r659669 * r659655;
        double r659672 = r659656 * r659671;
        double r659673 = r659670 * r659672;
        double r659674 = r659667 * r659673;
        double r659675 = 0.3333333333333333;
        double r659676 = r659638 * r659675;
        double r659677 = r659676 * r659639;
        double r659678 = cos(r659677);
        double r659679 = 3.0;
        double r659680 = pow(r659678, r659679);
        double r659681 = cbrt(r659680);
        double r659682 = r659655 * r659681;
        double r659683 = cos(r659649);
        double r659684 = r659682 * r659683;
        double r659685 = r659684 * r659656;
        double r659686 = r659674 + r659685;
        double r659687 = r659686 - r659663;
        double r659688 = r659646 ? r659664 : r659687;
        return r659688;
}

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))))