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

Average Error: 19.9 → 17.3

Time: 16.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99999999999984523:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \frac{t}{\sqrt[3]{3}}\right)\right) - \sin y \cdot \sin \left(-\left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{3}}} \cdot \sqrt[3]{\frac{\sqrt[3]{z}}{\sqrt[3]{3}}}\right) \cdot \frac{t}{\sqrt[3]{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r863406 = 2.0;
        double r863407 = x;
        double r863408 = sqrt(r863407);
        double r863409 = r863406 * r863408;
        double r863410 = y;
        double r863411 = z;
        double r863412 = t;
        double r863413 = r863411 * r863412;
        double r863414 = 3.0;
        double r863415 = r863413 / r863414;
        double r863416 = r863410 - r863415;
        double r863417 = cos(r863416);
        double r863418 = r863409 * r863417;
        double r863419 = a;
        double r863420 = b;
        double r863421 = r863420 * r863414;
        double r863422 = r863419 / r863421;
        double r863423 = r863418 - r863422;
        return r863423;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r863424 = y;
        double r863425 = z;
        double r863426 = t;
        double r863427 = r863425 * r863426;
        double r863428 = 3.0;
        double r863429 = r863427 / r863428;
        double r863430 = r863424 - r863429;
        double r863431 = cos(r863430);
        double r863432 = 0.9999999999998452;
        bool r863433 = r863431 <= r863432;
        double r863434 = 2.0;
        double r863435 = x;
        double r863436 = sqrt(r863435);
        double r863437 = r863434 * r863436;
        double r863438 = cos(r863424);
        double r863439 = cbrt(r863428);
        double r863440 = r863439 * r863439;
        double r863441 = r863425 / r863440;
        double r863442 = cbrt(r863441);
        double r863443 = r863442 * r863442;
        double r863444 = r863426 / r863439;
        double r863445 = r863442 * r863444;
        double r863446 = r863443 * r863445;
        double r863447 = cos(r863446);
        double r863448 = r863438 * r863447;
        double r863449 = sin(r863424);
        double r863450 = cbrt(r863425);
        double r863451 = r863450 * r863450;
        double r863452 = r863451 / r863439;
        double r863453 = cbrt(r863452);
        double r863454 = r863450 / r863439;
        double r863455 = cbrt(r863454);
        double r863456 = r863453 * r863455;
        double r863457 = r863456 * r863444;
        double r863458 = r863443 * r863457;
        double r863459 = -r863458;
        double r863460 = sin(r863459);
        double r863461 = r863449 * r863460;
        double r863462 = r863448 - r863461;
        double r863463 = r863437 * r863462;
        double r863464 = a;
        double r863465 = b;
        double r863466 = r863465 * r863428;
        double r863467 = r863464 / r863466;
        double r863468 = r863463 - r863467;
        double r863469 = 1.0;
        double r863470 = 0.5;
        double r863471 = 2.0;
        double r863472 = pow(r863424, r863471);
        double r863473 = r863470 * r863472;
        double r863474 = r863469 - r863473;
        double r863475 = r863437 * r863474;
        double r863476 = r863475 - r863467;
        double r863477 = r863433 ? r863468 : r863476;
        return r863477;
}

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	19.9
Target	18.1
Herbie	17.3

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999998452

   1. Initial program 19.1

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

   3. Applied add-cube-cbrt 19.1

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

   4. Applied times-frac 19.1

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

   6. Applied add-cube-cbrt 19.1

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

   7. Applied associate-*l* 19.1

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

   9. Applied sub-neg 19.1

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

   10. Applied cos-sum 18.5

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

   11. Simplified 18.5

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

   13. Applied add-cube-cbrt 18.5

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

   14. Applied times-frac 18.5

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

   15. Applied cbrt-prod 18.5

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

   if 0.9999999999998452 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.3

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

   2. Taylor expanded around 0 15.0

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

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99999999999984523:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \frac{t}{\sqrt[3]{3}}\right)\right) - \sin y \cdot \sin \left(-\left(\sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt[3]{\frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{3}}} \cdot \sqrt[3]{\frac{\sqrt[3]{z}}{\sqrt[3]{3}}}\right) \cdot \frac{t}{\sqrt[3]{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))))