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

Average Error: 20.8 → 17.9

Time: 11.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 5.7436459452120013 \cdot 10^{148}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r717613 = 2.0;
        double r717614 = x;
        double r717615 = sqrt(r717614);
        double r717616 = r717613 * r717615;
        double r717617 = y;
        double r717618 = z;
        double r717619 = t;
        double r717620 = r717618 * r717619;
        double r717621 = 3.0;
        double r717622 = r717620 / r717621;
        double r717623 = r717617 - r717622;
        double r717624 = cos(r717623);
        double r717625 = r717616 * r717624;
        double r717626 = a;
        double r717627 = b;
        double r717628 = r717627 * r717621;
        double r717629 = r717626 / r717628;
        double r717630 = r717625 - r717629;
        return r717630;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r717631 = 2.0;
        double r717632 = x;
        double r717633 = sqrt(r717632);
        double r717634 = r717631 * r717633;
        double r717635 = y;
        double r717636 = z;
        double r717637 = t;
        double r717638 = r717636 * r717637;
        double r717639 = 3.0;
        double r717640 = r717638 / r717639;
        double r717641 = r717635 - r717640;
        double r717642 = cos(r717641);
        double r717643 = r717634 * r717642;
        double r717644 = 5.743645945212001e+148;
        bool r717645 = r717643 <= r717644;
        double r717646 = cos(r717635);
        double r717647 = 0.3333333333333333;
        double r717648 = r717637 * r717636;
        double r717649 = r717647 * r717648;
        double r717650 = cos(r717649);
        double r717651 = cbrt(r717650);
        double r717652 = r717651 * r717651;
        double r717653 = r717652 * r717651;
        double r717654 = r717646 * r717653;
        double r717655 = r717634 * r717654;
        double r717656 = sin(r717635);
        double r717657 = sin(r717640);
        double r717658 = cbrt(r717657);
        double r717659 = r717658 * r717658;
        double r717660 = r717659 * r717658;
        double r717661 = r717656 * r717660;
        double r717662 = r717634 * r717661;
        double r717663 = r717655 + r717662;
        double r717664 = a;
        double r717665 = b;
        double r717666 = r717665 * r717639;
        double r717667 = r717664 / r717666;
        double r717668 = r717663 - r717667;
        double r717669 = 1.0;
        double r717670 = 0.5;
        double r717671 = 2.0;
        double r717672 = pow(r717635, r717671);
        double r717673 = r717670 * r717672;
        double r717674 = r717669 - r717673;
        double r717675 = r717634 * r717674;
        double r717676 = r717675 - r717667;
        double r717677 = r717645 ? r717668 : r717676;
        return r717677;
}

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	20.8
Target	18.7
Herbie	17.9

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) < 5.743645945212001e+148

   1. Initial program 14.4

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

   3. Applied cos-diff 13.8

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

   4. Applied distribute-lft-in 13.8

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

   5. Taylor expanded around inf 13.8

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 13.8

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 13.8

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right)}\right)\right) - \frac{a}{b \cdot 3}\]

   if 5.743645945212001e+148 < (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))

   1. Initial program 60.7

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

   2. Taylor expanded around 0 43.7

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 5.7436459452120013 \cdot 10^{148}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(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))))