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

Average Error: 20.4 → 17.7

Time: 10.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999998532562013431:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r783471 = 2.0;
        double r783472 = x;
        double r783473 = sqrt(r783472);
        double r783474 = r783471 * r783473;
        double r783475 = y;
        double r783476 = z;
        double r783477 = t;
        double r783478 = r783476 * r783477;
        double r783479 = 3.0;
        double r783480 = r783478 / r783479;
        double r783481 = r783475 - r783480;
        double r783482 = cos(r783481);
        double r783483 = r783474 * r783482;
        double r783484 = a;
        double r783485 = b;
        double r783486 = r783485 * r783479;
        double r783487 = r783484 / r783486;
        double r783488 = r783483 - r783487;
        return r783488;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r783489 = y;
        double r783490 = z;
        double r783491 = t;
        double r783492 = r783490 * r783491;
        double r783493 = 3.0;
        double r783494 = r783492 / r783493;
        double r783495 = r783489 - r783494;
        double r783496 = cos(r783495);
        double r783497 = 0.9999985325620134;
        bool r783498 = r783496 <= r783497;
        double r783499 = 2.0;
        double r783500 = x;
        double r783501 = sqrt(r783500);
        double r783502 = r783499 * r783501;
        double r783503 = cos(r783489);
        double r783504 = cos(r783494);
        double r783505 = 3.0;
        double r783506 = pow(r783504, r783505);
        double r783507 = cbrt(r783506);
        double r783508 = r783503 * r783507;
        double r783509 = r783502 * r783508;
        double r783510 = sin(r783489);
        double r783511 = 0.3333333333333333;
        double r783512 = r783491 * r783490;
        double r783513 = r783511 * r783512;
        double r783514 = sin(r783513);
        double r783515 = r783510 * r783514;
        double r783516 = r783502 * r783515;
        double r783517 = cbrt(r783516);
        double r783518 = r783517 * r783517;
        double r783519 = r783518 * r783517;
        double r783520 = r783509 + r783519;
        double r783521 = a;
        double r783522 = b;
        double r783523 = r783522 * r783493;
        double r783524 = r783521 / r783523;
        double r783525 = r783520 - r783524;
        double r783526 = 1.0;
        double r783527 = 0.5;
        double r783528 = 2.0;
        double r783529 = pow(r783489, r783528);
        double r783530 = r783527 * r783529;
        double r783531 = r783526 - r783530;
        double r783532 = r783502 * r783531;
        double r783533 = r783532 - r783524;
        double r783534 = r783498 ? r783525 : r783533;
        return r783534;
}

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	18.3
Herbie	17.7

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999985325620134

   1. Initial program 19.3

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

   3. Applied cos-diff 18.6

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

   4. Applied distribute-lft-in 18.6

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

   5. Taylor expanded around inf 18.7

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

   7. Applied add-cbrt-cube 18.7

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

   8. Simplified 18.7

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

   10. Applied add-cube-cbrt 18.7

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

   if 0.9999985325620134 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.2

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

   2. Taylor expanded around 0 16.0

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

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999998532562013431:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))