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

Average Error: 20.7 → 18.3

Time: 15.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999603707223627679923083633184433:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{t}{\sqrt[3]{3}}\right) - 0.3333333333333333148296162562473909929395 \cdot \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - a \cdot \frac{\frac{1}{b}}{3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r671261 = 2.0;
        double r671262 = x;
        double r671263 = sqrt(r671262);
        double r671264 = r671261 * r671263;
        double r671265 = y;
        double r671266 = z;
        double r671267 = t;
        double r671268 = r671266 * r671267;
        double r671269 = 3.0;
        double r671270 = r671268 / r671269;
        double r671271 = r671265 - r671270;
        double r671272 = cos(r671271);
        double r671273 = r671264 * r671272;
        double r671274 = a;
        double r671275 = b;
        double r671276 = r671275 * r671269;
        double r671277 = r671274 / r671276;
        double r671278 = r671273 - r671277;
        return r671278;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r671279 = y;
        double r671280 = z;
        double r671281 = t;
        double r671282 = r671280 * r671281;
        double r671283 = 3.0;
        double r671284 = r671282 / r671283;
        double r671285 = r671279 - r671284;
        double r671286 = cos(r671285);
        double r671287 = 0.9999999996037072;
        bool r671288 = r671286 <= r671287;
        double r671289 = 2.0;
        double r671290 = x;
        double r671291 = sqrt(r671290);
        double r671292 = r671289 * r671291;
        double r671293 = cbrt(r671283);
        double r671294 = r671293 * r671293;
        double r671295 = r671280 / r671294;
        double r671296 = r671281 / r671293;
        double r671297 = r671295 * r671296;
        double r671298 = r671279 - r671297;
        double r671299 = cos(r671298);
        double r671300 = r671292 * r671299;
        double r671301 = 0.3333333333333333;
        double r671302 = a;
        double r671303 = b;
        double r671304 = r671302 / r671303;
        double r671305 = r671301 * r671304;
        double r671306 = r671300 - r671305;
        double r671307 = 1.0;
        double r671308 = 0.5;
        double r671309 = 2.0;
        double r671310 = pow(r671279, r671309);
        double r671311 = r671308 * r671310;
        double r671312 = r671307 - r671311;
        double r671313 = r671292 * r671312;
        double r671314 = r671307 / r671303;
        double r671315 = r671314 / r671283;
        double r671316 = r671302 * r671315;
        double r671317 = r671313 - r671316;
        double r671318 = r671288 ? r671306 : r671317;
        return r671318;
}

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	20.7
Target	18.6
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 (cos (- y (/ (* z t) 3.0))) < 0.9999999996037072

   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 add-cube-cbrt 19.9

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

   4. Applied times-frac 19.8

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

   6. Applied associate-/r* 19.9

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

   7. Taylor expanded around 0 19.9

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

   if 0.9999999996037072 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.1

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

   3. Applied add-cube-cbrt 22.1

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

   4. Applied times-frac 22.1

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

   6. Applied associate-/r* 22.1

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

   8. Applied *-un-lft-identity 22.1

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

   9. Applied div-inv 22.1

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

   10. Applied times-frac 22.1

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

   11. Simplified 22.1

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

   12. Taylor expanded around 0 15.6

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

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999603707223627679923083633184433:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{t}{\sqrt[3]{3}}\right) - 0.3333333333333333148296162562473909929395 \cdot \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - a \cdot \frac{\frac{1}{b}}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))))