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

Average Error: 20.9 → 18.3

Time: 10.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 9.710637111596279340362321401812195925565 \cdot 10^{270}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r831359 = 2.0;
        double r831360 = x;
        double r831361 = sqrt(r831360);
        double r831362 = r831359 * r831361;
        double r831363 = y;
        double r831364 = z;
        double r831365 = t;
        double r831366 = r831364 * r831365;
        double r831367 = 3.0;
        double r831368 = r831366 / r831367;
        double r831369 = r831363 - r831368;
        double r831370 = cos(r831369);
        double r831371 = r831362 * r831370;
        double r831372 = a;
        double r831373 = b;
        double r831374 = r831373 * r831367;
        double r831375 = r831372 / r831374;
        double r831376 = r831371 - r831375;
        return r831376;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r831377 = z;
        double r831378 = t;
        double r831379 = r831377 * r831378;
        double r831380 = -inf.0;
        bool r831381 = r831379 <= r831380;
        double r831382 = 9.71063711159628e+270;
        bool r831383 = r831379 <= r831382;
        double r831384 = !r831383;
        bool r831385 = r831381 || r831384;
        double r831386 = 2.0;
        double r831387 = x;
        double r831388 = sqrt(r831387);
        double r831389 = r831386 * r831388;
        double r831390 = 1.0;
        double r831391 = 0.5;
        double r831392 = y;
        double r831393 = 2.0;
        double r831394 = pow(r831392, r831393);
        double r831395 = r831391 * r831394;
        double r831396 = r831390 - r831395;
        double r831397 = r831389 * r831396;
        double r831398 = a;
        double r831399 = b;
        double r831400 = 3.0;
        double r831401 = r831399 * r831400;
        double r831402 = r831398 / r831401;
        double r831403 = r831397 - r831402;
        double r831404 = cos(r831392);
        double r831405 = 0.3333333333333333;
        double r831406 = r831378 * r831377;
        double r831407 = r831405 * r831406;
        double r831408 = cos(r831407);
        double r831409 = r831404 * r831408;
        double r831410 = r831389 * r831409;
        double r831411 = sin(r831392);
        double r831412 = sin(r831407);
        double r831413 = r831411 * r831412;
        double r831414 = r831389 * r831413;
        double r831415 = cbrt(r831414);
        double r831416 = r831415 * r831415;
        double r831417 = r831416 * r831415;
        double r831418 = r831410 + r831417;
        double r831419 = r831418 - r831402;
        double r831420 = r831385 ? r831403 : r831419;
        return r831420;
}

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	20.9
Target	18.7
Herbie	18.3

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

Derivation

1. Split input into 2 regimes

2. if (* z t) < -inf.0 or 9.71063711159628e+270 < (* z t)

   1. Initial program 60.7

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

   2. Taylor expanded around 0 45.3

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

   if -inf.0 < (* z t) < 9.71063711159628e+270

   1. Initial program 14.5

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

   3. Applied cos-diff 13.9

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

   4. Applied distribute-lft-in 13.9

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

   5. Taylor expanded around inf 13.9

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   6. Taylor expanded around inf 13.9

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)}\right)\right) - \frac{a}{b \cdot 3}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 13.9

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right) + \color{blue}{\left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}}\right) - \frac{a}{b \cdot 3}\]
3. Recombined 2 regimes into one program.

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 9.710637111596279340362321401812195925565 \cdot 10^{270}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right) + \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.3333333333333333148296162562473909929395 \cdot \left(t \cdot z\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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))))