Average Error: 20.2 → 17.8
Time: 1.2m
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9984624378955296863935586770821828395128:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos y\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{t \cdot z}{3}\right)\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) - \frac{a}{3 \cdot b}\\ \end{array}\]
\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}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9984624378955296863935586770821828395128:\\
\;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos y\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{t \cdot z}{3}\right)\right)\right) - \frac{\frac{a}{b}}{3}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r31944428 = 2.0;
        double r31944429 = x;
        double r31944430 = sqrt(r31944429);
        double r31944431 = r31944428 * r31944430;
        double r31944432 = y;
        double r31944433 = z;
        double r31944434 = t;
        double r31944435 = r31944433 * r31944434;
        double r31944436 = 3.0;
        double r31944437 = r31944435 / r31944436;
        double r31944438 = r31944432 - r31944437;
        double r31944439 = cos(r31944438);
        double r31944440 = r31944431 * r31944439;
        double r31944441 = a;
        double r31944442 = b;
        double r31944443 = r31944442 * r31944436;
        double r31944444 = r31944441 / r31944443;
        double r31944445 = r31944440 - r31944444;
        return r31944445;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r31944446 = y;
        double r31944447 = t;
        double r31944448 = z;
        double r31944449 = r31944447 * r31944448;
        double r31944450 = 3.0;
        double r31944451 = r31944449 / r31944450;
        double r31944452 = r31944446 - r31944451;
        double r31944453 = cos(r31944452);
        double r31944454 = 0.9984624378955297;
        bool r31944455 = r31944453 <= r31944454;
        double r31944456 = 2.0;
        double r31944457 = x;
        double r31944458 = sqrt(r31944457);
        double r31944459 = r31944456 * r31944458;
        double r31944460 = 0.3333333333333333;
        double r31944461 = r31944460 * r31944449;
        double r31944462 = cos(r31944461);
        double r31944463 = cbrt(r31944462);
        double r31944464 = r31944463 * r31944463;
        double r31944465 = r31944464 * r31944463;
        double r31944466 = cos(r31944446);
        double r31944467 = r31944465 * r31944466;
        double r31944468 = r31944459 * r31944467;
        double r31944469 = sin(r31944446);
        double r31944470 = sin(r31944451);
        double r31944471 = r31944469 * r31944470;
        double r31944472 = r31944459 * r31944471;
        double r31944473 = r31944468 + r31944472;
        double r31944474 = a;
        double r31944475 = b;
        double r31944476 = r31944474 / r31944475;
        double r31944477 = r31944476 / r31944450;
        double r31944478 = r31944473 - r31944477;
        double r31944479 = 1.0;
        double r31944480 = r31944446 * r31944446;
        double r31944481 = 0.5;
        double r31944482 = r31944480 * r31944481;
        double r31944483 = r31944479 - r31944482;
        double r31944484 = r31944459 * r31944483;
        double r31944485 = r31944450 * r31944475;
        double r31944486 = r31944474 / r31944485;
        double r31944487 = r31944484 - r31944486;
        double r31944488 = r31944455 ? r31944478 : r31944487;
        return r31944488;
}

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

Original20.2
Target18.0
Herbie17.8
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos (- y (/ (* z t) 3.0))) < 0.9984624378955297

    1. Initial program 19.5

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

    3. Applied cos-diff18.8

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

    4. Applied distribute-rgt-in18.8

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

    5. Taylor expanded around inf 18.9

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

    7. Applied associate-/r*18.9

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

    9. Applied add-cube-cbrt18.9

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

    if 0.9984624378955297 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 21.3

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

    2. Taylor expanded around 0 16.2

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

    3. Simplified16.2

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9984624378955296863935586770821828395128:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos y\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{t \cdot z}{3}\right)\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \left(y \cdot y\right) \cdot \frac{1}{2}\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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))))