Average Error: 20.2 → 17.7
Time: 32.0s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.941728260032480555982451438125355174003 \cdot 10^{102} \lor \neg \left(z \le 8.906603696220313119083671611347589532707 \cdot 10^{80}\right):\\ \;\;\;\;\mathsf{fma}\left(1, \sqrt{x} \cdot 2, -\frac{\frac{a}{b}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y \cdot \cos \left(z \cdot \left(0.3333333333333333148296162562473909929395 \cdot t\right)\right) - \left(-\sin \left(\frac{t}{\frac{3}{z}}\right) \cdot \sin y\right), \sqrt{x} \cdot 2, -\frac{\frac{a}{3}}{b}\right)\\ \end{array}\]
\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 \le -6.941728260032480555982451438125355174003 \cdot 10^{102} \lor \neg \left(z \le 8.906603696220313119083671611347589532707 \cdot 10^{80}\right):\\
\;\;\;\;\mathsf{fma}\left(1, \sqrt{x} \cdot 2, -\frac{\frac{a}{b}}{3}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\cos y \cdot \cos \left(z \cdot \left(0.3333333333333333148296162562473909929395 \cdot t\right)\right) - \left(-\sin \left(\frac{t}{\frac{3}{z}}\right) \cdot \sin y\right), \sqrt{x} \cdot 2, -\frac{\frac{a}{3}}{b}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r1468589 = 2.0;
        double r1468590 = x;
        double r1468591 = sqrt(r1468590);
        double r1468592 = r1468589 * r1468591;
        double r1468593 = y;
        double r1468594 = z;
        double r1468595 = t;
        double r1468596 = r1468594 * r1468595;
        double r1468597 = 3.0;
        double r1468598 = r1468596 / r1468597;
        double r1468599 = r1468593 - r1468598;
        double r1468600 = cos(r1468599);
        double r1468601 = r1468592 * r1468600;
        double r1468602 = a;
        double r1468603 = b;
        double r1468604 = r1468603 * r1468597;
        double r1468605 = r1468602 / r1468604;
        double r1468606 = r1468601 - r1468605;
        return r1468606;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r1468607 = z;
        double r1468608 = -6.941728260032481e+102;
        bool r1468609 = r1468607 <= r1468608;
        double r1468610 = 8.906603696220313e+80;
        bool r1468611 = r1468607 <= r1468610;
        double r1468612 = !r1468611;
        bool r1468613 = r1468609 || r1468612;
        double r1468614 = 1.0;
        double r1468615 = x;
        double r1468616 = sqrt(r1468615);
        double r1468617 = 2.0;
        double r1468618 = r1468616 * r1468617;
        double r1468619 = a;
        double r1468620 = b;
        double r1468621 = r1468619 / r1468620;
        double r1468622 = 3.0;
        double r1468623 = r1468621 / r1468622;
        double r1468624 = -r1468623;
        double r1468625 = fma(r1468614, r1468618, r1468624);
        double r1468626 = y;
        double r1468627 = cos(r1468626);
        double r1468628 = 0.3333333333333333;
        double r1468629 = t;
        double r1468630 = r1468628 * r1468629;
        double r1468631 = r1468607 * r1468630;
        double r1468632 = cos(r1468631);
        double r1468633 = r1468627 * r1468632;
        double r1468634 = r1468622 / r1468607;
        double r1468635 = r1468629 / r1468634;
        double r1468636 = sin(r1468635);
        double r1468637 = sin(r1468626);
        double r1468638 = r1468636 * r1468637;
        double r1468639 = -r1468638;
        double r1468640 = r1468633 - r1468639;
        double r1468641 = r1468619 / r1468622;
        double r1468642 = r1468641 / r1468620;
        double r1468643 = -r1468642;
        double r1468644 = fma(r1468640, r1468618, r1468643);
        double r1468645 = r1468613 ? r1468625 : r1468644;
        return r1468645;
}

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

Original20.2
Target18.2
Herbie17.7
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.941728260032481e+102 or 8.906603696220313e+80 < z

    1. Initial program 37.0

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

    2. Simplified36.9

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

    3. Taylor expanded around 0 30.5

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

    if -6.941728260032481e+102 < z < 8.906603696220313e+80

    1. Initial program 11.2

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

    2. Simplified11.3

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

    4. Applied fma-udef11.3

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

    5. Applied cos-sum10.8

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

    6. Simplified10.8

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

    7. Simplified10.8

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

    8. Taylor expanded around inf 10.8

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

    9. Simplified10.8

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

    11. Applied neg-sub010.8

      \[\leadsto \mathsf{fma}\left(\cos \left(\left(t \cdot 0.3333333333333333148296162562473909929395\right) \cdot z\right) \cdot \cos y - \sin y \cdot \left(-\sin \left(\frac{t \cdot z}{3}\right)\right), 2 \cdot \sqrt{x}, \frac{\color{blue}{0 - \frac{a}{b}}}{3}\right)\]

    12. Applied div-sub10.8

      \[\leadsto \mathsf{fma}\left(\cos \left(\left(t \cdot 0.3333333333333333148296162562473909929395\right) \cdot z\right) \cdot \cos y - \sin y \cdot \left(-\sin \left(\frac{t \cdot z}{3}\right)\right), 2 \cdot \sqrt{x}, \color{blue}{\frac{0}{3} - \frac{\frac{a}{b}}{3}}\right)\]

    13. Simplified10.8

      \[\leadsto \mathsf{fma}\left(\cos \left(\left(t \cdot 0.3333333333333333148296162562473909929395\right) \cdot z\right) \cdot \cos y - \sin y \cdot \left(-\sin \left(\frac{t \cdot z}{3}\right)\right), 2 \cdot \sqrt{x}, \color{blue}{0} - \frac{\frac{a}{b}}{3}\right)\]

    14. Simplified10.8

      \[\leadsto \mathsf{fma}\left(\cos \left(\left(t \cdot 0.3333333333333333148296162562473909929395\right) \cdot z\right) \cdot \cos y - \sin y \cdot \left(-\sin \left(\frac{t \cdot z}{3}\right)\right), 2 \cdot \sqrt{x}, 0 - \color{blue}{\frac{\frac{a}{3}}{b}}\right)\]
    15. Using strategy rm

    16. Applied associate-/l*10.8

      \[\leadsto \mathsf{fma}\left(\cos \left(\left(t \cdot 0.3333333333333333148296162562473909929395\right) \cdot z\right) \cdot \cos y - \sin y \cdot \left(-\sin \color{blue}{\left(\frac{t}{\frac{3}{z}}\right)}\right), 2 \cdot \sqrt{x}, 0 - \frac{\frac{a}{3}}{b}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.941728260032480555982451438125355174003 \cdot 10^{102} \lor \neg \left(z \le 8.906603696220313119083671611347589532707 \cdot 10^{80}\right):\\ \;\;\;\;\mathsf{fma}\left(1, \sqrt{x} \cdot 2, -\frac{\frac{a}{b}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y \cdot \cos \left(z \cdot \left(0.3333333333333333148296162562473909929395 \cdot t\right)\right) - \left(-\sin \left(\frac{t}{\frac{3}{z}}\right) \cdot \sin y\right), \sqrt{x} \cdot 2, -\frac{\frac{a}{3}}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(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))))