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

Average Error: 20.5 → 17.9

Time: 10.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999000948875516:\\ \;\;\;\;\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right) \cdot \cos y\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right) \cdot \cos y\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \sin y\right) \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r823290 = 2.0;
        double r823291 = x;
        double r823292 = sqrt(r823291);
        double r823293 = r823290 * r823292;
        double r823294 = y;
        double r823295 = z;
        double r823296 = t;
        double r823297 = r823295 * r823296;
        double r823298 = 3.0;
        double r823299 = r823297 / r823298;
        double r823300 = r823294 - r823299;
        double r823301 = cos(r823300);
        double r823302 = r823293 * r823301;
        double r823303 = a;
        double r823304 = b;
        double r823305 = r823304 * r823298;
        double r823306 = r823303 / r823305;
        double r823307 = r823302 - r823306;
        return r823307;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r823308 = y;
        double r823309 = z;
        double r823310 = t;
        double r823311 = r823309 * r823310;
        double r823312 = 3.0;
        double r823313 = r823311 / r823312;
        double r823314 = r823308 - r823313;
        double r823315 = cos(r823314);
        double r823316 = 0.9999000948875516;
        bool r823317 = r823315 <= r823316;
        double r823318 = 2.0;
        double r823319 = x;
        double r823320 = sqrt(r823319);
        double r823321 = r823318 * r823320;
        double r823322 = cos(r823313);
        double r823323 = exp(r823322);
        double r823324 = sqrt(r823323);
        double r823325 = log(r823324);
        double r823326 = cos(r823308);
        double r823327 = r823325 * r823326;
        double r823328 = r823321 * r823327;
        double r823329 = r823328 + r823328;
        double r823330 = sin(r823308);
        double r823331 = r823321 * r823330;
        double r823332 = sin(r823313);
        double r823333 = r823331 * r823332;
        double r823334 = r823329 + r823333;
        double r823335 = a;
        double r823336 = b;
        double r823337 = r823336 * r823312;
        double r823338 = r823335 / r823337;
        double r823339 = r823334 - r823338;
        double r823340 = 1.0;
        double r823341 = 0.5;
        double r823342 = 2.0;
        double r823343 = pow(r823308, r823342);
        double r823344 = r823341 * r823343;
        double r823345 = r823340 - r823344;
        double r823346 = r823321 * r823345;
        double r823347 = r823346 - r823338;
        double r823348 = r823317 ? r823339 : r823347;
        return r823348;
}

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	20.5
Target	18.5
Herbie	17.9

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999000948875516

   1. Initial program 20.1

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

   3. Applied cos-diff 19.5

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

   4. Applied distribute-lft-in 19.5

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

   6. Applied associate-*r* 19.5

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

   8. Applied add-log-exp 19.5

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

   10. Applied add-sqr-sqrt 19.5

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

   11. Applied log-prod 19.5

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

   12. Applied distribute-lft-in 19.5

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

   13. Applied distribute-lft-in 19.5

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

   14. Simplified 19.5

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

   15. Simplified 19.5

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

   if 0.9999000948875516 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.1

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

   2. Taylor expanded around 0 15.1

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999000948875516:\\ \;\;\;\;\left(\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right) \cdot \cos y\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3}\right)}}\right) \cdot \cos y\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \sin y\right) \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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))))