Average Error: 20.4 → 17.9
Time: 11.3s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 9.58429520565278927 \cdot 10^{149}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) - \sin y \cdot \sin \left(-\frac{z \cdot t}{3}\right)\right) - \frac{1}{b} \cdot \frac{a}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 9.58429520565278927 \cdot 10^{149}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \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{1}{b} \cdot \frac{a}{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 r708421 = 2.0;
        double r708422 = x;
        double r708423 = sqrt(r708422);
        double r708424 = r708421 * r708423;
        double r708425 = y;
        double r708426 = z;
        double r708427 = t;
        double r708428 = r708426 * r708427;
        double r708429 = 3.0;
        double r708430 = r708428 / r708429;
        double r708431 = r708425 - r708430;
        double r708432 = cos(r708431);
        double r708433 = r708424 * r708432;
        double r708434 = a;
        double r708435 = b;
        double r708436 = r708435 * r708429;
        double r708437 = r708434 / r708436;
        double r708438 = r708433 - r708437;
        return r708438;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r708439 = 2.0;
        double r708440 = x;
        double r708441 = sqrt(r708440);
        double r708442 = r708439 * r708441;
        double r708443 = y;
        double r708444 = z;
        double r708445 = t;
        double r708446 = r708444 * r708445;
        double r708447 = 3.0;
        double r708448 = r708446 / r708447;
        double r708449 = r708443 - r708448;
        double r708450 = cos(r708449);
        double r708451 = r708442 * r708450;
        double r708452 = 9.58429520565279e+149;
        bool r708453 = r708451 <= r708452;
        double r708454 = cos(r708443);
        double r708455 = cos(r708448);
        double r708456 = r708454 * r708455;
        double r708457 = sin(r708443);
        double r708458 = -r708448;
        double r708459 = sin(r708458);
        double r708460 = r708457 * r708459;
        double r708461 = r708456 - r708460;
        double r708462 = r708442 * r708461;
        double r708463 = 1.0;
        double r708464 = b;
        double r708465 = r708463 / r708464;
        double r708466 = a;
        double r708467 = r708466 / r708447;
        double r708468 = r708465 * r708467;
        double r708469 = r708462 - r708468;
        double r708470 = 0.5;
        double r708471 = 2.0;
        double r708472 = pow(r708443, r708471);
        double r708473 = r708470 * r708472;
        double r708474 = r708463 - r708473;
        double r708475 = r708442 * r708474;
        double r708476 = r708464 * r708447;
        double r708477 = r708466 / r708476;
        double r708478 = r708475 - r708477;
        double r708479 = r708453 ? r708469 : r708478;
        return r708479;
}

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
Target18.7
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 (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) < 9.58429520565279e+149

    1. Initial program 14.4

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

    3. Applied sub-neg14.4

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

    4. Applied cos-sum13.9

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

    5. Simplified13.9

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

    7. Applied *-un-lft-identity13.9

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

    8. Applied times-frac13.9

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

    if 9.58429520565279e+149 < (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))

    1. Initial program 61.1

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

    2. Taylor expanded around 0 44.9

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 9.58429520565278927 \cdot 10^{149}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) - \sin y \cdot \sin \left(-\frac{z \cdot t}{3}\right)\right) - \frac{1}{b} \cdot \frac{a}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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))))