Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, K

Average Error: 21.1 → 17.8

Time: 11.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99998911849150296:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}}\right) + \sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r940246 = 2.0;
        double r940247 = x;
        double r940248 = sqrt(r940247);
        double r940249 = r940246 * r940248;
        double r940250 = y;
        double r940251 = z;
        double r940252 = t;
        double r940253 = r940251 * r940252;
        double r940254 = 3.0;
        double r940255 = r940253 / r940254;
        double r940256 = r940250 - r940255;
        double r940257 = cos(r940256);
        double r940258 = r940249 * r940257;
        double r940259 = a;
        double r940260 = b;
        double r940261 = r940260 * r940254;
        double r940262 = r940259 / r940261;
        double r940263 = r940258 - r940262;
        return r940263;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r940264 = y;
        double r940265 = z;
        double r940266 = t;
        double r940267 = r940265 * r940266;
        double r940268 = 3.0;
        double r940269 = r940267 / r940268;
        double r940270 = r940264 - r940269;
        double r940271 = cos(r940270);
        double r940272 = 0.999989118491503;
        bool r940273 = r940271 <= r940272;
        double r940274 = 2.0;
        double r940275 = x;
        double r940276 = sqrt(r940275);
        double r940277 = r940274 * r940276;
        double r940278 = cos(r940264);
        double r940279 = cos(r940269);
        double r940280 = cbrt(r940279);
        double r940281 = r940280 * r940280;
        double r940282 = r940281 * r940280;
        double r940283 = cbrt(r940282);
        double r940284 = r940281 * r940283;
        double r940285 = r940278 * r940284;
        double r940286 = sin(r940264);
        double r940287 = 0.3333333333333333;
        double r940288 = r940266 * r940265;
        double r940289 = r940287 * r940288;
        double r940290 = sin(r940289);
        double r940291 = r940286 * r940290;
        double r940292 = r940285 + r940291;
        double r940293 = r940277 * r940292;
        double r940294 = a;
        double r940295 = b;
        double r940296 = r940295 * r940268;
        double r940297 = r940294 / r940296;
        double r940298 = r940293 - r940297;
        double r940299 = 1.0;
        double r940300 = 0.5;
        double r940301 = 2.0;
        double r940302 = pow(r940264, r940301);
        double r940303 = r940300 * r940302;
        double r940304 = r940299 - r940303;
        double r940305 = r940277 * r940304;
        double r940306 = r940305 - r940297;
        double r940307 = r940273 ? r940298 : r940306;
        return r940307;
}

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

Original	21.1
Target	18.7
Herbie	17.8

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.999989118491503

   1. Initial program 20.0

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

   3. Applied cos-diff 19.3

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

   4. Taylor expanded around inf 19.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \color{blue}{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) - \frac{a}{b \cdot 3}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 19.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right)} + \sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) - \frac{a}{b \cdot 3}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 19.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}}}\right) + \sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) - \frac{a}{b \cdot 3}\]

   if 0.999989118491503 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.8

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

   2. Taylor expanded around 0 15.4

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99998911849150296:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}}\right) + \sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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))))