Average Error: 20.6 → 18.2
Time: 30.0s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -2.134950766792070736054264212885483840549 \cdot 10^{298} \lor \neg \left(z \cdot t \le 2.609088471259102629340660177783840039975 \cdot 10^{231}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right) + \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 -2.134950766792070736054264212885483840549 \cdot 10^{298} \lor \neg \left(z \cdot t \le 2.609088471259102629340660177783840039975 \cdot 10^{231}\right):\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - \frac{a}{b \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right) + \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r389518 = 2.0;
        double r389519 = x;
        double r389520 = sqrt(r389519);
        double r389521 = r389518 * r389520;
        double r389522 = y;
        double r389523 = z;
        double r389524 = t;
        double r389525 = r389523 * r389524;
        double r389526 = 3.0;
        double r389527 = r389525 / r389526;
        double r389528 = r389522 - r389527;
        double r389529 = cos(r389528);
        double r389530 = r389521 * r389529;
        double r389531 = a;
        double r389532 = b;
        double r389533 = r389532 * r389526;
        double r389534 = r389531 / r389533;
        double r389535 = r389530 - r389534;
        return r389535;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r389536 = z;
        double r389537 = t;
        double r389538 = r389536 * r389537;
        double r389539 = -2.1349507667920707e+298;
        bool r389540 = r389538 <= r389539;
        double r389541 = 2.6090884712591026e+231;
        bool r389542 = r389538 <= r389541;
        double r389543 = !r389542;
        bool r389544 = r389540 || r389543;
        double r389545 = 2.0;
        double r389546 = x;
        double r389547 = sqrt(r389546);
        double r389548 = r389545 * r389547;
        double r389549 = -0.5;
        double r389550 = y;
        double r389551 = 2.0;
        double r389552 = pow(r389550, r389551);
        double r389553 = 1.0;
        double r389554 = fma(r389549, r389552, r389553);
        double r389555 = r389548 * r389554;
        double r389556 = a;
        double r389557 = b;
        double r389558 = 3.0;
        double r389559 = r389557 * r389558;
        double r389560 = r389556 / r389559;
        double r389561 = r389555 - r389560;
        double r389562 = cos(r389550);
        double r389563 = 0.3333333333333333;
        double r389564 = r389537 * r389536;
        double r389565 = r389563 * r389564;
        double r389566 = cos(r389565);
        double r389567 = log1p(r389566);
        double r389568 = expm1(r389567);
        double r389569 = r389562 * r389568;
        double r389570 = sin(r389565);
        double r389571 = sin(r389550);
        double r389572 = r389570 * r389571;
        double r389573 = r389569 + r389572;
        double r389574 = r389548 * r389573;
        double r389575 = r389574 - r389560;
        double r389576 = r389544 ? r389561 : r389575;
        return r389576;
}

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

Original20.6
Target18.6
Herbie18.2
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z t) < -2.1349507667920707e+298 or 2.6090884712591026e+231 < (* z t)

    1. Initial program 56.4

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

    2. Taylor expanded around 0 44.8

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

    3. Simplified44.8

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

    if -2.1349507667920707e+298 < (* z t) < 2.6090884712591026e+231

    1. Initial program 13.4

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

    3. Applied cos-diff12.9

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

    4. Taylor expanded around inf 12.9

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

    5. Taylor expanded around inf 12.9

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

    7. Applied expm1-log1p-u12.9

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right)} + \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -2.134950766792070736054264212885483840549 \cdot 10^{298} \lor \neg \left(z \cdot t \le 2.609088471259102629340660177783840039975 \cdot 10^{231}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)\right) + \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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 -1.379333748723514e129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.333333333333333315 z) t)))) (/ (/ a 3) b)) (if (< z 3.51629061355598715e106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.333333333333333315 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))