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

Average Error: 20.6 → 17.7

Time: 28.6s

Precision: 64

Math

\[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3.0}\right) \le 0.9999999999997474:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \left(\cos y \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{t \cdot z}{3.0}\right)}}\right) + \log \left(\sqrt{e^{\cos \left(\frac{t \cdot z}{3.0}\right)}}\right)\right) + \sin y \cdot \sin \left(\frac{t \cdot z}{3.0}\right)\right) - \frac{\frac{a}{b}}{3.0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \left(1 - y \cdot \left(\frac{1}{2} \cdot y\right)\right) - \frac{a}{b \cdot 3.0}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r36697412 = 2.0;
        double r36697413 = x;
        double r36697414 = sqrt(r36697413);
        double r36697415 = r36697412 * r36697414;
        double r36697416 = y;
        double r36697417 = z;
        double r36697418 = t;
        double r36697419 = r36697417 * r36697418;
        double r36697420 = 3.0;
        double r36697421 = r36697419 / r36697420;
        double r36697422 = r36697416 - r36697421;
        double r36697423 = cos(r36697422);
        double r36697424 = r36697415 * r36697423;
        double r36697425 = a;
        double r36697426 = b;
        double r36697427 = r36697426 * r36697420;
        double r36697428 = r36697425 / r36697427;
        double r36697429 = r36697424 - r36697428;
        return r36697429;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r36697430 = y;
        double r36697431 = t;
        double r36697432 = z;
        double r36697433 = r36697431 * r36697432;
        double r36697434 = 3.0;
        double r36697435 = r36697433 / r36697434;
        double r36697436 = r36697430 - r36697435;
        double r36697437 = cos(r36697436);
        double r36697438 = 0.9999999999997474;
        bool r36697439 = r36697437 <= r36697438;
        double r36697440 = x;
        double r36697441 = sqrt(r36697440);
        double r36697442 = 2.0;
        double r36697443 = r36697441 * r36697442;
        double r36697444 = cos(r36697430);
        double r36697445 = cos(r36697435);
        double r36697446 = exp(r36697445);
        double r36697447 = sqrt(r36697446);
        double r36697448 = log(r36697447);
        double r36697449 = r36697448 + r36697448;
        double r36697450 = r36697444 * r36697449;
        double r36697451 = sin(r36697430);
        double r36697452 = sin(r36697435);
        double r36697453 = r36697451 * r36697452;
        double r36697454 = r36697450 + r36697453;
        double r36697455 = r36697443 * r36697454;
        double r36697456 = a;
        double r36697457 = b;
        double r36697458 = r36697456 / r36697457;
        double r36697459 = r36697458 / r36697434;
        double r36697460 = r36697455 - r36697459;
        double r36697461 = 1.0;
        double r36697462 = 0.5;
        double r36697463 = r36697462 * r36697430;
        double r36697464 = r36697430 * r36697463;
        double r36697465 = r36697461 - r36697464;
        double r36697466 = r36697443 * r36697465;
        double r36697467 = r36697457 * r36697434;
        double r36697468 = r36697456 / r36697467;
        double r36697469 = r36697466 - r36697468;
        double r36697470 = r36697439 ? r36697460 : r36697469;
        return r36697470;
}

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	20.6
Target	18.6
Herbie	17.7

\[\begin{array}{l} \mathbf{if}\;z \lt -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \cos \left(y - \frac{t}{3.0} \cdot z\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2.0 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3.0}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999997474

   1. Initial program 19.5

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
   2. Using strategy rm

   3. Applied cos-diff 18.8

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3.0}\right)\right)} - \frac{a}{b \cdot 3.0}\]
   4. Using strategy rm

   5. Applied associate-/r* 18.9

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3.0}\right)\right) - \color{blue}{\frac{\frac{a}{b}}{3.0}}\]
   6. Using strategy rm

   7. Applied add-log-exp 18.9

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\log \left(e^{\cos \left(\frac{z \cdot t}{3.0}\right)}\right)} + \sin y \cdot \sin \left(\frac{z \cdot t}{3.0}\right)\right) - \frac{\frac{a}{b}}{3.0}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 18.9

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \log \color{blue}{\left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3.0}\right)}} \cdot \sqrt{e^{\cos \left(\frac{z \cdot t}{3.0}\right)}}\right)} + \sin y \cdot \sin \left(\frac{z \cdot t}{3.0}\right)\right) - \frac{\frac{a}{b}}{3.0}\]

   10. Applied log-prod 18.9

       \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\left(\log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3.0}\right)}}\right) + \log \left(\sqrt{e^{\cos \left(\frac{z \cdot t}{3.0}\right)}}\right)\right)} + \sin y \cdot \sin \left(\frac{z \cdot t}{3.0}\right)\right) - \frac{\frac{a}{b}}{3.0}\]

   if 0.9999999999997474 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.6

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

   2. Taylor expanded around 0 15.6

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3.0}\]

   3. Simplified 15.6

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot y\right) \cdot y\right)} - \frac{a}{b \cdot 3.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3.0}\right) \le 0.9999999999997474:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \left(\cos y \cdot \left(\log \left(\sqrt{e^{\cos \left(\frac{t \cdot z}{3.0}\right)}}\right) + \log \left(\sqrt{e^{\cos \left(\frac{t \cdot z}{3.0}\right)}}\right)\right) + \sin y \cdot \sin \left(\frac{t \cdot z}{3.0}\right)\right) - \frac{\frac{a}{b}}{3.0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \left(1 - y \cdot \left(\frac{1}{2} \cdot y\right)\right) - \frac{a}{b \cdot 3.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))