Average Error: 20.8 → 17.9
Time: 16.3s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999481199109:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\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}\\

\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 r126312 = 2.0;
        double r126313 = x;
        double r126314 = sqrt(r126313);
        double r126315 = r126312 * r126314;
        double r126316 = y;
        double r126317 = z;
        double r126318 = t;
        double r126319 = r126317 * r126318;
        double r126320 = 3.0;
        double r126321 = r126319 / r126320;
        double r126322 = r126316 - r126321;
        double r126323 = cos(r126322);
        double r126324 = r126315 * r126323;
        double r126325 = a;
        double r126326 = b;
        double r126327 = r126326 * r126320;
        double r126328 = r126325 / r126327;
        double r126329 = r126324 - r126328;
        return r126329;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r126330 = y;
        double r126331 = z;
        double r126332 = t;
        double r126333 = r126331 * r126332;
        double r126334 = 3.0;
        double r126335 = r126333 / r126334;
        double r126336 = r126330 - r126335;
        double r126337 = cos(r126336);
        double r126338 = 0.9999999994811991;
        bool r126339 = r126337 <= r126338;
        double r126340 = 2.0;
        double r126341 = x;
        double r126342 = sqrt(r126341);
        double r126343 = r126340 * r126342;
        double r126344 = cos(r126330);
        double r126345 = cos(r126335);
        double r126346 = 3.0;
        double r126347 = pow(r126345, r126346);
        double r126348 = cbrt(r126347);
        double r126349 = cbrt(r126348);
        double r126350 = r126349 * r126349;
        double r126351 = cbrt(r126345);
        double r126352 = r126350 * r126351;
        double r126353 = r126344 * r126352;
        double r126354 = sin(r126330);
        double r126355 = sin(r126335);
        double r126356 = r126354 * r126355;
        double r126357 = r126353 + r126356;
        double r126358 = r126343 * r126357;
        double r126359 = a;
        double r126360 = b;
        double r126361 = r126360 * r126334;
        double r126362 = r126359 / r126361;
        double r126363 = r126358 - r126362;
        double r126364 = 1.0;
        double r126365 = 0.5;
        double r126366 = 2.0;
        double r126367 = pow(r126330, r126366);
        double r126368 = r126365 * r126367;
        double r126369 = r126364 - r126368;
        double r126370 = r126343 * r126369;
        double r126371 = r126370 - r126362;
        double r126372 = r126339 ? r126363 : r126371;
        return r126372;
}

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

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

    11. Simplified19.5

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

    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

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

Reproduce

herbie shell --seed 2020045 +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))))