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

Average Error: 5.2 → 5.7

Time: 7.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.23687781110933502 \cdot 10^{-231}:\\ \;\;\;\;t \cdot \left({\left(\left(z \cdot y\right) \cdot \left(x \cdot 18\right)\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le -3.94513555421917625 \cdot 10^{-297}:\\ \;\;\;\;0 + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le 6.13398898620058475 \cdot 10^{-138}:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{elif}\;t \le 2.7075755514912269 \cdot 10^{-99}:\\ \;\;\;\;0 + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r158288 = x;
        double r158289 = 18.0;
        double r158290 = r158288 * r158289;
        double r158291 = y;
        double r158292 = r158290 * r158291;
        double r158293 = z;
        double r158294 = r158292 * r158293;
        double r158295 = t;
        double r158296 = r158294 * r158295;
        double r158297 = a;
        double r158298 = 4.0;
        double r158299 = r158297 * r158298;
        double r158300 = r158299 * r158295;
        double r158301 = r158296 - r158300;
        double r158302 = b;
        double r158303 = c;
        double r158304 = r158302 * r158303;
        double r158305 = r158301 + r158304;
        double r158306 = r158288 * r158298;
        double r158307 = i;
        double r158308 = r158306 * r158307;
        double r158309 = r158305 - r158308;
        double r158310 = j;
        double r158311 = 27.0;
        double r158312 = r158310 * r158311;
        double r158313 = k;
        double r158314 = r158312 * r158313;
        double r158315 = r158309 - r158314;
        return r158315;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r158316 = t;
        double r158317 = -4.236877811109335e-231;
        bool r158318 = r158316 <= r158317;
        double r158319 = z;
        double r158320 = y;
        double r158321 = r158319 * r158320;
        double r158322 = x;
        double r158323 = 18.0;
        double r158324 = r158322 * r158323;
        double r158325 = r158321 * r158324;
        double r158326 = 1.0;
        double r158327 = pow(r158325, r158326);
        double r158328 = a;
        double r158329 = 4.0;
        double r158330 = r158328 * r158329;
        double r158331 = r158327 - r158330;
        double r158332 = r158316 * r158331;
        double r158333 = b;
        double r158334 = c;
        double r158335 = r158333 * r158334;
        double r158336 = r158322 * r158329;
        double r158337 = i;
        double r158338 = r158336 * r158337;
        double r158339 = j;
        double r158340 = 27.0;
        double r158341 = r158339 * r158340;
        double r158342 = k;
        double r158343 = r158341 * r158342;
        double r158344 = r158338 + r158343;
        double r158345 = r158335 - r158344;
        double r158346 = r158332 + r158345;
        double r158347 = -3.945135554219176e-297;
        bool r158348 = r158316 <= r158347;
        double r158349 = 0.0;
        double r158350 = r158349 + r158345;
        double r158351 = 6.133988986200585e-138;
        bool r158352 = r158316 <= r158351;
        double r158353 = r158323 * r158320;
        double r158354 = r158322 * r158353;
        double r158355 = r158354 * r158319;
        double r158356 = r158355 - r158330;
        double r158357 = r158316 * r158356;
        double r158358 = r158340 * r158342;
        double r158359 = r158339 * r158358;
        double r158360 = r158338 + r158359;
        double r158361 = r158335 - r158360;
        double r158362 = r158357 + r158361;
        double r158363 = 2.707575551491227e-99;
        bool r158364 = r158316 <= r158363;
        double r158365 = r158364 ? r158350 : r158362;
        double r158366 = r158352 ? r158362 : r158365;
        double r158367 = r158348 ? r158350 : r158366;
        double r158368 = r158318 ? r158346 : r158367;
        return r158368;
}

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 t < -4.236877811109335e-231

   1. Initial program 4.3

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

   2. Simplified 4.3

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

   4. Applied add-cube-cbrt 4.4

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

   5. Applied associate-*r* 4.4

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

   7. Applied pow1 4.4

      \[\leadsto t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot \color{blue}{{z}^{1}} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   8. Applied pow1 4.4

      \[\leadsto t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   9. Applied pow1 4.4

      \[\leadsto t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}\right)\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   10. Applied pow1 4.4

       \[\leadsto t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{1}} \cdot {\left(\sqrt[3]{y}\right)}^{1}\right)\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   11. Applied pow-prod-down 4.4

       \[\leadsto t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot \color{blue}{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   12. Applied pow1 4.4

       \[\leadsto t \cdot \left(\left(\left(\left(x \cdot \color{blue}{{18}^{1}}\right) \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   13. Applied pow1 4.4

       \[\leadsto t \cdot \left(\left(\left(\left(\color{blue}{{x}^{1}} \cdot {18}^{1}\right) \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   14. Applied pow-prod-down 4.4

       \[\leadsto t \cdot \left(\left(\left(\color{blue}{{\left(x \cdot 18\right)}^{1}} \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   15. Applied pow-prod-down 4.4

       \[\leadsto t \cdot \left(\left(\color{blue}{{\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)}^{1}} \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   16. Applied pow-prod-down 4.4

       \[\leadsto t \cdot \left(\color{blue}{{\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right)}^{1}} \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   17. Applied pow-prod-down 4.4

       \[\leadsto t \cdot \left(\color{blue}{{\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot z\right)}^{1}} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   18. Simplified 5.0

       \[\leadsto t \cdot \left({\color{blue}{\left(\left(z \cdot y\right) \cdot \left(x \cdot 18\right)\right)}}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   if -4.236877811109335e-231 < t < -3.945135554219176e-297 or 6.133988986200585e-138 < t < 2.707575551491227e-99

   1. Initial program 8.4

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

   2. Simplified 8.4

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

   3. Taylor expanded around 0 10.2

      \[\leadsto \color{blue}{0} + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]

   if -3.945135554219176e-297 < t < 6.133988986200585e-138 or 2.707575551491227e-99 < t

   1. Initial program 5.2

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

   2. Simplified 5.2

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

   4. Applied associate-*l* 5.3

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

   6. Applied associate-*l* 5.3

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

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.23687781110933502 \cdot 10^{-231}:\\ \;\;\;\;t \cdot \left({\left(\left(z \cdot y\right) \cdot \left(x \cdot 18\right)\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le -3.94513555421917625 \cdot 10^{-297}:\\ \;\;\;\;0 + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le 6.13398898620058475 \cdot 10^{-138}:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{elif}\;t \le 2.7075755514912269 \cdot 10^{-99}:\\ \;\;\;\;0 + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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)))