Average Error: 20.8 → 19.4
Time: 29.7s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3390462636455431410687387521523908608:\\ \;\;\;\;\left(\cos \left(\frac{t}{3} \cdot z\right) \cdot \left(\cos y \cdot \left(\sqrt{x} \cdot 2\right)\right) + 2 \cdot \left(\sqrt{x} \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right)\right)\right) - a \cdot \frac{0.3333333333333333148296162562473909929395}{b}\\ \mathbf{elif}\;y \le 3.194138595283726197995626607277139252794 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]
\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}\;y \le -3390462636455431410687387521523908608:\\
\;\;\;\;\left(\cos \left(\frac{t}{3} \cdot z\right) \cdot \left(\cos y \cdot \left(\sqrt{x} \cdot 2\right)\right) + 2 \cdot \left(\sqrt{x} \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right)\right)\right) - a \cdot \frac{0.3333333333333333148296162562473909929395}{b}\\

\mathbf{elif}\;y \le 3.194138595283726197995626607277139252794 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right) - \frac{a}{3 \cdot b}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r465450 = 2.0;
        double r465451 = x;
        double r465452 = sqrt(r465451);
        double r465453 = r465450 * r465452;
        double r465454 = y;
        double r465455 = z;
        double r465456 = t;
        double r465457 = r465455 * r465456;
        double r465458 = 3.0;
        double r465459 = r465457 / r465458;
        double r465460 = r465454 - r465459;
        double r465461 = cos(r465460);
        double r465462 = r465453 * r465461;
        double r465463 = a;
        double r465464 = b;
        double r465465 = r465464 * r465458;
        double r465466 = r465463 / r465465;
        double r465467 = r465462 - r465466;
        return r465467;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r465468 = y;
        double r465469 = -3.3904626364554314e+36;
        bool r465470 = r465468 <= r465469;
        double r465471 = t;
        double r465472 = 3.0;
        double r465473 = r465471 / r465472;
        double r465474 = z;
        double r465475 = r465473 * r465474;
        double r465476 = cos(r465475);
        double r465477 = cos(r465468);
        double r465478 = x;
        double r465479 = sqrt(r465478);
        double r465480 = 2.0;
        double r465481 = r465479 * r465480;
        double r465482 = r465477 * r465481;
        double r465483 = r465476 * r465482;
        double r465484 = r465471 * r465474;
        double r465485 = r465484 / r465472;
        double r465486 = sin(r465485);
        double r465487 = sin(r465468);
        double r465488 = r465486 * r465487;
        double r465489 = r465479 * r465488;
        double r465490 = r465480 * r465489;
        double r465491 = r465483 + r465490;
        double r465492 = a;
        double r465493 = 0.3333333333333333;
        double r465494 = b;
        double r465495 = r465493 / r465494;
        double r465496 = r465492 * r465495;
        double r465497 = r465491 - r465496;
        double r465498 = 3.194138595283726e-09;
        bool r465499 = r465468 <= r465498;
        double r465500 = -0.5;
        double r465501 = r465468 * r465468;
        double r465502 = 1.0;
        double r465503 = fma(r465500, r465501, r465502);
        double r465504 = r465503 * r465481;
        double r465505 = r465472 * r465494;
        double r465506 = r465492 / r465505;
        double r465507 = r465504 - r465506;
        double r465508 = r465502 / r465472;
        double r465509 = r465508 * r465484;
        double r465510 = r465468 - r465509;
        double r465511 = cos(r465510);
        double r465512 = r465481 * r465511;
        double r465513 = r465512 - r465506;
        double r465514 = r465499 ? r465507 : r465513;
        double r465515 = r465470 ? r465497 : r465514;
        return r465515;
}

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

Original20.8
Target18.7
Herbie19.4
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -3.3904626364554314e+36

    1. Initial program 21.2

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

    3. Applied div-inv21.2

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

    4. Simplified21.2

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

    5. Taylor expanded around 0 21.2

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

    7. Applied cos-diff20.4

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

    8. Applied distribute-lft-in20.4

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

    9. Simplified20.3

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

    10. Simplified20.3

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

    if -3.3904626364554314e+36 < y < 3.194138595283726e-09

    1. Initial program 20.8

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

    2. Taylor expanded around 0 18.5

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

    3. Simplified18.5

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

    if 3.194138595283726e-09 < y

    1. Initial program 20.6

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

    3. Applied div-inv20.6

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

  4. Final simplification19.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3390462636455431410687387521523908608:\\ \;\;\;\;\left(\cos \left(\frac{t}{3} \cdot z\right) \cdot \left(\cos y \cdot \left(\sqrt{x} \cdot 2\right)\right) + 2 \cdot \left(\sqrt{x} \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin y\right)\right)\right) - a \cdot \frac{0.3333333333333333148296162562473909929395}{b}\\ \mathbf{elif}\;y \le 3.194138595283726197995626607277139252794 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))))