Average Error: 20.4 → 17.9
Time: 12.9s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -4.4548286609804394 \cdot 10^{283} \lor \neg \left(z \cdot t \le 3.41570283314918143 \cdot 10^{303}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \sqrt[3]{\sqrt[3]{{\left({\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}\right)}^{3}}} + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 -4.4548286609804394 \cdot 10^{283} \lor \neg \left(z \cdot t \le 3.41570283314918143 \cdot 10^{303}\right):\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r769655 = 2.0;
        double r769656 = x;
        double r769657 = sqrt(r769656);
        double r769658 = r769655 * r769657;
        double r769659 = y;
        double r769660 = z;
        double r769661 = t;
        double r769662 = r769660 * r769661;
        double r769663 = 3.0;
        double r769664 = r769662 / r769663;
        double r769665 = r769659 - r769664;
        double r769666 = cos(r769665);
        double r769667 = r769658 * r769666;
        double r769668 = a;
        double r769669 = b;
        double r769670 = r769669 * r769663;
        double r769671 = r769668 / r769670;
        double r769672 = r769667 - r769671;
        return r769672;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r769673 = z;
        double r769674 = t;
        double r769675 = r769673 * r769674;
        double r769676 = -4.454828660980439e+283;
        bool r769677 = r769675 <= r769676;
        double r769678 = 3.4157028331491814e+303;
        bool r769679 = r769675 <= r769678;
        double r769680 = !r769679;
        bool r769681 = r769677 || r769680;
        double r769682 = 2.0;
        double r769683 = x;
        double r769684 = sqrt(r769683);
        double r769685 = r769682 * r769684;
        double r769686 = 1.0;
        double r769687 = 0.5;
        double r769688 = y;
        double r769689 = 2.0;
        double r769690 = pow(r769688, r769689);
        double r769691 = r769687 * r769690;
        double r769692 = r769686 - r769691;
        double r769693 = r769685 * r769692;
        double r769694 = a;
        double r769695 = b;
        double r769696 = 3.0;
        double r769697 = r769695 * r769696;
        double r769698 = r769694 / r769697;
        double r769699 = r769693 - r769698;
        double r769700 = cos(r769688);
        double r769701 = r769675 / r769696;
        double r769702 = cos(r769701);
        double r769703 = 3.0;
        double r769704 = pow(r769702, r769703);
        double r769705 = pow(r769704, r769703);
        double r769706 = cbrt(r769705);
        double r769707 = cbrt(r769706);
        double r769708 = r769700 * r769707;
        double r769709 = sin(r769688);
        double r769710 = sin(r769701);
        double r769711 = r769709 * r769710;
        double r769712 = r769708 + r769711;
        double r769713 = r769685 * r769712;
        double r769714 = r769713 - r769698;
        double r769715 = r769681 ? r769699 : r769714;
        return r769715;
}

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

Original20.4
Target18.4
Herbie17.9
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z t) < -4.454828660980439e+283 or 3.4157028331491814e+303 < (* z t)

    1. Initial program 61.3

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

    2. Taylor expanded around 0 46.6

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

    if -4.454828660980439e+283 < (* z t) < 3.4157028331491814e+303

    1. Initial program 13.9

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

    3. Applied cos-diff13.3

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

    5. Applied add-cbrt-cube13.3

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

    6. Simplified13.3

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

    8. Applied add-cbrt-cube13.3

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

    9. Simplified13.3

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -4.4548286609804394 \cdot 10^{283} \lor \neg \left(z \cdot t \le 3.41570283314918143 \cdot 10^{303}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \sqrt[3]{\sqrt[3]{{\left({\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}\right)}^{3}}} + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))))