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

Average Error: 20.3 → 17.8

Time: 10.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99990718372146614:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r583313 = 2.0;
        double r583314 = x;
        double r583315 = sqrt(r583314);
        double r583316 = r583313 * r583315;
        double r583317 = y;
        double r583318 = z;
        double r583319 = t;
        double r583320 = r583318 * r583319;
        double r583321 = 3.0;
        double r583322 = r583320 / r583321;
        double r583323 = r583317 - r583322;
        double r583324 = cos(r583323);
        double r583325 = r583316 * r583324;
        double r583326 = a;
        double r583327 = b;
        double r583328 = r583327 * r583321;
        double r583329 = r583326 / r583328;
        double r583330 = r583325 - r583329;
        return r583330;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r583331 = y;
        double r583332 = z;
        double r583333 = t;
        double r583334 = r583332 * r583333;
        double r583335 = 3.0;
        double r583336 = r583334 / r583335;
        double r583337 = r583331 - r583336;
        double r583338 = cos(r583337);
        double r583339 = 0.9999071837214661;
        bool r583340 = r583338 <= r583339;
        double r583341 = 2.0;
        double r583342 = x;
        double r583343 = sqrt(r583342);
        double r583344 = r583341 * r583343;
        double r583345 = cos(r583331);
        double r583346 = 0.3333333333333333;
        double r583347 = r583333 * r583332;
        double r583348 = r583346 * r583347;
        double r583349 = cos(r583348);
        double r583350 = r583345 * r583349;
        double r583351 = r583344 * r583350;
        double r583352 = sin(r583348);
        double r583353 = sin(r583331);
        double r583354 = r583352 * r583353;
        double r583355 = r583344 * r583354;
        double r583356 = cbrt(r583355);
        double r583357 = r583356 * r583356;
        double r583358 = r583357 * r583356;
        double r583359 = r583351 + r583358;
        double r583360 = a;
        double r583361 = b;
        double r583362 = r583361 * r583335;
        double r583363 = r583360 / r583362;
        double r583364 = r583359 - r583363;
        double r583365 = 1.0;
        double r583366 = 0.5;
        double r583367 = 2.0;
        double r583368 = pow(r583331, r583367);
        double r583369 = r583366 * r583368;
        double r583370 = r583365 - r583369;
        double r583371 = r583344 * r583370;
        double r583372 = r583371 - r583363;
        double r583373 = r583340 ? r583364 : r583372;
        return r583373;
}

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	20.3
Target	18.4
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))) < 0.9999071837214661

   1. Initial program 19.9

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

   3. Applied cos-diff 19.4

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

   4. Applied distribute-lft-in 19.4

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

   5. Taylor expanded around inf 19.4

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

   6. Taylor expanded around inf 19.3

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

   8. Applied add-cube-cbrt 19.3

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

   if 0.9999071837214661 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 20.9

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

   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 simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99990718372146614:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))