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

Average Error: 20.4 → 17.9

Time: 10.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 9.58429520565278927 \cdot 10^{149}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{\frac{z \cdot t}{\sqrt{3}}}{\sqrt{3}}\right) - \sin y \cdot \sin \left(-0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r768349 = 2.0;
        double r768350 = x;
        double r768351 = sqrt(r768350);
        double r768352 = r768349 * r768351;
        double r768353 = y;
        double r768354 = z;
        double r768355 = t;
        double r768356 = r768354 * r768355;
        double r768357 = 3.0;
        double r768358 = r768356 / r768357;
        double r768359 = r768353 - r768358;
        double r768360 = cos(r768359);
        double r768361 = r768352 * r768360;
        double r768362 = a;
        double r768363 = b;
        double r768364 = r768363 * r768357;
        double r768365 = r768362 / r768364;
        double r768366 = r768361 - r768365;
        return r768366;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r768367 = 2.0;
        double r768368 = x;
        double r768369 = sqrt(r768368);
        double r768370 = r768367 * r768369;
        double r768371 = y;
        double r768372 = z;
        double r768373 = t;
        double r768374 = r768372 * r768373;
        double r768375 = 3.0;
        double r768376 = r768374 / r768375;
        double r768377 = r768371 - r768376;
        double r768378 = cos(r768377);
        double r768379 = r768370 * r768378;
        double r768380 = 9.58429520565279e+149;
        bool r768381 = r768379 <= r768380;
        double r768382 = cos(r768371);
        double r768383 = sqrt(r768375);
        double r768384 = r768374 / r768383;
        double r768385 = r768384 / r768383;
        double r768386 = cos(r768385);
        double r768387 = r768382 * r768386;
        double r768388 = sin(r768371);
        double r768389 = 0.3333333333333333;
        double r768390 = r768373 * r768372;
        double r768391 = r768389 * r768390;
        double r768392 = -r768391;
        double r768393 = sin(r768392);
        double r768394 = r768388 * r768393;
        double r768395 = r768387 - r768394;
        double r768396 = r768370 * r768395;
        double r768397 = a;
        double r768398 = b;
        double r768399 = r768397 / r768398;
        double r768400 = r768399 / r768375;
        double r768401 = r768396 - r768400;
        double r768402 = 1.0;
        double r768403 = 0.5;
        double r768404 = 2.0;
        double r768405 = pow(r768371, r768404);
        double r768406 = r768403 * r768405;
        double r768407 = r768402 - r768406;
        double r768408 = r768370 * r768407;
        double r768409 = r768398 * r768375;
        double r768410 = r768397 / r768409;
        double r768411 = r768408 - r768410;
        double r768412 = r768381 ? r768401 : r768411;
        return r768412;
}

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	20.4
Target	18.7
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 (* (* 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-neg 14.4

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

   4. Applied cos-sum 13.9

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

   5. Simplified 13.9

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

   7. Applied associate-/r* 13.9

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

   8. Taylor expanded around inf 13.9

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

   10. Applied add-sqr-sqrt 13.9

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

   11. Applied associate-/r* 13.9

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

   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 simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 9.58429520565278927 \cdot 10^{149}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{\frac{z \cdot t}{\sqrt{3}}}{\sqrt{3}}\right) - \sin y \cdot \sin \left(-0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))))