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

Average Error: 20.8 → 16.5

Time: 41.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9974097663181321626879594077763613313437:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \left(\cos y \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(-z\right) \cdot \frac{t}{3}\right)\right)\right) - \left(-\sin y\right) \cdot \sin \left(\frac{z}{\frac{3}{t}}\right)\right) - \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(1, y, \left(-z\right) \cdot \frac{t}{3}\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r30423300 = 2.0;
        double r30423301 = x;
        double r30423302 = sqrt(r30423301);
        double r30423303 = r30423300 * r30423302;
        double r30423304 = y;
        double r30423305 = z;
        double r30423306 = t;
        double r30423307 = r30423305 * r30423306;
        double r30423308 = 3.0;
        double r30423309 = r30423307 / r30423308;
        double r30423310 = r30423304 - r30423309;
        double r30423311 = cos(r30423310);
        double r30423312 = r30423303 * r30423311;
        double r30423313 = a;
        double r30423314 = b;
        double r30423315 = r30423314 * r30423308;
        double r30423316 = r30423313 / r30423315;
        double r30423317 = r30423312 - r30423316;
        return r30423317;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r30423318 = y;
        double r30423319 = t;
        double r30423320 = z;
        double r30423321 = r30423319 * r30423320;
        double r30423322 = 3.0;
        double r30423323 = r30423321 / r30423322;
        double r30423324 = r30423318 - r30423323;
        double r30423325 = cos(r30423324);
        double r30423326 = 0.9974097663181322;
        bool r30423327 = r30423325 <= r30423326;
        double r30423328 = r30423319 / r30423322;
        double r30423329 = -r30423328;
        double r30423330 = r30423328 * r30423320;
        double r30423331 = fma(r30423329, r30423320, r30423330);
        double r30423332 = cos(r30423331);
        double r30423333 = cos(r30423318);
        double r30423334 = -r30423320;
        double r30423335 = r30423334 * r30423328;
        double r30423336 = cos(r30423335);
        double r30423337 = expm1(r30423336);
        double r30423338 = log1p(r30423337);
        double r30423339 = r30423333 * r30423338;
        double r30423340 = sin(r30423318);
        double r30423341 = -r30423340;
        double r30423342 = r30423322 / r30423319;
        double r30423343 = r30423320 / r30423342;
        double r30423344 = sin(r30423343);
        double r30423345 = r30423341 * r30423344;
        double r30423346 = r30423339 - r30423345;
        double r30423347 = r30423332 * r30423346;
        double r30423348 = sin(r30423331);
        double r30423349 = 1.0;
        double r30423350 = fma(r30423349, r30423318, r30423335);
        double r30423351 = sin(r30423350);
        double r30423352 = r30423348 * r30423351;
        double r30423353 = r30423347 - r30423352;
        double r30423354 = x;
        double r30423355 = sqrt(r30423354);
        double r30423356 = 2.0;
        double r30423357 = r30423355 * r30423356;
        double r30423358 = r30423353 * r30423357;
        double r30423359 = a;
        double r30423360 = b;
        double r30423361 = r30423322 * r30423360;
        double r30423362 = r30423359 / r30423361;
        double r30423363 = r30423358 - r30423362;
        double r30423364 = -0.5;
        double r30423365 = r30423318 * r30423318;
        double r30423366 = fma(r30423364, r30423365, r30423349);
        double r30423367 = r30423366 * r30423357;
        double r30423368 = r30423367 - r30423362;
        double r30423369 = r30423327 ? r30423363 : r30423368;
        return r30423369;
}

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	20.8
Target	18.7
Herbie	16.5

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9974097663181322

   1. Initial program 20.1

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

   3. Applied *-un-lft-identity 20.1

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

   4. Applied times-frac 20.1

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

   5. Applied *-un-lft-identity 20.1

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

   6. Applied prod-diff 20.1

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

   7. Applied cos-sum 16.8

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

   9. Applied fma-udef 16.8

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

   10. Applied cos-sum 16.0

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

   12. Applied neg-sub0 16.0

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(1 \cdot y\right) \cdot \cos \left(-\frac{t}{3} \cdot \frac{z}{1}\right) - \sin \left(1 \cdot y\right) \cdot \sin \color{blue}{\left(0 - \frac{t}{3} \cdot \frac{z}{1}\right)}\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   13. Applied sin-diff 16.0

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(1 \cdot y\right) \cdot \cos \left(-\frac{t}{3} \cdot \frac{z}{1}\right) - \sin \left(1 \cdot y\right) \cdot \color{blue}{\left(\sin 0 \cdot \cos \left(\frac{t}{3} \cdot \frac{z}{1}\right) - \cos 0 \cdot \sin \left(\frac{t}{3} \cdot \frac{z}{1}\right)\right)}\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   14. Simplified 16.0

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(1 \cdot y\right) \cdot \cos \left(-\frac{t}{3} \cdot \frac{z}{1}\right) - \sin \left(1 \cdot y\right) \cdot \left(\color{blue}{0} - \cos 0 \cdot \sin \left(\frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   15. Simplified 16.0

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(1 \cdot y\right) \cdot \cos \left(-\frac{t}{3} \cdot \frac{z}{1}\right) - \sin \left(1 \cdot y\right) \cdot \left(0 - \color{blue}{\sin \left(\frac{z}{\frac{3}{t}}\right)}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   16. Using strategy rm

   17. Applied log1p-expm1-u 16.0

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(1 \cdot y\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(-\frac{t}{3} \cdot \frac{z}{1}\right)\right)\right)} - \sin \left(1 \cdot y\right) \cdot \left(0 - \sin \left(\frac{z}{\frac{3}{t}}\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   if 0.9974097663181322 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.0

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

   2. Taylor expanded around 0 17.2

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

   3. Simplified 17.2

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

4. Final simplification 16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9974097663181321626879594077763613313437:\\ \;\;\;\;\left(\cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \left(\cos y \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\left(-z\right) \cdot \frac{t}{3}\right)\right)\right) - \left(-\sin y\right) \cdot \sin \left(\frac{z}{\frac{3}{t}}\right)\right) - \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, \frac{t}{3} \cdot z\right)\right) \cdot \sin \left(\mathsf{fma}\left(1, y, \left(-z\right) \cdot \frac{t}{3}\right)\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))))