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

Average Error: 5.9 → 1.5

Time: 29.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le -3.562733702540698501769975042751072620323 \cdot 10^{286} \lor \neg \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 8.107231729667045342202036618227366964577 \cdot 10^{306}\right):\\ \;\;\;\;\left(\left(\left(\left(y \cdot \left(t \cdot z\right)\right) \cdot \left(x \cdot 18\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{\left(27 \cdot k\right) \cdot j} \cdot \sqrt[3]{\left(27 \cdot k\right) \cdot j}\right) \cdot \left(\sqrt[3]{27 \cdot k} \cdot \sqrt[3]{j}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r193231 = x;
        double r193232 = 18.0;
        double r193233 = r193231 * r193232;
        double r193234 = y;
        double r193235 = r193233 * r193234;
        double r193236 = z;
        double r193237 = r193235 * r193236;
        double r193238 = t;
        double r193239 = r193237 * r193238;
        double r193240 = a;
        double r193241 = 4.0;
        double r193242 = r193240 * r193241;
        double r193243 = r193242 * r193238;
        double r193244 = r193239 - r193243;
        double r193245 = b;
        double r193246 = c;
        double r193247 = r193245 * r193246;
        double r193248 = r193244 + r193247;
        double r193249 = r193231 * r193241;
        double r193250 = i;
        double r193251 = r193249 * r193250;
        double r193252 = r193248 - r193251;
        double r193253 = j;
        double r193254 = 27.0;
        double r193255 = r193253 * r193254;
        double r193256 = k;
        double r193257 = r193255 * r193256;
        double r193258 = r193252 - r193257;
        return r193258;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r193259 = t;
        double r193260 = x;
        double r193261 = 18.0;
        double r193262 = r193260 * r193261;
        double r193263 = y;
        double r193264 = r193262 * r193263;
        double r193265 = z;
        double r193266 = r193264 * r193265;
        double r193267 = r193259 * r193266;
        double r193268 = a;
        double r193269 = 4.0;
        double r193270 = r193268 * r193269;
        double r193271 = r193270 * r193259;
        double r193272 = r193267 - r193271;
        double r193273 = c;
        double r193274 = b;
        double r193275 = r193273 * r193274;
        double r193276 = r193272 + r193275;
        double r193277 = r193260 * r193269;
        double r193278 = i;
        double r193279 = r193277 * r193278;
        double r193280 = r193276 - r193279;
        double r193281 = -3.5627337025406985e+286;
        bool r193282 = r193280 <= r193281;
        double r193283 = 8.107231729667045e+306;
        bool r193284 = r193280 <= r193283;
        double r193285 = !r193284;
        bool r193286 = r193282 || r193285;
        double r193287 = r193259 * r193265;
        double r193288 = r193263 * r193287;
        double r193289 = r193288 * r193262;
        double r193290 = r193289 - r193271;
        double r193291 = r193290 + r193275;
        double r193292 = r193291 - r193279;
        double r193293 = 27.0;
        double r193294 = j;
        double r193295 = r193293 * r193294;
        double r193296 = k;
        double r193297 = r193295 * r193296;
        double r193298 = r193292 - r193297;
        double r193299 = r193263 * r193260;
        double r193300 = r193299 * r193261;
        double r193301 = r193265 * r193300;
        double r193302 = r193301 * r193259;
        double r193303 = r193302 - r193271;
        double r193304 = r193275 + r193303;
        double r193305 = r193304 - r193279;
        double r193306 = r193293 * r193296;
        double r193307 = r193306 * r193294;
        double r193308 = cbrt(r193307);
        double r193309 = r193308 * r193308;
        double r193310 = cbrt(r193306);
        double r193311 = cbrt(r193294);
        double r193312 = r193310 * r193311;
        double r193313 = r193309 * r193312;
        double r193314 = r193305 - r193313;
        double r193315 = r193286 ? r193298 : r193314;
        return r193315;
}

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 2 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -3.5627337025406985e+286 or 8.107231729667045e+306 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 47.1

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

   3. Applied pow1 47.1

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

   4. Applied pow1 47.1

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

   5. Applied pow1 47.1

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

   6. Applied pow1 47.1

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

   7. Applied pow1 47.1

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

   8. Applied pow-prod-down 47.1

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

   9. Applied pow-prod-down 47.1

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

   10. Applied pow-prod-down 47.1

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

   11. Applied pow-prod-down 47.1

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

   12. Simplified 7.7

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

   if -3.5627337025406985e+286 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 8.107231729667045e+306

   1. Initial program 0.4

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

   3. Applied add-cube-cbrt 0.6

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

   4. Simplified 0.7

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

   5. Simplified 0.7

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

   7. Applied cbrt-prod 0.6

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

   8. Simplified 0.6

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

   10. Applied *-un-lft-identity 0.6

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

   11. Applied associate-*r* 0.6

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

   12. Simplified 0.6

       \[\leadsto \left(\left(\left(\left(\color{blue}{\left(\left(y \cdot x\right) \cdot 18\right)} \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{j \cdot \left(k \cdot 27\right)} \cdot \sqrt[3]{j \cdot \left(k \cdot 27\right)}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{27 \cdot k}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le -3.562733702540698501769975042751072620323 \cdot 10^{286} \lor \neg \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 8.107231729667045342202036618227366964577 \cdot 10^{306}\right):\\ \;\;\;\;\left(\left(\left(\left(y \cdot \left(t \cdot z\right)\right) \cdot \left(x \cdot 18\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{\left(27 \cdot k\right) \cdot j} \cdot \sqrt[3]{\left(27 \cdot k\right) \cdot j}\right) \cdot \left(\sqrt[3]{27 \cdot k} \cdot \sqrt[3]{j}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))