Average Error: 20.3 → 18.3
Time: 30.7s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -4.557684899971343275948588241655457690987 \cdot 10^{175}:\\ \;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;z \cdot t \le 2.650669621479893038998696276370616139907 \cdot 10^{306}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) \cdot \cos y + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \end{array}\]
\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}\;z \cdot t \le -4.557684899971343275948588241655457690987 \cdot 10^{175}:\\
\;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\

\mathbf{elif}\;z \cdot t \le 2.650669621479893038998696276370616139907 \cdot 10^{306}:\\
\;\;\;\;\left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) \cdot \cos y + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r38693504 = 2.0;
        double r38693505 = x;
        double r38693506 = sqrt(r38693505);
        double r38693507 = r38693504 * r38693506;
        double r38693508 = y;
        double r38693509 = z;
        double r38693510 = t;
        double r38693511 = r38693509 * r38693510;
        double r38693512 = 3.0;
        double r38693513 = r38693511 / r38693512;
        double r38693514 = r38693508 - r38693513;
        double r38693515 = cos(r38693514);
        double r38693516 = r38693507 * r38693515;
        double r38693517 = a;
        double r38693518 = b;
        double r38693519 = r38693518 * r38693512;
        double r38693520 = r38693517 / r38693519;
        double r38693521 = r38693516 - r38693520;
        return r38693521;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r38693522 = z;
        double r38693523 = t;
        double r38693524 = r38693522 * r38693523;
        double r38693525 = -4.557684899971343e+175;
        bool r38693526 = r38693524 <= r38693525;
        double r38693527 = 1.0;
        double r38693528 = y;
        double r38693529 = r38693528 * r38693528;
        double r38693530 = 0.5;
        double r38693531 = r38693529 * r38693530;
        double r38693532 = r38693527 - r38693531;
        double r38693533 = x;
        double r38693534 = sqrt(r38693533);
        double r38693535 = 2.0;
        double r38693536 = r38693534 * r38693535;
        double r38693537 = r38693532 * r38693536;
        double r38693538 = a;
        double r38693539 = 3.0;
        double r38693540 = b;
        double r38693541 = r38693539 * r38693540;
        double r38693542 = r38693538 / r38693541;
        double r38693543 = r38693537 - r38693542;
        double r38693544 = 2.650669621479893e+306;
        bool r38693545 = r38693524 <= r38693544;
        double r38693546 = r38693524 / r38693539;
        double r38693547 = cos(r38693546);
        double r38693548 = cbrt(r38693547);
        double r38693549 = r38693548 * r38693548;
        double r38693550 = r38693548 * r38693549;
        double r38693551 = cos(r38693528);
        double r38693552 = r38693550 * r38693551;
        double r38693553 = sin(r38693528);
        double r38693554 = sin(r38693546);
        double r38693555 = r38693553 * r38693554;
        double r38693556 = r38693552 + r38693555;
        double r38693557 = r38693556 * r38693536;
        double r38693558 = r38693557 - r38693542;
        double r38693559 = r38693545 ? r38693558 : r38693543;
        double r38693560 = r38693526 ? r38693543 : r38693559;
        return r38693560;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.3
Target18.4
Herbie18.3
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z t) < -4.557684899971343e+175 or 2.650669621479893e+306 < (* z t)

    1. Initial program 52.6

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

    2. Taylor expanded around 0 44.3

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

    3. Simplified44.3

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

    if -4.557684899971343e+175 < (* z t) < 2.650669621479893e+306

    1. Initial program 12.8

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

    3. Applied cos-diff12.3

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

    5. Applied add-cube-cbrt12.3

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

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -4.557684899971343275948588241655457690987 \cdot 10^{175}:\\ \;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;z \cdot t \le 2.650669621479893038998696276370616139907 \cdot 10^{306}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) \cdot \cos y + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))