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

Average Error: 21.3 → 18.5

Time: 30.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999915456516674794329446740448475:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) - \sin y \cdot \sin \left(-\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right) \cdot \left(\sqrt[3]{\frac{1}{3}} \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r463258 = 2.0;
        double r463259 = x;
        double r463260 = sqrt(r463259);
        double r463261 = r463258 * r463260;
        double r463262 = y;
        double r463263 = z;
        double r463264 = t;
        double r463265 = r463263 * r463264;
        double r463266 = 3.0;
        double r463267 = r463265 / r463266;
        double r463268 = r463262 - r463267;
        double r463269 = cos(r463268);
        double r463270 = r463261 * r463269;
        double r463271 = a;
        double r463272 = b;
        double r463273 = r463272 * r463266;
        double r463274 = r463271 / r463273;
        double r463275 = r463270 - r463274;
        return r463275;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r463276 = y;
        double r463277 = z;
        double r463278 = t;
        double r463279 = r463277 * r463278;
        double r463280 = 3.0;
        double r463281 = r463279 / r463280;
        double r463282 = r463276 - r463281;
        double r463283 = cos(r463282);
        double r463284 = 0.9999999999915457;
        bool r463285 = r463283 <= r463284;
        double r463286 = 2.0;
        double r463287 = x;
        double r463288 = sqrt(r463287);
        double r463289 = r463286 * r463288;
        double r463290 = cos(r463276);
        double r463291 = cos(r463281);
        double r463292 = r463290 * r463291;
        double r463293 = sin(r463276);
        double r463294 = 1.0;
        double r463295 = r463294 / r463280;
        double r463296 = cbrt(r463295);
        double r463297 = r463296 * r463296;
        double r463298 = r463278 * r463277;
        double r463299 = r463296 * r463298;
        double r463300 = r463297 * r463299;
        double r463301 = -r463300;
        double r463302 = sin(r463301);
        double r463303 = r463293 * r463302;
        double r463304 = r463292 - r463303;
        double r463305 = r463289 * r463304;
        double r463306 = a;
        double r463307 = b;
        double r463308 = r463307 * r463280;
        double r463309 = r463306 / r463308;
        double r463310 = r463305 - r463309;
        double r463311 = 0.5;
        double r463312 = 2.0;
        double r463313 = pow(r463276, r463312);
        double r463314 = r463311 * r463313;
        double r463315 = r463294 - r463314;
        double r463316 = r463289 * r463315;
        double r463317 = r463316 - r463309;
        double r463318 = r463285 ? r463310 : r463317;
        return r463318;
}

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	21.3
Target	19.4
Herbie	18.5

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999915457

   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 sub-neg 20.5

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

   4. Applied cos-sum 19.8

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

   5. Simplified 19.8

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

   7. Applied clear-num 19.9

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

   9. Applied div-inv 19.9

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

   10. Applied associate-/r* 19.9

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

   12. Applied *-un-lft-identity 19.9

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

   13. Applied add-cube-cbrt 19.9

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

   14. Applied times-frac 19.9

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

   15. Simplified 19.9

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

   16. Simplified 19.9

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

   if 0.9999999999915457 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.7

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

   2. Taylor expanded around 0 16.1

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

4. Final simplification 18.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999915456516674794329446740448475:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) - \sin y \cdot \sin \left(-\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right) \cdot \left(\sqrt[3]{\frac{1}{3}} \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.333333333333333315 z) t)))) (/ (/ a 3) b)) (if (< z 3.51629061355598715e106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.333333333333333315 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))