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

Average Error: 5.6 → 3.2

Time: 28.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.2131406689924034 \cdot 10^{45}:\\ \;\;\;\;\left(\left({\left(\sqrt[3]{\sqrt[3]{z}}\right)}^{3} \cdot \left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right) + \left(-a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{elif}\;z \le 1.86064292666481138 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \sqrt{z}\right) \cdot \sqrt{z} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r219237 = x;
        double r219238 = 18.0;
        double r219239 = r219237 * r219238;
        double r219240 = y;
        double r219241 = r219239 * r219240;
        double r219242 = z;
        double r219243 = r219241 * r219242;
        double r219244 = t;
        double r219245 = r219243 * r219244;
        double r219246 = a;
        double r219247 = 4.0;
        double r219248 = r219246 * r219247;
        double r219249 = r219248 * r219244;
        double r219250 = r219245 - r219249;
        double r219251 = b;
        double r219252 = c;
        double r219253 = r219251 * r219252;
        double r219254 = r219250 + r219253;
        double r219255 = r219237 * r219247;
        double r219256 = i;
        double r219257 = r219255 * r219256;
        double r219258 = r219254 - r219257;
        double r219259 = j;
        double r219260 = 27.0;
        double r219261 = r219259 * r219260;
        double r219262 = k;
        double r219263 = r219261 * r219262;
        double r219264 = r219258 - r219263;
        return r219264;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r219265 = z;
        double r219266 = -7.213140668992403e+45;
        bool r219267 = r219265 <= r219266;
        double r219268 = cbrt(r219265);
        double r219269 = cbrt(r219268);
        double r219270 = 3.0;
        double r219271 = pow(r219269, r219270);
        double r219272 = t;
        double r219273 = x;
        double r219274 = 18.0;
        double r219275 = r219273 * r219274;
        double r219276 = y;
        double r219277 = r219275 * r219276;
        double r219278 = r219268 * r219268;
        double r219279 = r219277 * r219278;
        double r219280 = r219272 * r219279;
        double r219281 = r219271 * r219280;
        double r219282 = a;
        double r219283 = 4.0;
        double r219284 = r219282 * r219283;
        double r219285 = -r219284;
        double r219286 = r219285 * r219272;
        double r219287 = r219281 + r219286;
        double r219288 = b;
        double r219289 = c;
        double r219290 = r219288 * r219289;
        double r219291 = r219287 + r219290;
        double r219292 = r219273 * r219283;
        double r219293 = i;
        double r219294 = r219292 * r219293;
        double r219295 = j;
        double r219296 = 27.0;
        double r219297 = k;
        double r219298 = r219296 * r219297;
        double r219299 = r219295 * r219298;
        double r219300 = r219294 + r219299;
        double r219301 = r219291 - r219300;
        double r219302 = 1.8606429266648114e-39;
        bool r219303 = r219265 <= r219302;
        double r219304 = r219265 * r219276;
        double r219305 = r219273 * r219304;
        double r219306 = r219272 * r219305;
        double r219307 = r219274 * r219306;
        double r219308 = r219282 * r219272;
        double r219309 = r219283 * r219308;
        double r219310 = r219307 - r219309;
        double r219311 = r219310 + r219290;
        double r219312 = r219311 - r219300;
        double r219313 = sqrt(r219265);
        double r219314 = r219277 * r219313;
        double r219315 = r219314 * r219313;
        double r219316 = r219315 - r219284;
        double r219317 = r219272 * r219316;
        double r219318 = r219317 + r219290;
        double r219319 = r219295 * r219296;
        double r219320 = r219319 * r219297;
        double r219321 = r219294 + r219320;
        double r219322 = r219318 - r219321;
        double r219323 = r219303 ? r219312 : r219322;
        double r219324 = r219267 ? r219301 : r219323;
        return r219324;
}

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

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if z < -7.213140668992403e+45

   1. Initial program 6.6

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   2. Simplified 6.6

      \[\leadsto \color{blue}{\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)}\]
   3. Using strategy rm

   4. Applied associate-*l* 6.5

      \[\leadsto \left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \color{blue}{j \cdot \left(27 \cdot k\right)}\right)\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 6.7

      \[\leadsto \left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]

   7. Applied associate-*r* 6.7

      \[\leadsto \left(t \cdot \left(\color{blue}{\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 6.8

      \[\leadsto \left(t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}\right) \cdot \sqrt[3]{\sqrt[3]{z}}\right)} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]
   10. Using strategy rm

   11. Applied sub-neg 6.8

       \[\leadsto \left(t \cdot \color{blue}{\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}\right) \cdot \sqrt[3]{\sqrt[3]{z}}\right) + \left(-a \cdot 4\right)\right)} + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]

   12. Applied distribute-lft-in 6.8

       \[\leadsto \left(\color{blue}{\left(t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}\right) \cdot \sqrt[3]{\sqrt[3]{z}}\right)\right) + t \cdot \left(-a \cdot 4\right)\right)} + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]

   13. Simplified 3.9

       \[\leadsto \left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{z}}\right)}^{3} \cdot \left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right)} + t \cdot \left(-a \cdot 4\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]

   14. Simplified 3.9

       \[\leadsto \left(\left({\left(\sqrt[3]{\sqrt[3]{z}}\right)}^{3} \cdot \left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right) + \color{blue}{\left(-a \cdot 4\right) \cdot t}\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]

   if -7.213140668992403e+45 < z < 1.8606429266648114e-39

   1. Initial program 4.9

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   2. Simplified 4.9

      \[\leadsto \color{blue}{\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)}\]
   3. Using strategy rm

   4. Applied associate-*l* 4.8

      \[\leadsto \left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \color{blue}{j \cdot \left(27 \cdot k\right)}\right)\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 4.8

      \[\leadsto \left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]

   7. Applied associate-*r* 4.8

      \[\leadsto \left(t \cdot \left(\color{blue}{\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 4.8

      \[\leadsto \left(t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}\right) \cdot \sqrt[3]{\sqrt[3]{z}}\right)} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]

   10. Taylor expanded around inf 1.4

       \[\leadsto \left(\color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right)} + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]

   if 1.8606429266648114e-39 < z

   1. Initial program 6.5

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   2. Simplified 6.5

      \[\leadsto \color{blue}{\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 6.5

      \[\leadsto \left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\]

   5. Applied associate-*r* 6.5

      \[\leadsto \left(t \cdot \left(\color{blue}{\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \sqrt{z}\right) \cdot \sqrt{z}} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.2131406689924034 \cdot 10^{45}:\\ \;\;\;\;\left(\left({\left(\sqrt[3]{\sqrt[3]{z}}\right)}^{3} \cdot \left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right) + \left(-a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{elif}\;z \le 1.86064292666481138 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - 4 \cdot \left(a \cdot t\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \sqrt{z}\right) \cdot \sqrt{z} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))