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

Average Error: 20.3 → 15.7

Time: 4.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 3.623770893491818076037760003088588811179 \cdot 10^{147}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\left(\left(\sqrt[3]{\cos \left(\left(-\frac{t}{3}\right) \cdot z\right)} \cdot \sqrt[3]{\cos \left(\left(-\frac{t}{3}\right) \cdot z\right)}\right) \cdot \sqrt[3]{\cos \left(\left(-\frac{t}{3}\right) \cdot z\right)}\right) \cdot \cos y - \sin y \cdot \sin \left(\left(-\frac{t}{3}\right) \cdot z\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) - \sin \left(\mathsf{fma}\left(1, y, \left(-\frac{t}{3}\right) \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r194155389 = 2.0;
        double r194155390 = x;
        double r194155391 = sqrt(r194155390);
        double r194155392 = r194155389 * r194155391;
        double r194155393 = y;
        double r194155394 = z;
        double r194155395 = t;
        double r194155396 = r194155394 * r194155395;
        double r194155397 = 3.0;
        double r194155398 = r194155396 / r194155397;
        double r194155399 = r194155393 - r194155398;
        double r194155400 = cos(r194155399);
        double r194155401 = r194155392 * r194155400;
        double r194155402 = a;
        double r194155403 = b;
        double r194155404 = r194155403 * r194155397;
        double r194155405 = r194155402 / r194155404;
        double r194155406 = r194155401 - r194155405;
        return r194155406;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r194155407 = 2.0;
        double r194155408 = x;
        double r194155409 = sqrt(r194155408);
        double r194155410 = r194155407 * r194155409;
        double r194155411 = y;
        double r194155412 = z;
        double r194155413 = t;
        double r194155414 = r194155412 * r194155413;
        double r194155415 = 3.0;
        double r194155416 = r194155414 / r194155415;
        double r194155417 = r194155411 - r194155416;
        double r194155418 = cos(r194155417);
        double r194155419 = r194155410 * r194155418;
        double r194155420 = 3.623770893491818e+147;
        bool r194155421 = r194155419 <= r194155420;
        double r194155422 = r194155413 / r194155415;
        double r194155423 = -r194155422;
        double r194155424 = r194155423 * r194155412;
        double r194155425 = cos(r194155424);
        double r194155426 = cbrt(r194155425);
        double r194155427 = r194155426 * r194155426;
        double r194155428 = r194155427 * r194155426;
        double r194155429 = cos(r194155411);
        double r194155430 = r194155428 * r194155429;
        double r194155431 = sin(r194155411);
        double r194155432 = sin(r194155424);
        double r194155433 = r194155431 * r194155432;
        double r194155434 = r194155430 - r194155433;
        double r194155435 = r194155422 * r194155412;
        double r194155436 = fma(r194155423, r194155412, r194155435);
        double r194155437 = cos(r194155436);
        double r194155438 = r194155434 * r194155437;
        double r194155439 = 1.0;
        double r194155440 = fma(r194155439, r194155411, r194155424);
        double r194155441 = sin(r194155440);
        double r194155442 = sin(r194155436);
        double r194155443 = r194155441 * r194155442;
        double r194155444 = r194155438 - r194155443;
        double r194155445 = r194155410 * r194155444;
        double r194155446 = a;
        double r194155447 = b;
        double r194155448 = r194155447 * r194155415;
        double r194155449 = r194155446 / r194155448;
        double r194155450 = r194155445 - r194155449;
        double r194155451 = -0.5;
        double r194155452 = r194155411 * r194155411;
        double r194155453 = fma(r194155451, r194155452, r194155439);
        double r194155454 = r194155410 * r194155453;
        double r194155455 = r194155454 - r194155449;
        double r194155456 = r194155421 ? r194155450 : r194155455;
        return r194155456;
}

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	15.7

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

Derivation

1. Split input into 2 regimes

2. if (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) < 3.623770893491818e+147

   1. Initial program 14.1

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

   3. Applied *-un-lft-identity 14.1

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

   4. Applied times-frac 14.1

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

   5. Applied *-un-lft-identity 14.1

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

   6. Applied prod-diff 14.1

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

   7. Applied cos-sum 12.0

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

   9. Applied fma-udef 12.0

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

   10. Applied cos-sum 11.3

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

   12. Applied add-cube-cbrt 11.4

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

   if 3.623770893491818e+147 < (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))

   1. Initial program 59.6

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

   2. Taylor expanded around 0 43.9

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

   3. Simplified 43.9

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

4. Final simplification 15.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \le 3.623770893491818076037760003088588811179 \cdot 10^{147}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\left(\left(\sqrt[3]{\cos \left(\left(-\frac{t}{3}\right) \cdot z\right)} \cdot \sqrt[3]{\cos \left(\left(-\frac{t}{3}\right) \cdot z\right)}\right) \cdot \sqrt[3]{\cos \left(\left(-\frac{t}{3}\right) \cdot z\right)}\right) \cdot \cos y - \sin y \cdot \sin \left(\left(-\frac{t}{3}\right) \cdot z\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) - \sin \left(\mathsf{fma}\left(1, y, \left(-\frac{t}{3}\right) \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +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))))