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

Average Error: 5.8 → 4.1

Time: 22.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.24424295411286357321282707929428675159 \cdot 10^{-146} \lor \neg \left(z \le 128162218108037256959754240\right):\\ \;\;\;\;\left(t \cdot \left(z \cdot \left(x \cdot \left(18 \cdot y\right)\right) - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(27 \cdot k\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(27 \cdot k\right) \cdot j\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r635276 = x;
        double r635277 = 18.0;
        double r635278 = r635276 * r635277;
        double r635279 = y;
        double r635280 = r635278 * r635279;
        double r635281 = z;
        double r635282 = r635280 * r635281;
        double r635283 = t;
        double r635284 = r635282 * r635283;
        double r635285 = a;
        double r635286 = 4.0;
        double r635287 = r635285 * r635286;
        double r635288 = r635287 * r635283;
        double r635289 = r635284 - r635288;
        double r635290 = b;
        double r635291 = c;
        double r635292 = r635290 * r635291;
        double r635293 = r635289 + r635292;
        double r635294 = r635276 * r635286;
        double r635295 = i;
        double r635296 = r635294 * r635295;
        double r635297 = r635293 - r635296;
        double r635298 = j;
        double r635299 = 27.0;
        double r635300 = r635298 * r635299;
        double r635301 = k;
        double r635302 = r635300 * r635301;
        double r635303 = r635297 - r635302;
        return r635303;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r635304 = z;
        double r635305 = -3.2442429541128636e-146;
        bool r635306 = r635304 <= r635305;
        double r635307 = 1.2816221810803726e+26;
        bool r635308 = r635304 <= r635307;
        double r635309 = !r635308;
        bool r635310 = r635306 || r635309;
        double r635311 = t;
        double r635312 = x;
        double r635313 = 18.0;
        double r635314 = y;
        double r635315 = r635313 * r635314;
        double r635316 = r635312 * r635315;
        double r635317 = r635304 * r635316;
        double r635318 = a;
        double r635319 = 4.0;
        double r635320 = r635318 * r635319;
        double r635321 = r635317 - r635320;
        double r635322 = r635311 * r635321;
        double r635323 = b;
        double r635324 = c;
        double r635325 = r635323 * r635324;
        double r635326 = r635322 + r635325;
        double r635327 = r635312 * r635319;
        double r635328 = i;
        double r635329 = r635327 * r635328;
        double r635330 = 27.0;
        double r635331 = k;
        double r635332 = r635330 * r635331;
        double r635333 = j;
        double r635334 = r635332 * r635333;
        double r635335 = r635329 + r635334;
        double r635336 = r635326 - r635335;
        double r635337 = r635312 * r635313;
        double r635338 = r635314 * r635304;
        double r635339 = r635337 * r635338;
        double r635340 = r635339 - r635320;
        double r635341 = r635311 * r635340;
        double r635342 = r635341 + r635325;
        double r635343 = r635342 - r635335;
        double r635344 = r635310 ? r635336 : r635343;
        return r635344;
}

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

Target

Original	5.8
Target	1.6
Herbie	4.1

\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -3.2442429541128636e-146 or 1.2816221810803726e+26 < z

   1. Initial program 6.4

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

   2. Simplified 6.4

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

   4. Applied pow1 6.4

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

   5. Applied pow1 6.4

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

   6. Applied pow1 6.4

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

   7. Applied pow-prod-down 6.4

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

   8. Applied pow-prod-down 6.4

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

   9. Simplified 6.4

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

   11. Applied associate-*r* 6.5

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

   13. Applied pow1 6.5

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

   14. Applied pow1 6.5

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

   15. Applied pow1 6.5

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

   16. Applied pow-prod-down 6.5

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

   17. Applied pow-prod-down 6.5

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

   18. Simplified 6.6

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

   if -3.2442429541128636e-146 < z < 1.2816221810803726e+26

   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

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

   4. Applied pow1 5.2

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

   5. Applied pow1 5.2

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

   6. Applied pow1 5.2

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

   7. Applied pow-prod-down 5.2

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

   8. Applied pow-prod-down 5.2

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

   9. Simplified 5.1

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

   11. Applied associate-*r* 5.2

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

   13. Applied associate-*l* 1.1

       \[\leadsto \left(t \cdot \left(\color{blue}{\left(x \cdot 18\right) \cdot \left(y \cdot z\right)} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + {\left(\left(27 \cdot k\right) \cdot j\right)}^{1}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.24424295411286357321282707929428675159 \cdot 10^{-146} \lor \neg \left(z \le 128162218108037256959754240\right):\\ \;\;\;\;\left(t \cdot \left(z \cdot \left(x \cdot \left(18 \cdot y\right)\right) - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(27 \cdot k\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(27 \cdot k\right) \cdot j\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.680279438052224) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))