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

Average Error: 20.6 → 17.8

Time: 10.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999429473585:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \log \left(e^{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r800341 = 2.0;
        double r800342 = x;
        double r800343 = sqrt(r800342);
        double r800344 = r800341 * r800343;
        double r800345 = y;
        double r800346 = z;
        double r800347 = t;
        double r800348 = r800346 * r800347;
        double r800349 = 3.0;
        double r800350 = r800348 / r800349;
        double r800351 = r800345 - r800350;
        double r800352 = cos(r800351);
        double r800353 = r800344 * r800352;
        double r800354 = a;
        double r800355 = b;
        double r800356 = r800355 * r800349;
        double r800357 = r800354 / r800356;
        double r800358 = r800353 - r800357;
        return r800358;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r800359 = y;
        double r800360 = z;
        double r800361 = t;
        double r800362 = r800360 * r800361;
        double r800363 = 3.0;
        double r800364 = r800362 / r800363;
        double r800365 = r800359 - r800364;
        double r800366 = cos(r800365);
        double r800367 = 0.9999999429473585;
        bool r800368 = r800366 <= r800367;
        double r800369 = 2.0;
        double r800370 = x;
        double r800371 = sqrt(r800370);
        double r800372 = r800369 * r800371;
        double r800373 = cos(r800359);
        double r800374 = cos(r800364);
        double r800375 = exp(r800374);
        double r800376 = log(r800375);
        double r800377 = r800373 * r800376;
        double r800378 = r800372 * r800377;
        double r800379 = sin(r800359);
        double r800380 = 0.3333333333333333;
        double r800381 = r800361 * r800360;
        double r800382 = r800380 * r800381;
        double r800383 = sin(r800382);
        double r800384 = cbrt(r800383);
        double r800385 = r800384 * r800384;
        double r800386 = r800385 * r800384;
        double r800387 = r800379 * r800386;
        double r800388 = r800372 * r800387;
        double r800389 = r800378 + r800388;
        double r800390 = a;
        double r800391 = b;
        double r800392 = r800391 * r800363;
        double r800393 = r800390 / r800392;
        double r800394 = r800389 - r800393;
        double r800395 = 1.0;
        double r800396 = 0.5;
        double r800397 = 2.0;
        double r800398 = pow(r800359, r800397);
        double r800399 = r800396 * r800398;
        double r800400 = r800395 - r800399;
        double r800401 = r800372 * r800400;
        double r800402 = r800401 - r800393;
        double r800403 = r800368 ? r800394 : r800402;
        return r800403;
}

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	20.6
Target	18.7
Herbie	17.8

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999429473585

   1. Initial program 19.9

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

   3. Applied cos-diff 19.2

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

   4. Applied distribute-lft-in 19.2

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

   5. Taylor expanded around inf 19.2

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right) - \frac{a}{b \cdot 3}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 19.2

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)}\right)\right) - \frac{a}{b \cdot 3}\]
   8. Using strategy rm

   9. Applied add-log-exp 19.2

      \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\log \left(e^{\cos \left(\frac{z \cdot t}{3}\right)}\right)}\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   if 0.9999999429473585 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.8

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

   2. Taylor expanded around 0 15.5

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999429473585:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \log \left(e^{\cos \left(\frac{z \cdot t}{3}\right)}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \left(\left(\sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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))))