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

Average Error: 21.0 → 18.8

Time: 13.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.995537973235021245:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) - \sin y \cdot \sin \left(-\left(\sqrt[3]{\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}}}\right)\right) \cdot \sqrt[3]{\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r663469 = 2.0;
        double r663470 = x;
        double r663471 = sqrt(r663470);
        double r663472 = r663469 * r663471;
        double r663473 = y;
        double r663474 = z;
        double r663475 = t;
        double r663476 = r663474 * r663475;
        double r663477 = 3.0;
        double r663478 = r663476 / r663477;
        double r663479 = r663473 - r663478;
        double r663480 = cos(r663479);
        double r663481 = r663472 * r663480;
        double r663482 = a;
        double r663483 = b;
        double r663484 = r663483 * r663477;
        double r663485 = r663482 / r663484;
        double r663486 = r663481 - r663485;
        return r663486;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r663487 = y;
        double r663488 = z;
        double r663489 = t;
        double r663490 = r663488 * r663489;
        double r663491 = 3.0;
        double r663492 = r663490 / r663491;
        double r663493 = r663487 - r663492;
        double r663494 = cos(r663493);
        double r663495 = 0.9955379732350212;
        bool r663496 = r663494 <= r663495;
        double r663497 = 2.0;
        double r663498 = x;
        double r663499 = sqrt(r663498);
        double r663500 = r663497 * r663499;
        double r663501 = cos(r663487);
        double r663502 = cbrt(r663491);
        double r663503 = r663502 * r663502;
        double r663504 = r663490 / r663503;
        double r663505 = r663504 / r663502;
        double r663506 = cos(r663505);
        double r663507 = r663501 * r663506;
        double r663508 = sin(r663487);
        double r663509 = cbrt(r663505);
        double r663510 = cbrt(r663509);
        double r663511 = r663510 * r663510;
        double r663512 = r663511 * r663510;
        double r663513 = r663509 * r663512;
        double r663514 = r663513 * r663509;
        double r663515 = -r663514;
        double r663516 = sin(r663515);
        double r663517 = r663508 * r663516;
        double r663518 = r663507 - r663517;
        double r663519 = r663500 * r663518;
        double r663520 = a;
        double r663521 = b;
        double r663522 = r663521 * r663491;
        double r663523 = r663520 / r663522;
        double r663524 = r663519 - r663523;
        double r663525 = 1.0;
        double r663526 = 0.5;
        double r663527 = 2.0;
        double r663528 = pow(r663487, r663527);
        double r663529 = r663526 * r663528;
        double r663530 = r663525 - r663529;
        double r663531 = r663500 * r663530;
        double r663532 = r663531 - r663523;
        double r663533 = r663496 ? r663524 : r663532;
        return r663533;
}

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	21.0
Target	19.0
Herbie	18.8

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9955379732350212

   1. Initial program 20.3

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

   3. Applied add-cube-cbrt 20.3

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

   4. Applied associate-/r* 20.2

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

   6. Applied sub-neg 20.2

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

   7. Applied cos-sum 19.7

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

   8. Simplified 19.7

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

   10. Applied add-cube-cbrt 19.6

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

   12. Applied add-cube-cbrt 19.7

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

   if 0.9955379732350212 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.1

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

   2. Taylor expanded around 0 17.5

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

4. Final simplification 18.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.995537973235021245:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}\right) - \sin y \cdot \sin \left(-\left(\sqrt[3]{\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}}}\right)\right) \cdot \sqrt[3]{\frac{\frac{z \cdot t}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt[3]{3}}}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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))))