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

Average Error: 21.1 → 18.7

Time: 19.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -1.428478097956767831303888965158296736042 \cdot 10^{226} \lor \neg \left(z \cdot t \le 7.274902941706250528826933509874936047753 \cdot 10^{294}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos y \cdot \left(\left(\sqrt[3]{\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r493440 = 2.0;
        double r493441 = x;
        double r493442 = sqrt(r493441);
        double r493443 = r493440 * r493442;
        double r493444 = y;
        double r493445 = z;
        double r493446 = t;
        double r493447 = r493445 * r493446;
        double r493448 = 3.0;
        double r493449 = r493447 / r493448;
        double r493450 = r493444 - r493449;
        double r493451 = cos(r493450);
        double r493452 = r493443 * r493451;
        double r493453 = a;
        double r493454 = b;
        double r493455 = r493454 * r493448;
        double r493456 = r493453 / r493455;
        double r493457 = r493452 - r493456;
        return r493457;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r493458 = z;
        double r493459 = t;
        double r493460 = r493458 * r493459;
        double r493461 = -1.4284780979567678e+226;
        bool r493462 = r493460 <= r493461;
        double r493463 = 7.27490294170625e+294;
        bool r493464 = r493460 <= r493463;
        double r493465 = !r493464;
        bool r493466 = r493462 || r493465;
        double r493467 = 2.0;
        double r493468 = x;
        double r493469 = sqrt(r493468);
        double r493470 = r493467 * r493469;
        double r493471 = 1.0;
        double r493472 = 0.5;
        double r493473 = y;
        double r493474 = 2.0;
        double r493475 = pow(r493473, r493474);
        double r493476 = r493472 * r493475;
        double r493477 = r493471 - r493476;
        double r493478 = r493470 * r493477;
        double r493479 = a;
        double r493480 = b;
        double r493481 = 3.0;
        double r493482 = r493480 * r493481;
        double r493483 = r493479 / r493482;
        double r493484 = r493478 - r493483;
        double r493485 = cos(r493473);
        double r493486 = 0.3333333333333333;
        double r493487 = r493459 * r493458;
        double r493488 = r493486 * r493487;
        double r493489 = cos(r493488);
        double r493490 = cbrt(r493489);
        double r493491 = r493490 * r493490;
        double r493492 = r493491 * r493490;
        double r493493 = cbrt(r493492);
        double r493494 = r493493 * r493490;
        double r493495 = r493494 * r493490;
        double r493496 = r493485 * r493495;
        double r493497 = r493496 * r493470;
        double r493498 = sin(r493473);
        double r493499 = r493460 / r493481;
        double r493500 = sin(r493499);
        double r493501 = r493498 * r493500;
        double r493502 = r493470 * r493501;
        double r493503 = r493497 + r493502;
        double r493504 = r493503 - r493483;
        double r493505 = r493466 ? r493484 : r493504;
        return r493505;
}

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	19.0
Herbie	18.7

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* z t) < -1.4284780979567678e+226 or 7.27490294170625e+294 < (* z t)

   1. Initial program 57.2

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

   2. Taylor expanded around 0 45.8

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

   if -1.4284780979567678e+226 < (* z t) < 7.27490294170625e+294

   1. Initial program 13.8

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

   3. Applied cos-diff 13.2

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

   4. Applied distribute-lft-in 13.2

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

   5. Simplified 13.2

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

   6. Taylor expanded around inf 13.2

      \[\leadsto \left(\left(\cos y \cdot \color{blue}{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 13.2

      \[\leadsto \left(\left(\cos y \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)}\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 13.2

       \[\leadsto \left(\left(\cos y \cdot \left(\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}}} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]
3. Recombined 2 regimes into one program.

4. Final simplification 18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -1.428478097956767831303888965158296736042 \cdot 10^{226} \lor \neg \left(z \cdot t \le 7.274902941706250528826933509874936047753 \cdot 10^{294}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos y \cdot \left(\left(\sqrt[3]{\left(\sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}} \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.333333333333333315 z) t)))) (/ (/ a 3) b)) (if (< z 3.51629061355598715e106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.333333333333333315 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))