Average Error: 20.8 → 17.9
Time: 14.3s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999481199109:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999481199109:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r216494 = 2.0;
        double r216495 = x;
        double r216496 = sqrt(r216495);
        double r216497 = r216494 * r216496;
        double r216498 = y;
        double r216499 = z;
        double r216500 = t;
        double r216501 = r216499 * r216500;
        double r216502 = 3.0;
        double r216503 = r216501 / r216502;
        double r216504 = r216498 - r216503;
        double r216505 = cos(r216504);
        double r216506 = r216497 * r216505;
        double r216507 = a;
        double r216508 = b;
        double r216509 = r216508 * r216502;
        double r216510 = r216507 / r216509;
        double r216511 = r216506 - r216510;
        return r216511;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r216512 = y;
        double r216513 = z;
        double r216514 = t;
        double r216515 = r216513 * r216514;
        double r216516 = 3.0;
        double r216517 = r216515 / r216516;
        double r216518 = r216512 - r216517;
        double r216519 = cos(r216518);
        double r216520 = 0.9999999994811991;
        bool r216521 = r216519 <= r216520;
        double r216522 = 2.0;
        double r216523 = x;
        double r216524 = sqrt(r216523);
        double r216525 = r216522 * r216524;
        double r216526 = cos(r216512);
        double r216527 = cos(r216517);
        double r216528 = cbrt(r216527);
        double r216529 = 3.0;
        double r216530 = pow(r216527, r216529);
        double r216531 = cbrt(r216530);
        double r216532 = cbrt(r216531);
        double r216533 = r216528 * r216532;
        double r216534 = r216533 * r216532;
        double r216535 = r216526 * r216534;
        double r216536 = sin(r216512);
        double r216537 = sin(r216517);
        double r216538 = r216536 * r216537;
        double r216539 = r216535 + r216538;
        double r216540 = r216525 * r216539;
        double r216541 = a;
        double r216542 = b;
        double r216543 = r216542 * r216516;
        double r216544 = r216541 / r216543;
        double r216545 = r216540 - r216544;
        double r216546 = 1.0;
        double r216547 = 0.5;
        double r216548 = 2.0;
        double r216549 = pow(r216512, r216548);
        double r216550 = r216547 * r216549;
        double r216551 = r216546 - r216550;
        double r216552 = r216525 * r216551;
        double r216553 = r216552 - r216544;
        double r216554 = r216521 ? r216545 : r216553;
        return r216554;
}

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

Original20.8
Target18.6
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 (cos (- y (/ (* z t) 3.0))) < 0.9999999994811991

    1. Initial program 20.3

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

    3. Applied cos-diff19.5

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

    5. Applied add-cube-cbrt19.5

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

    7. Applied add-cbrt-cube19.5

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

    8. Simplified19.5

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

    10. Applied add-cbrt-cube19.5

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

    11. Simplified19.5

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

    if 0.9999999994811991 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 21.7

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

    2. Taylor expanded around 0 15.2

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999481199109:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))