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

Average Error: 5.7 → 0.8

Time: 48.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 1.332226899097801573653352800247938704487 \cdot 10^{292}\right):\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(k \cdot 27\right) \cdot j\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r645254 = x;
        double r645255 = 18.0;
        double r645256 = r645254 * r645255;
        double r645257 = y;
        double r645258 = r645256 * r645257;
        double r645259 = z;
        double r645260 = r645258 * r645259;
        double r645261 = t;
        double r645262 = r645260 * r645261;
        double r645263 = a;
        double r645264 = 4.0;
        double r645265 = r645263 * r645264;
        double r645266 = r645265 * r645261;
        double r645267 = r645262 - r645266;
        double r645268 = b;
        double r645269 = c;
        double r645270 = r645268 * r645269;
        double r645271 = r645267 + r645270;
        double r645272 = r645254 * r645264;
        double r645273 = i;
        double r645274 = r645272 * r645273;
        double r645275 = r645271 - r645274;
        double r645276 = j;
        double r645277 = 27.0;
        double r645278 = r645276 * r645277;
        double r645279 = k;
        double r645280 = r645278 * r645279;
        double r645281 = r645275 - r645280;
        return r645281;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r645282 = t;
        double r645283 = x;
        double r645284 = 18.0;
        double r645285 = r645283 * r645284;
        double r645286 = y;
        double r645287 = r645285 * r645286;
        double r645288 = z;
        double r645289 = r645287 * r645288;
        double r645290 = r645282 * r645289;
        double r645291 = a;
        double r645292 = 4.0;
        double r645293 = r645291 * r645292;
        double r645294 = r645293 * r645282;
        double r645295 = r645290 - r645294;
        double r645296 = c;
        double r645297 = b;
        double r645298 = r645296 * r645297;
        double r645299 = r645295 + r645298;
        double r645300 = r645283 * r645292;
        double r645301 = i;
        double r645302 = r645300 * r645301;
        double r645303 = r645299 - r645302;
        double r645304 = -inf.0;
        bool r645305 = r645303 <= r645304;
        double r645306 = 1.3322268990978016e+292;
        bool r645307 = r645303 <= r645306;
        double r645308 = !r645307;
        bool r645309 = r645305 || r645308;
        double r645310 = r645282 * r645288;
        double r645311 = r645283 * r645310;
        double r645312 = r645286 * r645311;
        double r645313 = r645312 * r645284;
        double r645314 = r645313 - r645294;
        double r645315 = r645298 + r645314;
        double r645316 = r645315 - r645302;
        double r645317 = k;
        double r645318 = j;
        double r645319 = 27.0;
        double r645320 = r645318 * r645319;
        double r645321 = r645317 * r645320;
        double r645322 = r645316 - r645321;
        double r645323 = r645317 * r645319;
        double r645324 = r645323 * r645318;
        double r645325 = r645303 - r645324;
        double r645326 = r645309 ? r645322 : r645325;
        return r645326;
}

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	5.7
Target	1.8
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -inf.0 or 1.3322268990978016e+292 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 50.4

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

   2. Taylor expanded around inf 32.1

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

   3. Simplified 7.7

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

   5. Applied *-un-lft-identity 7.7

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

   6. Applied associate-*r* 7.7

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

   7. Simplified 5.1

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

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 1.3322268990978016e+292

   1. Initial program 0.3

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

   3. Applied pow1 0.3

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

   4. Applied pow1 0.3

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

   5. Applied pow1 0.3

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

   6. Applied pow-prod-down 0.3

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

   7. Applied pow-prod-down 0.3

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

   8. Simplified 0.3

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 1.332226899097801573653352800247938704487 \cdot 10^{292}\right):\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(k \cdot 27\right) \cdot j\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))