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

Average Error: 5.9 → 2.2

Time: 25.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.776984382821296984516260974198729562529 \cdot 10^{-79}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot 18 - 4 \cdot a\right) \cdot t\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;t \le 7.289027731463515002673382915959472142239 \cdot 10^{-83}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(18 \cdot x\right) \cdot \left(y \cdot \left(t \cdot z\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \left(k \cdot j\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - \left(4 \cdot a\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r39529321 = x;
        double r39529322 = 18.0;
        double r39529323 = r39529321 * r39529322;
        double r39529324 = y;
        double r39529325 = r39529323 * r39529324;
        double r39529326 = z;
        double r39529327 = r39529325 * r39529326;
        double r39529328 = t;
        double r39529329 = r39529327 * r39529328;
        double r39529330 = a;
        double r39529331 = 4.0;
        double r39529332 = r39529330 * r39529331;
        double r39529333 = r39529332 * r39529328;
        double r39529334 = r39529329 - r39529333;
        double r39529335 = b;
        double r39529336 = c;
        double r39529337 = r39529335 * r39529336;
        double r39529338 = r39529334 + r39529337;
        double r39529339 = r39529321 * r39529331;
        double r39529340 = i;
        double r39529341 = r39529339 * r39529340;
        double r39529342 = r39529338 - r39529341;
        double r39529343 = j;
        double r39529344 = 27.0;
        double r39529345 = r39529343 * r39529344;
        double r39529346 = k;
        double r39529347 = r39529345 * r39529346;
        double r39529348 = r39529342 - r39529347;
        return r39529348;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r39529349 = t;
        double r39529350 = -3.776984382821297e-79;
        bool r39529351 = r39529349 <= r39529350;
        double r39529352 = c;
        double r39529353 = b;
        double r39529354 = r39529352 * r39529353;
        double r39529355 = x;
        double r39529356 = y;
        double r39529357 = z;
        double r39529358 = r39529356 * r39529357;
        double r39529359 = r39529355 * r39529358;
        double r39529360 = 18.0;
        double r39529361 = r39529359 * r39529360;
        double r39529362 = 4.0;
        double r39529363 = a;
        double r39529364 = r39529362 * r39529363;
        double r39529365 = r39529361 - r39529364;
        double r39529366 = r39529365 * r39529349;
        double r39529367 = r39529354 + r39529366;
        double r39529368 = r39529355 * r39529362;
        double r39529369 = i;
        double r39529370 = r39529368 * r39529369;
        double r39529371 = r39529367 - r39529370;
        double r39529372 = j;
        double r39529373 = 27.0;
        double r39529374 = k;
        double r39529375 = r39529373 * r39529374;
        double r39529376 = r39529372 * r39529375;
        double r39529377 = r39529371 - r39529376;
        double r39529378 = 7.289027731463515e-83;
        bool r39529379 = r39529349 <= r39529378;
        double r39529380 = r39529360 * r39529355;
        double r39529381 = r39529349 * r39529357;
        double r39529382 = r39529356 * r39529381;
        double r39529383 = r39529380 * r39529382;
        double r39529384 = r39529364 * r39529349;
        double r39529385 = r39529383 - r39529384;
        double r39529386 = r39529354 + r39529385;
        double r39529387 = r39529386 - r39529370;
        double r39529388 = r39529374 * r39529372;
        double r39529389 = r39529388 * r39529373;
        double r39529390 = r39529387 - r39529389;
        double r39529391 = r39529349 * r39529359;
        double r39529392 = r39529360 * r39529391;
        double r39529393 = r39529392 - r39529384;
        double r39529394 = r39529393 + r39529354;
        double r39529395 = r39529394 - r39529370;
        double r39529396 = r39529395 - r39529376;
        double r39529397 = r39529379 ? r39529390 : r39529396;
        double r39529398 = r39529351 ? r39529377 : r39529397;
        return r39529398;
}

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	5.9
Target	1.7
Herbie	2.2

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

Derivation

1. Split input into 3 regimes

2. if t < -3.776984382821297e-79

   1. Initial program 3.0

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

   3. Applied associate-*l* 8.2

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

   5. Applied associate-*l* 9.4

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

   7. Applied associate-*l* 9.3

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

   8. Taylor expanded around inf 3.0

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

   9. Simplified 3.0

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

   if -3.776984382821297e-79 < t < 7.289027731463515e-83

   1. Initial program 9.0

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

   3. Applied associate-*l* 4.6

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

   5. Applied associate-*l* 1.3

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

   7. Applied associate-*l* 1.3

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

   8. Taylor expanded around 0 1.2

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

   if 7.289027731463515e-83 < t

   1. Initial program 3.0

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

   3. Applied associate-*l* 7.1

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

   5. Applied associate-*l* 8.8

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

   7. Applied associate-*l* 8.9

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

   8. Taylor expanded around inf 3.0

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.776984382821296984516260974198729562529 \cdot 10^{-79}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(x \cdot \left(y \cdot z\right)\right) \cdot 18 - 4 \cdot a\right) \cdot t\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;t \le 7.289027731463515002673382915959472142239 \cdot 10^{-83}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(18 \cdot x\right) \cdot \left(y \cdot \left(t \cdot z\right)\right) - \left(4 \cdot a\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \left(k \cdot j\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - \left(4 \cdot a\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))