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

Average Error: 20.3 → 16.9

Time: 52.6s

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 -1.799335942069011867122479081012577649284 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \le -5554585690422355:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{t}{3} \cdot \left(-z\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(\frac{-t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{t}{3} \cdot \left(-z\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{-t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \le 2.650669621479893038998696276370616139907 \cdot 10^{306}:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(\frac{-\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(1, y, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{-\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right) - \sin \left(\mathsf{fma}\left(\frac{-\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(1, y, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{-\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(\sqrt{x} \cdot 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 r28890406 = 2.0;
        double r28890407 = x;
        double r28890408 = sqrt(r28890407);
        double r28890409 = r28890406 * r28890408;
        double r28890410 = y;
        double r28890411 = z;
        double r28890412 = t;
        double r28890413 = r28890411 * r28890412;
        double r28890414 = 3.0;
        double r28890415 = r28890413 / r28890414;
        double r28890416 = r28890410 - r28890415;
        double r28890417 = cos(r28890416);
        double r28890418 = r28890409 * r28890417;
        double r28890419 = a;
        double r28890420 = b;
        double r28890421 = r28890420 * r28890414;
        double r28890422 = r28890419 / r28890421;
        double r28890423 = r28890418 - r28890422;
        return r28890423;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r28890424 = z;
        double r28890425 = t;
        double r28890426 = r28890424 * r28890425;
        double r28890427 = -1.799335942069012e+305;
        bool r28890428 = r28890426 <= r28890427;
        double r28890429 = -0.5;
        double r28890430 = y;
        double r28890431 = r28890430 * r28890430;
        double r28890432 = 1.0;
        double r28890433 = fma(r28890429, r28890431, r28890432);
        double r28890434 = x;
        double r28890435 = sqrt(r28890434);
        double r28890436 = 2.0;
        double r28890437 = r28890435 * r28890436;
        double r28890438 = r28890433 * r28890437;
        double r28890439 = a;
        double r28890440 = b;
        double r28890441 = 3.0;
        double r28890442 = r28890440 * r28890441;
        double r28890443 = r28890439 / r28890442;
        double r28890444 = r28890438 - r28890443;
        double r28890445 = -5554585690422355.0;
        bool r28890446 = r28890426 <= r28890445;
        double r28890447 = cbrt(r28890430);
        double r28890448 = r28890447 * r28890447;
        double r28890449 = r28890425 / r28890441;
        double r28890450 = -r28890424;
        double r28890451 = r28890449 * r28890450;
        double r28890452 = fma(r28890448, r28890447, r28890451);
        double r28890453 = cos(r28890452);
        double r28890454 = -r28890425;
        double r28890455 = r28890454 / r28890441;
        double r28890456 = r28890424 * r28890449;
        double r28890457 = fma(r28890455, r28890424, r28890456);
        double r28890458 = cos(r28890457);
        double r28890459 = r28890453 * r28890458;
        double r28890460 = sin(r28890452);
        double r28890461 = sin(r28890457);
        double r28890462 = r28890460 * r28890461;
        double r28890463 = r28890459 - r28890462;
        double r28890464 = r28890463 * r28890437;
        double r28890465 = r28890464 - r28890443;
        double r28890466 = 2.650669621479893e+306;
        bool r28890467 = r28890426 <= r28890466;
        double r28890468 = cbrt(r28890441);
        double r28890469 = r28890425 / r28890468;
        double r28890470 = -r28890469;
        double r28890471 = cbrt(r28890468);
        double r28890472 = r28890470 / r28890471;
        double r28890473 = r28890424 / r28890468;
        double r28890474 = r28890471 * r28890471;
        double r28890475 = r28890473 / r28890474;
        double r28890476 = r28890469 / r28890471;
        double r28890477 = r28890476 * r28890475;
        double r28890478 = fma(r28890472, r28890475, r28890477);
        double r28890479 = cos(r28890478);
        double r28890480 = -r28890473;
        double r28890481 = r28890480 / r28890474;
        double r28890482 = r28890476 * r28890481;
        double r28890483 = fma(r28890432, r28890430, r28890482);
        double r28890484 = cos(r28890483);
        double r28890485 = r28890479 * r28890484;
        double r28890486 = sin(r28890478);
        double r28890487 = sin(r28890483);
        double r28890488 = r28890486 * r28890487;
        double r28890489 = r28890485 - r28890488;
        double r28890490 = r28890489 * r28890437;
        double r28890491 = r28890490 - r28890443;
        double r28890492 = r28890467 ? r28890491 : r28890444;
        double r28890493 = r28890446 ? r28890465 : r28890492;
        double r28890494 = r28890428 ? r28890444 : r28890493;
        return r28890494;
}

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	20.3
Target	18.4
Herbie	16.9

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

Derivation

1. Split input into 3 regimes

2. if (* z t) < -1.799335942069012e+305 or 2.650669621479893e+306 < (* z t)

   1. Initial program 63.6

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

   2. Taylor expanded around 0 44.5

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

   3. Simplified 44.5

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

   if -1.799335942069012e+305 < (* z t) < -5554585690422355.0

   1. Initial program 33.6

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

   3. Applied *-un-lft-identity 33.6

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

   4. Applied times-frac 33.7

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

   5. Applied add-cube-cbrt 33.8

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

   6. Applied prod-diff 33.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, -\frac{t}{3} \cdot \frac{z}{1}\right) + \mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)} - \frac{a}{b \cdot 3}\]

   7. Applied cos-sum 27.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right)} - \frac{a}{b \cdot 3}\]

   if -5554585690422355.0 < (* z t) < 2.650669621479893e+306

   1. Initial program 9.4

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

   3. Applied add-cube-cbrt 9.4

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

   4. Applied associate-/r* 9.4

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

   6. Applied add-cube-cbrt 9.4

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

   7. Applied times-frac 9.4

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

   8. Applied times-frac 9.4

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

   9. Applied *-un-lft-identity 9.4

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

   10. Applied prod-diff 9.4

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(1, y, -\frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right) + \mathsf{fma}\left(-\frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right)} - \frac{a}{b \cdot 3}\]

   11. Applied cos-sum 9.4

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(1, y, -\frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right)\right)} - \frac{a}{b \cdot 3}\]
3. Recombined 3 regimes into one program.

4. Final simplification 16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -1.799335942069011867122479081012577649284 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \le -5554585690422355:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{t}{3} \cdot \left(-z\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(\frac{-t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{t}{3} \cdot \left(-z\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{-t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \le 2.650669621479893038998696276370616139907 \cdot 10^{306}:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(\frac{-\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(1, y, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{-\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right) - \sin \left(\mathsf{fma}\left(\frac{-\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(1, y, \frac{\frac{t}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}}} \cdot \frac{-\frac{z}{\sqrt[3]{3}}}{\sqrt[3]{\sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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 (- (/ 1.0 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))))