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

Average Error: 20.6 → 17.9

Time: 32.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9997953791044875693216908985050395131111:\\ \;\;\;\;\left(\left(\sin y \cdot \sin \left(\frac{z}{\frac{3}{t}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\cos y \cdot \cos \left(\frac{z}{\frac{3}{t}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b} \cdot 0.3333333333333333148296162562473909929395\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{2} + 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r32114371 = 2.0;
        double r32114372 = x;
        double r32114373 = sqrt(r32114372);
        double r32114374 = r32114371 * r32114373;
        double r32114375 = y;
        double r32114376 = z;
        double r32114377 = t;
        double r32114378 = r32114376 * r32114377;
        double r32114379 = 3.0;
        double r32114380 = r32114378 / r32114379;
        double r32114381 = r32114375 - r32114380;
        double r32114382 = cos(r32114381);
        double r32114383 = r32114374 * r32114382;
        double r32114384 = a;
        double r32114385 = b;
        double r32114386 = r32114385 * r32114379;
        double r32114387 = r32114384 / r32114386;
        double r32114388 = r32114383 - r32114387;
        return r32114388;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r32114389 = y;
        double r32114390 = t;
        double r32114391 = z;
        double r32114392 = r32114390 * r32114391;
        double r32114393 = 3.0;
        double r32114394 = r32114392 / r32114393;
        double r32114395 = r32114389 - r32114394;
        double r32114396 = cos(r32114395);
        double r32114397 = 0.9997953791044876;
        bool r32114398 = r32114396 <= r32114397;
        double r32114399 = sin(r32114389);
        double r32114400 = r32114393 / r32114390;
        double r32114401 = r32114391 / r32114400;
        double r32114402 = sin(r32114401);
        double r32114403 = r32114399 * r32114402;
        double r32114404 = 2.0;
        double r32114405 = x;
        double r32114406 = sqrt(r32114405);
        double r32114407 = r32114404 * r32114406;
        double r32114408 = r32114403 * r32114407;
        double r32114409 = cos(r32114389);
        double r32114410 = cos(r32114401);
        double r32114411 = r32114409 * r32114410;
        double r32114412 = r32114411 * r32114407;
        double r32114413 = r32114408 + r32114412;
        double r32114414 = a;
        double r32114415 = b;
        double r32114416 = r32114414 / r32114415;
        double r32114417 = 0.3333333333333333;
        double r32114418 = r32114416 * r32114417;
        double r32114419 = r32114413 - r32114418;
        double r32114420 = r32114389 * r32114389;
        double r32114421 = -0.5;
        double r32114422 = r32114420 * r32114421;
        double r32114423 = 1.0;
        double r32114424 = r32114422 + r32114423;
        double r32114425 = r32114407 * r32114424;
        double r32114426 = r32114415 * r32114393;
        double r32114427 = r32114414 / r32114426;
        double r32114428 = r32114425 - r32114427;
        double r32114429 = r32114398 ? r32114419 : r32114428;
        return r32114429;
}

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	17.9

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9997953791044876

   1. Initial program 20.5

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

   3. Applied associate-/l* 20.5

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

   5. Applied cos-diff 19.7

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

   6. Applied distribute-rgt-in 19.7

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

   7. Taylor expanded around 0 19.7

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

   if 0.9997953791044876 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 20.7

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

   3. Applied associate-/l* 20.6

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

   4. Taylor expanded around 0 14.8

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

   5. Simplified 14.8

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9997953791044875693216908985050395131111:\\ \;\;\;\;\left(\left(\sin y \cdot \sin \left(\frac{z}{\frac{3}{t}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\cos y \cdot \cos \left(\frac{z}{\frac{3}{t}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b} \cdot 0.3333333333333333148296162562473909929395\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{2} + 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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))))